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.
pictures
any thing you want i will devolpe
fix pictures clen up edit photoshop
i work with indesian ai and photoshop


Gaussian, Lorentzian and pseudo-Voigt line shapes.
y = lshape(x,x0,fwhm) y = lshape(x,x0,fwhm,diff) y = lshape(x,x0,fwhm,diff,alpha) y = lshape(x,x0,fwhm,diff,alpha,phase)
lshape
evaluates normalised line shapes over a given abscissa
vector x
. x0
is the centre of the line shape, fwhm
denotes the line shpe's full width at half height (FWHM).
If diff
is given, it specifies the derivative wanted: 1 for first
and 2 for second derivative. 0 means no derivative, which is the
default. -1 indicates the integral with as lower limit.
If alpha
is given, it determines the line shape function.
alpha = 1
is a pure Gaussian (the default), and
alpha = 0
a pure Lorentzian. Any value
in between gives a hybrid pseudo-Voigt function
For the pseudo-Voigt function either one or two widths can be
specified. If fwhm
is a scalar, it is used for both components,
if it is a 2-element vector, the first element gives the line width for
the Gaussian and the second that for the Lorentzian component.
phase
determines the phase of the Lorentzian component. 0 is pure absorption, and pi/2
is pure dispersion.
The code
x = -100:100; centre = -30; y = lshape(x,-30,20,1,0); plot(x,y)
produces a plot of the first derivative of a Lorentzian, a very common line shape in CW EPR spectroscopy.
For the mathematical expressions for the various line shapes, see gaussian and lorentzian.
convspec, gaussian, hilberttrans, lorentzian, voigtian


After downloading a binary release suitable for your system, please follow the installation instructions.
If you are building from source, follow the source installation instructions.
See the release history for more information about Go releases.
 See https://proxy.golang.org/privacy for privacy information about these services and the go command documentation for configuration details including how to disable the use of these servers or use different ones.
This is an unstable version of Go. Use with caution.


...Proper recording of your chief complain, dental examination, Digital X-ray if necessary to confirm your diagnosis. Special emphasis to record any debilitative conditions which may require medical consultation to ensure minimum post operative complications and maximum comfort. Proper guidance regarding the type of treatment required.
...Dr Vidyesh Nadkerny provides various orthodontic options to correct your crooked teeth and give you a beautiful smile and a perfect bite. Being a “full time” orthodontist he is able to give you appointments tailored to your convenience and ensure that you are treated with maximum attention and comfort. Braces range from metal, ceramic (tooth colour) and Aligners. Both labial and lingual orthodontic options are available. Orthodontic Options facilitating least accumulation of plaque( deposits) on the teeth and providing ease of maintaining good oral hygiene are available. Various payment options with easy instalment facilities are provided for your convenience. Any breakage, bracket fixing etc is treated immediately without waiting time and discomfort.
...Dr Purnima Nadkerny provides painless periodontal treatment which includes scaling and polishing, periodontal flap surgeries with bone grafts, mucogingival surgeries to cover loss of gum and exposure of roots. LASER assisted surgeries which cause minimal bleeding and discomfort are an available option. Cosmetic Gingival Depigmentation is also carried out. Full mouth disinfection and proper maintenance to prevent recurrence is stressed on.
...Any tooth cavity is properly evaluated, cleaned thoroughly and restored. Various tooth fillings are available which are strong and durable which will provide best cosmetic results and allow you to bite on your favourite foods efficiently.
...A Root canal treatment is usually required when you have deep caries (decay) in your teeth or a periodontal (gum) infection which causes inflammation or infection in the roots of your tooth. During the root canal treatment the pulp inside the tooth is removed, the root canals are cleaned, shaped, disinfected and then filled to seal the space. The procedure is carried out at our clinic with minimum pain and may require local anaesthesia to make you comfortable. It may be instituted in more than one sitting depending on the infection in your tooth.
...Treatments range from Veneers, Crowns, Bleaching (teeth whitening) , full mouth rehabilitation and smile make over.
...Replacement of missing teeth with titanium Screws following a surgical procedure. Various options are available. Successful implants are the best option for replacing missing teeth in a healthy patient.
...This is probably the most feared dental procedure by most patients. We institute painless local anaesthesia and carry out the procedure with least trauma. Patients are discharged from the clinic with proper post operative instructions and monitoring to evaluate cessation of post extraction bleeding.
...All children are treated with lots of patience and care. We try to instil proper oral hygiene habits in children and educate and counsel parents regarding the maintenance of the same in their children. Restorations, Fluoride treatment, Adverse oral habit correction and interceptive orthodontics is stressed on.
...Replacement of teeth which have been lost can be efficiently replaced with various treatment options such as removable partial dentures, cast partial dentures, complete dentures. Fixed teeth are a very comfortable and convenient option. Various options such as metal, full ceramic and metal free crowns are available. You can efficiently bite and smile with fixed teeth.


A Tale of Two States: The Effects of a Statewide Mask Mandate
November 15, 2020
by Juliet Jeske
The U.S. has left state governments to plan their own strategies on how to contain the spread of COVID-19. This patchwork approach has led to drastically different outcomes. In the case of two states with similarly sized populations the disparity is dramatic.
Vermont issued a statewide mask mandate on Aug. 1st.
To date Wyoming only has a mask mandate in four of its 23 counties wih two of those mandates coming as late as early Nov.
Face masks aren't the only way to control the spread of COVID-19 but they are one major factor.


Ivy's Blog about DBC
And how she survives it
About
Home
Projects
Contact
This is a Blog Template for my Website!
January 10th, 2016
Here's where all my paragraphs and pictures and experiences and soup will go.
Previous Post
Next Post


Risk-neutral type probabilities of default for the constituents of the S&P500 Financials Index.
Investor Sentiment Indices based on Herculano & Lutkebohmert (2022).


App Academy Instructors on React & Redux
By
- It helps you write applications that behave consistently, run in different environments (client, server, and native), and are easy to test. On top of that, it provides a great developer experience, such as live code editing combined with a time traveling debugger.
- The whole state of your app is stored in an object tree inside a single store. The only way to change the state tree is to emit an action, an object describing what happened. To specify how the actions transform the state tree, you write pure reducers.


Art Health Barbecue Party Tips For A Truly Amazing Event June 14, 2015 by Author Doe Once upon a time, in an age before the Internet, the only way to meet people was to leave your home and interact with humanity.
Art The Principles Of Scientific Cooking June 14, 2015 by Author Doe If you're shy about going out alone, bring a book or a journal at first, and take small steps to get comfortable with the feeling.
Art How To Meet That Special Someone June 14, 2015 by Author Doe Once upon a time, in an age before the Internet, the only way to meet people was to leave your home and interact with humanity.
Art How To Meet That Special Someone June 14, 2015 by Author Doe Considering how long the human race existed before online dating, it looks like meeting people in real life was really working for them.
Art Lifestyle How To Meet That Special Someone June 14, 2015 by Author Doe Once upon a time, in an age before the Internet, the only way to meet people was to leave your home and interact with humanity.
Travel How To Meet That Special Someone June 14, 2015 by Author Doe Considering how long the human race existed before online dating, it looks like meeting people in real life was really working for them.
Art How To Meet That Special Someone June 14, 2015 by Author Doe Once upon a time, in an age before the Internet, the only way to meet people was to leave your home and interact with humanity.


Note: To complete these procedures, you must be assigned a role having the necessary permissions. To determine your role, follow the directions in Participant roles. For a basic understanding of roles and permissions, see Permissions and roles: Overview.
A web address you copy from your browser's address bar when working in your site may not always take you back to the particular content you were viewing; in some cases, it will only take you back to that tool. In Forums, you can generate a persistent link to a particular message:
Forums
.
Copy link
. A URL will appear below; you
can copy this and paste it into another application (e.g., an email
message).


The bit about what I do.
The process is a journey
A great website balances the needs of your clients with your image, what story do you want to tell, what are your client's needs and how are you going to meet them? These are all considerations for something truly beautiful.
Responsive Design
I work primarily with responsive design to reflect the fact that mobile devices are increasingly driving web traffic, with some estimates as high as 52% and growing it's important that your site is ready.
I Play Nice
With some of the most popular technologies and frameworks out there, and I'm always learning the new stuff too!
HTML5, CSS3, JavaScript, jQuery...
and responsive frameworks such as Bootstrap and PureCSS.io
I build what you want
but more importantly, I build what your users need. I take the time to learn what user experience will be best for your clients and deliver it, and always make sure that it fits the look you want.


eweb is a program which expands a literate Erlang program. Literate Erlang programs are stored in files with a .lit extension.
| Installing |   |
Create a directory where you will store the
eweb program. Fetch the files eweb.erl
and
eweb.lit and store them in this directory.
Compile eweb.erl with your favorite Erlang. evaluate the expression eweb:file("eweb"). Evaluating this expression will cause a number of files to be created eweb.tex, Makefile, eweb etc. Give the command Make doc to make the documentation. This will create the file eweb.dvi. Note: for this to work you must have installed LaTeX on your machine. It is assumed that you have already done this :-)
|
Running |   | If you look in the installation directory you will
find a script called eweb. Edit the path in this script to
point to the installation directory and put this file somewhere in you
path.
The command eweb scans the current directory and looks for any .lit files which have a later date stamp that the corresponding .tex files. It then runs eweb:file on any such files.
|
Make clean |   | Make clean can be run to remove any
unwanted files which were created by running eweb
|


--- layout: post title: Renovate Your Retrospectives date: 2015-06-25 16:30:13.000000000 -05:00 categories: - Agile tags: - agile - kanban - kcdc - retrospectives - scrum - session talks status: publish type: post published: true author: Adam N England comments: true ---
Today I had the pleasure of presenting a session at the Kansas City Developer Conference.


Now, you can also add desktop computers to your Google's 'My Devices' page
As the official blog notes, the page now includes both mobile and desktop devices
by ANIGoogle has updated its My Devices page to add desktop computers.
As the official blog notes, the page now includes both mobile and desktop devices. Google will also redirect visitors to the new mydevices.google.com site.
The updated page continues to offer detailed device information and the options to manage device access and wipe the device if desired. The new My Devices page is being rolled out to everyone.


Student Details
Parent/Guardian Details
Emergency Contact & Siblings Details
School Details
Other Details
Uploads & Declaration
3. Emergency Contact and Siblings Details
3.1 Emergency Contact Details
Choose Photo
3.2 Siblings Details
Number Of Siblings
1
2
3
Gender
Male
Female
Current Class
PreKG
LKG
UKG
1
2
3
4
5
6
7
8
9
10
11
12
Others
School
Same School
Other School
Gender
Male
Female
Current Class
PreKG
LKG
UKG
1
2
3
4
5
6
7
8
9
10
11
12
Others
School
Same School
Other School
Gender
Male
Female
Current Class
PreKG
LKG
UKG
1
2
3
4
5
6
7
8
9
10
11
12
Others
School
Same School
Other School
--Next--


Alice tells a sad story. Bob tells an even sadder story to cheer Alice up. If You Thought That Was Bad is the phrase people often use before they tell an even sadder story than the previous. Surprisingly, it usually cheers up the first person by making them realize their situation isn't that bad. Sometimes it makes things go From Bad to Worse.
Compare From Bad to Worse for when things actually got worse. Contrast Misery Poker, where the point is not to cheer the other person up but to keep their problems from overshadowing yours.
"If you think that story about the child molester was bad, let me tell you about..."
Examples:
- Also used in the last story in Night on Earth. Several guys in a cab discuss how one of them has been laid off from his job, and then his wife divorced him. Then the cab driver tells them his even more depressing story, which makes them feel like the laid-off-and-divorced guy is just whining about nothing.
- Sue Sylvester from Glee says things like "You think this is hard? Try living with hepatitis! That's hard!" or "You think this is hard? Try being told by Baywatch that they want to go in another direction, that's hard!"
- Supernatural: when Chuck is told that he's a prophet and understandably wigs out, Castiel just says "You should've seen Luke."
- On one episode of That '70s Show Eric walks in on his parents having sex, and then has difficulty concentrating on his girlfriend, Donna, when they are alone. When he confesses to Donna why he is distracted, she tries to cheer him up with a story about how she walked OUT on her parents... in a hammock... in broad daylight. Eric is cheered by the story, which has a lot to do with Donna's mother being quite attractive, but it is shortlived as Donna is so traumatised by recounting the story that she is no longer in the mood.
- Done inadvertantly by Oscar the Grouch in one episode of Sesame Street. His goal was to make everybody miserable, but it backfired, resulting in this trope. Nice Job Fixing It, Villain! is kind of inevitable with him.
- Subverted in The Young Ones. In the final episode, Rick receives a message that his parents have been killed. While he's trying to get his head around this news, Neil immediately moans "You think that's bad?", but before he can contribute his worse story Rick immediately snaps back "Yes I do actually, pissface!"
- Played with in Warhammer 40,000, where it usually goes "If you think that was bad..." and the something worse is in fact about to attack.
- Subverted in The Nostalgia Critic's review of Bloodrayne:
- The stinger by the KGB agent for the ad for the parody board game Communist Monopoly, after the "winner" is left sobbing since his three friends were, in order, shot by said agent for questioning how come a space that used to say Leningrad now said Stalingrad, taken away by more secret police to be sent to jail despite not actually having landed on the "Go To Jail" space, and then losing a game of Russian Roulette after landing on Chance.
"You think this is bad? You should play Hungry Hungry Bolsheviks."
- Turned into Peter's catch phrase in the South Park parodies of Family Guy, where it's what he says to introduce all their Cutaway Gags.
- In a first season bumper of The Beatles cartoon, the boys are in a desert dying of lack of water. Ringo accidentally does a rain dance which saves them—but it's way too much and he doesn't know how to stop it. John turns to the camera:
John: (to us) If you think we're up to our necks in hot water now, wait till you see next week's stories!
- The Simpsons: At the end of "Bart the Daredevil", Homer says this to Lance Murdock when the two are both in traction.
Homer: You think you've got guts? Try raising my kids.


Core losses can be separated into eddy current and hysteresis losses. The total core losses can thus be expressed as
p = PRef * (ratioHysteresis * (wRef / w) + 1 - ratioHysteresis) * (v / VRef)^2
where w
is the actual angular velocity and
v
is the actual voltage. The term
ratioHysteresis
is the ratio of the hysteresis losses
with respect to the total core losses for reference inner voltage
and reference angular velocity.
For the voltage and angular velocity range with respect to
Fig. 1, the dependency of total core losses on the parameter
ratioHysteresis
is depicted in Fig. 2.
| Fig. 1: Voltage versus angular velocity |
Fig. 2: Core losses versus angular velocity
with parameter ratioHysteresis |
In the current implementation it is assumed that
ratioHysteresis = 0
. This parameter cannot be changed
due to compatibility reasons.


title: Crate and Docker: Scale your database like your application description: In 20 minutes use Docker to build and scale a Crate database cluster. heroImg: wrapper-hero-frontpage og_image: https://cdn.crate.io./media/1503/crate-logo-165.png {% extends "base.html" %} {% block header %}
Build and scale any application powered by a backend that changes with it
Get your cluster upCrate is a distributed database for Docker that allows you to create a flexible and scalable backend suited to a wide variety of purposes. By deploying and managing your Crate cluster using Docker, you benefit from scalable capacity and resilience, no matter how complex or large your application becomes.
Docker is an open platform for developers and sysadmins to build, ship, and run distributed applications. Containerized applications help to isolate dependencies, standardize the operating environment, and foster portability.
Data can be moved anywhere, no matter the dependencies or platform.
Crate has an official Docker image available along with over 13,000 other pre-built apps. Start developing with a Crate database today.
Crate clusters grow seamlessly with your application. Quickly and easily add capacity as your customers increase.
Machine is a Crate-compatible tool for easily provisioning a Docker host in numerous environments. With one command docker-machine will create new a host, install Docker and then enable you to use your local client to manage the instance.
Swarm is a tool for native clustering of Docker containers. It can be used to create and scale a Crate cluster across multiple hosts. It supports several different platforms and integrates with other third party tools like Mesosphere and RancherOS.
Compose is a Docker orchestration tool for defining and spinning up multiple container applications. Because Compose provides an easy way to define volumes, links, and configurations in a single file it is a great fit for deploying a Crate cluster.


Preparation time
30 mins to 1 hour
Cooking time
1 to 2 hours
Serves
Makes 16-20
Shami kebabs were apparently invented by a highly skilled chef for a toothless Nawab of Lucknow.  So a kebab was made so fine that it required no teeth to eat it. When I hear stories like that I’m inclined to think, ‘If you believe that, you’ll believe anything.’ But then again, it’s a nice story, and so are the kebabs – silky smooth and stuffed with just a little finely chopped onion, mint and green chilli.
Drain the split peas and put them aside.
Put the onions, garlic, ginger and two tablespoons of water in a mini food processor and blend to a paste.
Heat the ghee or oil in a heavy-based saucepan over a medium heat, add the onion paste and cook for about 10 minutes. Stir in the lamb mince and split peas, pour over enough water to just cover the meat (roughly 400ml/14fl oz), add half a teaspoon of salt, partially cover with a lid and bring to a simmer. Cook for 20 minutes then remove the lid and cook for a further 5–10 minutes, or until the meat is just starting to brown and catch on the bottom. It’s important that any excess moisture has evaporated. Transfer to a plate and leave to cool for about 15 minutes.
Tip the mixture into a food processor and blend into a smooth paste. Add the green chillies, coriander, garam masala, chilli powder, cumin, the remaining salt and the lime juice to the lamb mixture and blend again, then gradually add enough of the egg to bind the mixture without making it too wet. Transfer the mix into a bowl and stir in the flour.
In a separate bowl, mix together all the filling ingredients and drain off any excess liquid.
To shape the kebabs, wet your hands and divide the mixture into about 16–20 portions. (If the mixture a little too wet to shape into patties, then add another tablespoon of flour.) Shape one portion into a patty about 4cm/1½in in diameter and 5mm thick. Spoon three-quarters of a teaspoon of the filling in the middle of the patty, and draw the edges around and over it to encase the filling and form a rough ball. Then, carefully, flatten it into a 5mm-thick patty. Place on a tray and repeat with the remaining mixture. Refrigerate for an hour.
Heat a few tablespoons of oil in a heavy-based frying pan over a medium heat. Fry the patties in batches for 2–3 minutes on each side, or until golden-brown and cooked through. Remove the patties from the pan and add the lime quarters for 1–2 minutes until warm.
Sprinkle the kebabs with a little salt and serve with the lime wedges, red onion rings and green chutney on the side.


Info Edge posts net loss of Rs 111.8 cr in Sep qtr
The company's billing at Rs 300.5 crore was up by 15.3 percent over the corresponding quarter in FY2018-19, a statement said.
by PTIInfo Edge (India), which runs job portal naukri.com, on November 12 reported a wider net loss of Rs 111.8 crore for the September 2019 quarter.
The company's net loss stood at Rs 40.7 crore in the year-ago period, Info Edge said in a regulatory filing.
However, its revenue from operations rose 17.6 percent to Rs 329.5 crore in the quarter under review from Rs 280 crore in the September 2018 quarter, it added.
Recruitment solutions and property portal 99acres contributed Rs 231.1 crore and Rs 57 crore, respectively to the revenue.
The company's billing at Rs 300.5 crore was up by 15.3 percent over the corresponding quarter in FY2018-19, a statement said.
"We continue to witness strong quarterly operational performance in terms of billing, revenue, total income and operating EBITDA growth, which are in high teens on year-on-year basis, supported by 99acres which grew over 20 percent in terms of billing and revenue during the quarter on y-o-y basis," Info Edge (India) Chief Financial Officer Chintan Thakkar said.
He added that the company continues to leverage its strong cash surplus position to invest in technology and branding.


This component demonstrates fetching data from the server.
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| Date | Title | Amount | Category | Personnel | PaidFrom | Settled Date |
|---|---|---|---|---|---|---|
| {{ expense.date }} | {{ expense.title }} | {{ expense.amount }} | {{ expense.item }} | {{ expense.personnel }} | {{ expense.paidfrom }} | {{ expense.settledDate }} |


See what it's like living life in the future, today!
It all starts here...
So by now, you've probably been contemplating, researching or both on going smart with your home. You have had conversations with co-workers, friends and families who have made the leap to begin their smart home journey, so why not you??
Maybe it is intimidating to uproot a way of life you have always known, or it may be an issue of expense or even a concern of security! All of these are legitimate concerns, but do not even think about getting discouraged! With our rapidly growing technology today, advancements are happening every second and the demand for smart home technology grows along side it making it more affordable, more realistic and as secure for your home and family as you might imagine! Let's dive right in and see why your future home might be your new present..!


Artifical Intelligence Resources Collections by jesliwang on May 8, 2016 Share on: 游戏ai的学习网站 http://aigamedev.com http://www.gameai.com


--- title: polarCoordinates sidebar: function_reference_sidebar permalink: sphericalRegion.polarCoordinates.html folder: function_reference toc: false ---
compute polar coordinates of with respect to a spherical region
| sR | sphericalRegion |
| v | vector3d |
| center | vector3d |
| r | relative distance to the center |
| rho | angle with respect to the center |


"MOTH" Hebrews ‘ash, from a root meaning “to fall away,” as moth-eaten garments fall to pieces (Job 4:19; 13:28; Isaiah 50:9; 51:8; Hos. 5:12). Gr. ses, thus rendered in Matthew 6:19, 20; Luke 12:33. Allusion is thus made to the destruction of clothing by the larvae of the clothes-moth. This is the only lepidopterous insect referred to in Scripture.
"MOTH" Hebrews ‘ash, from a root meaning “to fall away,” as moth-eaten garments fall to pieces (Job 4:19; 13:28; Isaiah 50:9; 51:8; Hos. 5:12).
Gr. ses, thus rendered in Matthew 6:19, 20; Luke 12:33. Allusion is thus made to the destruction of clothing by the larvae of the clothes-moth. This is the only lepidopterous insect referred to in Scripture.


@Falkvinge The idea could be good only if conditioned by user using smart contract - so that when I die (my condition would be no tx for 1 year) my funds would go to family, friends or organization I support. Not sure if BSV can achieve that. I believe ETH, EOS, Lisk or KM...


Chapter 6 Descriptive Statistics
6.1 Introduction to Statistics
6.1.1 Some Basic Terminologies Used in Statistics
i Population
- The set of all possible elements in the universe of interest to the researcher
ii Sample
- A Sample is a subset (a portion or part) of the population of interest
- The sample must be a representative of the population of interest
iii Element
- Element is an entity or object which the information is collected.
- Eg: Student, household, farm, company, tomato plant
iv Variable
- A variable is a feature characteristic which has different ‘values’ or categories for different elements (items/subjects/individuals)
- Eg: Gender of client, brand of mobile phones, risk level, number of emails received per day, age of client, income of client
v Data
- Data are measurements or facts that are collected from a statistical unit/entity of interest
- We collect data on variables
- Data are raw numbers or facts that must be processed (analysed) to get useful information.
We get information from data.
Eg:
Variable: Age (in years) of client
Data: 21, 45, 18, 32, 30, 22, 23, 27
Information:
The mean age is 27.25 years
The minimum age is 18 years
The range of ages is 18-45
The percentage of clients below 25 years of age: 50%
vi Statistic
- Characteristic of a sample
- The value which calculated based on sample data
vii Parameter
- Characteristic of a population
- The value which calculated based on population data
viii Census
- When a researcher gathers data from the whole population for a given measurement, it is called a census
ix Sampling
- When a researcher gathers data from a sample of the population for a given measurement, it is called sampling
- The process of selecting a sample is also called sampling
Why take a sample instead of studying every member of the population ?
- Prohibitive cost of census
- Destruction of item being studied may be required
- Not possible to test or inspect all members of a population being studied.
6.1.2 Branches of Statistics
i Descriptive Statistics
- Descriptive statistics consists of organizing, summarizing and presenting data in an informative way.
- The main purpose of descriptive statistics is to provide an overview of the data collected.
- Descriptive statistics describes the data collected through frequency tables, graphs and summary measures (mean, variance, quartiles, etc.).
ii Inferential Statistics
- In inferential statistics sample data are used to draw inferences (i.e. derive conclusions) or make predictions about the populations from which the sample has been taken.
- This includes methods used to make decisions, estimates, predictions or generalizations about a population based on a sample.
- This includes point estimations, interval estimation, test of hypotheses, regression analysis, time series analysis, multivariate analysis, etc.
6.1.3 Types of Variables
<IP_ADDRESS> Qualitative / Quantitative Variables
i Qualitative variable (Categorical variable)
- The characteristic is a quality.
- The data are categories.
- They cannot be given numerical values.
- However, it may be given a numerical label
- Qualitative variables are sometimes referred as categorical variables.
- Eg:
Gender:
Age group:
Education level:
A/L stream:
Degree type:
Hair colour:
FIT student batch:
Undergraduate level:
Grade that you can obtain for CM 1110/ CM1130
ii Quantitative variable
- The characteristic is a quantity
- The data are numbers
- Quantitative data require numeric values that indicate how much or how many.
- They are obtained by counting or measuring with some scale
- Eg:
Number of family members:
Number of emails received per day:
Weight of a student:
Age:
Credit balance in the SIM card:
Time remaining in class:
Temperature:
Marks
<IP_ADDRESS> Discrete/ Continuous Variables
- Quantitative variables can be classified as either discrete or continuous.
i Discrete Variables
- Quantitative
- Usually the data are obtained by counting
- There are impossible values between any two possible values
- Eg:
Number of family members:
Number of emails received per day:
ii Continuous Variables
- Quantitative
- Usually, the data are obtained by measuring with a scale
- There are no impossible values between any two possible values.(any value between any two possible values is also a possible value)
- i.e a continuous variable can take any value within a specified range.
- Eg:
Weight of a student:
Age:
Credit balance in the SIM card:
Time remaining in class:
Temperature:
Marks
6.1.4 Scales of Measurements
- There are four levels of measurements called, nominal, ordinal, interval and ratio.
- Each levels has its own rules and restrictions
- Different levels of measurement contains different amount of information with respect to whatever the data are measuring
i Nominal Scale
- Qualitative
- No order or ranking in categories.
- These categories have to be mutually exclusive, i.e. it should not be possible to place an individual or object in more than one category
- A name of a category can be substituted by a number, but it will be mere label and have no numerical meaning
ii Ordinal Scale
- Qualitative
- Categories can be ordered or ranked
- A name of a category can be substituted by a number, but such a sequence does not indicate absolute quantities.
- Difference between any two numbers on the scale does not have a numerical meaningful.
- It cannot be assumed that the differences between adjacent numbers on the scale are equal.
iii Interval Scale
- Quantitative
- Data can be ordered or ranked
- There is no absolute zero point. Zero is only an arbitrary point with which other values can compare
- Difference between two numbers is a meaningful numerical value
- Ration of two numbers is not a meaningful numerical value.
iv Ratio Scale
- Quantitative
- Highest level of measurement
- There exist an absolute zero point (It has a true zero point)
- Ratio between different measurements is meaningful
6.2 Presentation of Data
The sinking of the Titanic is one of the most infamous shipwrecks in history.
On April 15, 1912, during her maiden voyage, the widely considered “unsinkable” RMS Titanic sank after colliding with an iceberg. Unfortunately, there weren’t enough lifeboats for everyone onboard, resulting in the death of 1502 out of 2224 passengers and crew
Here’s a quick summary of our variables:
| Variable Name | Description |
|---|---|
| PassengerID | Passenger ID (just a row number, so obviously not useful for prediction) |
| Survived | Survived (1) or died (0) |
| Pclass | Passenger class (first, second or third) |
| Name | Passenger name |
| Gender | Passenger Gender |
| Age | Passenger age |
| SibSp | Number of siblings/spouses aboard |
| Parch | Number of parents/children aboard |
| Ticket | Ticket number |
| Fare | Fare |
| Cabin | Cabin |
| Embarked | Port of embarkation (S = Southampton, C = Cherbourg, Q = Queenstown) |
6.2.1 Tabular Presentations of Data
Raw Data
- Raw data are collected data that have not been organized numerically
- Eg: Passenger age
## PassengerId Survived Pclass
## 1 1 0 3
## 2 2 1 1
## 3 3 1 3
## 4 4 1 1
## 5 5 0 3
## 6 6 0 3
## Name Sex Age SibSp Parch
## 1 Braund, Mr. Owen Harris male 22 1 0
## 2 Cumings, Mrs. John Bradley (Florence Briggs Thayer) female 38 1 0
## 3 Heikkinen, Miss. Laina female 26 0 0
## 4 Futrelle, Mrs. Jacques Heath (Lily May Peel) female 35 1 0
## 5 Allen, Mr. William Henry male 35 0 0
## 6 Moran, Mr. James male NA 0 0
## Ticket Fare Cabin Embarked
## 1 A/5 21171 7.2500 S
## 2 PC 17599 71.2833 C85 C
## 3 STON/O2. 3101282 7.9250 S
## 4 113803 53.1000 C123 S
## 5 373450 8.0500 S
## 6 330877 8.4583 Q
## [1] 22 38 26 35 35 NA 54 2 27 14 4 58 20 39 14 55 2 NA 31 NA 35 34 15 28 8
## [26] 38 NA 19 NA NA 40 NA NA 66 28 42 NA 21 18 14
An array
- An array is an arrangement of raw numerical data in ascending or descending order of magnitude.
- Eg: Passenger age
## [1] 2 2 4 8 14 14 14 15 18 19 20 21 22 26 27 28 28 31 34 35 35 35 38 38 39
## [26] 40 42 54 55 58 66
Frequency Table (Frequency Distributions)
- A frequency table (frequency distribution) is a listing of the values a variable takes in a data set, along with how often (frequently) each value occurs
- frequency can be recorded as a
- frequency or count: the number of times a value occurs, or
- percentage frequency: the percentage of times a value occurs
- Percentage frequency can be calculated as,
\[Percentage frequency = \frac{a}{b} \times100 \%\]
- The objective of constructing a frequency table are as follows
- to organize the data in a meaningful manner
- to determine the nature or shape of the distribution
- to draw charts and graphs for the presentation of data
- to facilitate computational procedures for measures of average and spread
- to make comparisons between different data sets
- There are two basic types of frequency tables
- Simple frequency tables (Ungrouped frequency distribution)
- Grouped frequency distribution
<IP_ADDRESS> Simple frequency table (Ungrouped frequency distribution)
- Each possible value or category is taken as a class
- More suitable for
- Qualitative variables
- Discrete variables
- Sometimes construct for continuous variables when there is a small number of possible values between the minimum and maximum.
Examples:
CASE I:
Example 1
The native countries of 56 students from a certain education institute are as follows:
## [1] "SL" "BD" "SL" "SL" "SL" "SL" "IN" "SL" "SL" "SL" "BD" "SL" "SL" "SL" "IN"
## [16] "SL" "SL" "BD" "SL" "SL" "SL" "SL" "SL" "SL" "SL" "SL" "SL" "MD" "SL" "SL"
## [31] "SL" "SL" "SL" "SL" "PK" "MD" "PK" "SL" "SL" "SL" "SL" "SL" "PK" "MD" "SL"
## [46] "SL" "SL" "SL" "SL" "SL" "SL" "SL" "SL" "SL" "MD" "MD"
BD- Bangladesh, IN-India, MD-Maldives, PK-Pakistan, SL- Sri Lanka
Construct a frequency table
## Native Country Count Percentage (%)
## Bangladesh 3 5.357
## India 2 3.571
## Maldives 5 8.929
## Pakistan 3 5.357
## Sri Lanka 43 76.786
## Total 56 100.000
CASE II:
Example 2
The grades of 30 students for Statistics are as follows:
## [1] "B" "C" "B" "D" "B" "C" "C" "A" "B" "C" "C" "B" "E" "B" "B" "D" "D" "F" "B"
## [20] "D" "D" "A" "B" "A" "B" "C" "E" "A" "A"
Construct a frequency table
## Grade Count Percentage (%)
## A 5 17.241
## B 10 34.483
## C 6 20.690
## D 5 17.241
## E 2 6.897
## F 1 3.448
## Total 29 100.000
CASE III:
Example 3
The number of family members of a sample of undergraduates of Batch 19 are as follows:
## [1] 7 5 3 4 5 4 3 6 4 4 5 2 7 4 5 6 4 4 3 5
Construct a frequency table
## # A tibble: 6 x 3
## `Number of family members` Count `Percentage (%)`
## <dbl> <int> <dbl>
## 1 2 1 5
## 2 3 3 15
## 3 4 7 35
## 4 5 5 25
## 5 6 2 10
## 6 7 2 10
CASE IV:
Example 4
The ages (in years) of a sample of undergraduates of Batch 19 are as follows:
## [1] 21 22 22 23 22 24 24 23 21 22 23 22 22 23 21 21 22 23 22 23
Construct a frequency table
## # A tibble: 4 x 3
## `Age (years)` Count `Percentage (%)`
## <dbl> <int> <dbl>
## 1 21 4 20
## 2 22 8 40
## 3 23 6 30
## 4 24 2 10
<IP_ADDRESS> Grouped frequency distribution
- A grouped frequency distribution (table) is obtained by constructing classes (or intervals) for the data and then listing the corresponding number of values in each interval.
- Suitable for quantitative variables with large number of possible values in the range of data.
- Note that when items have been grouped in this way, their individual values are lost.
- When studying about frequency distributions it is very important to know the meaning of the following terms
i Class intervals
- In a frequency distribution the total range of the observations are divided into a number of classes. Those are called class intervals
- Eg: Class intervals: 10-14, 15-19, 20-24, …, 40-44
ii Class limits
- Class limits are the smallest and largest piece of data value that can fall into a given class.
- In the class interval 10-14, the end numbers, 10 and 14, are called class limits
- The smaller number (10) is the lower class limit
- The larger number (14) is the upper class limit
iii Class boundaries
- Class boundaries are obtained by adding the upper limit of one class interval to the lower limit of the next-higher class interval and dividing by 2.
- Class boundaries are also called True class limits
- Class boundaries should not coincide with actual observations
| Class interval | Class boundaries |
|---|---|
| 10 - 14 | 9.5 – 14.5 |
| 15 - 19 | 14.5 – 19.5 |
| 20 - 24 | 19.5 – 24.5 |
| 25 - 29 | 24.5 – 29.5 |
| 30 - 34 | 29.5 – 34.5 |
| 35 - 39 | 34.5 – 39.5 |
| 40 - 44 | 39.5 – 44.5 |
iv The size or width of a class interval
- The size or width of a class interval is the difference between the lower and upper class boundaries
- It is also referred to as the class width, class size, or class length
- Eg: The class width for the class 10-14 is = 14.5-9.5 = 5
v The class mark ( Midpoint of the class)
- Midpoint of the class
- Also called as class midpoint
- \(\text{Midpoint of the class} = \frac{\text{Lower limit} + \text{Upper limit}}{2}\)
or
- \(\text{Midpoint of the class} = \frac{\text{Lower boundary} + \text{Upper boundary}}{2}\)
vi Open class intervals
A class interval that, at least theoretically, has either no upper class limit or no lower class limit indicated is called an open class interval
For example, referring to age groups of individuals, the class interval “65 year and over” is an open class interval
Rules and Practices for constructing grouped frequency tables
- Every data value should be in an interval
- The intervals should be mutually exclusive
- The classes of the distribution must be arrayed in size order.
- The number of classes not less than 5 or not greater than 15 is recommended.
- The following formula is often used to determine the number of classes: If n is the number of observations, then
\[\text{Number of classes} = \sqrt{n}\]
\[\text{Width of the class interval} = \frac{Range}{\sqrt{n}}= \frac{Min-Max}{\sqrt{n}}\]
- Data should be represented within classes having limits which the data can attain
- Classes should be continuous
- By convention, the beginning of the interval is given the appropriate exact value, rather than the end.
Eg: intervals of 0-49, 50-99,100-149 would be preferred over the intervals 1-50, 51-100, 101-150 etc. - The number f observations falling into each category or class interval (class frequency) can be easily found using tally marks.
Examples:
In a grouped frequency distribution, class intervals can be constructed in different ways
Example 1
| Class interval | Number of students |
|---|---|
| 10 - 14 | 4 |
| 15 - 19 | 5 |
| 20 - 24 | 11 |
| 25 - 29 | 9 |
| 30 - 34 | 6 |
| 35 - 39 | 3 |
| 40 - 44 | 2 |
Example 2
| Salary | Number of employees |
|---|---|
| 0 – 1999 | 1 |
| 2000 – 3999 | 31 |
| 4000 – 5999 | 18 |
| 6000 – 7999 | 4 |
| 8000 – 9999 | 2 |
| 10000 - 11999 | 1 |
| 12000 – 13999 | 0 |
| 14000 – 15999 | 0 |
| 16000 – 17999 | 1 |
| 18000 -19999 | 1 |
| 20000-21999 | 1 |
| Salary | Number of employees |
|---|---|
| 0 – 1999 | 1 |
| 2000 – 3999 | 31 |
| 4000 – 5999 | 18 |
| 6000 – 7999 | 4 |
| 8000 – 9999 | 2 |
| 10000 - 15999 | 1 |
| 16000 – 21999 | 3 |
| Total | 60 |
Example 3
| Salary | Number of employees |
|---|---|
| Less than 2000 | 1 |
| 2000 – 2999 | 11 |
| 3000 – 3999 | 20 |
| 4000 – 5999 | 18 |
| 6000 – 9999 | 6 |
| Greater than or equal to 10000 | 4 |
| Total | 60 |
<IP_ADDRESS> Two-way frequency table
- Cross tabulation, Cross classification table, Contingency table, Two-way table
- Display the relationship between two or more qualitative variables (categorical variables (nominal or ordinal))
## # A tibble: 2 x 4
## Survived First Second Third
## <chr> <dbl> <dbl> <dbl>
## 1 died 80 97 372
## 2 Survived 136 87 119
## # A tibble: 2 x 4
## Survived First Second Third
## <chr> <dbl> <dbl> <dbl>
## 1 died 0.37 0.53 0.76
## 2 Survived 0.63 0.47 0.24
6.2.2 Graphic Presentations of Data
- A diagram is a visual form for presentation of statistical data.
- The form of the diagram varies according to the nature of the data
<IP_ADDRESS> Describing Qualitative Data
- Bar chart / Pie chart
- Suitable for
- Qualitative variables (nominal or ordinal)
- Discrete variables (when the number of bars or number of different values is small)
I Bar Chart
- A bar graph uses bars to represent discrete categories of data
- It can be drawn either on horizontal (more common) or vertical base
- A rectangle of equal width is drawn for each category
- The height (in vertical bar chart) or the length (in horizontal bar chart) of the rectangle is equal to the category’s frequency or percentage.
i Simple Bar Chart
- Only one categorical variable can be presented
- Often used in conjunction with simple frequency tables
- The bars do not touch each other
- The gaps between adjacent bars are same in length
| Passenger class | Count | Percentage |
|---|---|---|
| First | 216 | 24.242 |
| Second | 184 | 20.651 |
| Third | 491 | 55.107 |
ii Component Bar Chart
- Sub divided bar chart/ Stacked bar chart
- Use to compare two or more qualitative variables (nominal or ordinal)
- Often used in conjunction with two way tables
- Start by drawing a simple bar chart with the total figures.
- The bars are then divided into the component parts
- Can be drawn on absolute figures or percentages
- The various components should be kept in the same order in each bar
- To distinguish different components from one another, different colours or shades can be used
| Survived | First | Second | Third |
|---|---|---|---|
| died | 80 | 97 | 372 |
| Survived | 136 | 87 | 119 |
Percentage component bar chart
- When sub-divided bar chart is drawn on percentage basis it is called percentage bar chart
- The various components are expressed as percentage to the total
- All bars are equal in height
| Survived | First | Second | Third |
|---|---|---|---|
| died | 0.3703704 | 0.5271739 | 0.7576375 |
| Survived | 0.6296296 | 0.4728261 | 0.2423625 |
iii Multiple Bar Chart
- Compound bar chart/ Cluster bar chart
- Use to compare two or more qualitative variables (nominal or ordinal)
- Often used in conjunction with two way tables
- These bar charts are drawn side by side
| Survived | First | Second | Third |
|---|---|---|---|
| died | 37.04 | 52.72 | 75.76 |
| Survived | 62.96 | 47.28 | 24.24 |
<IP_ADDRESS> Describing Quantitative Data
- Histogram/ Dot plot / Box plot/ Scatter plot
II Histogram
- Histogram looks similar to bar chart since it also has bars.
- However, it is different from a bar chart in a number of aspects.
- One main difference is that in the histogram, the bars are drawn attached to each other; there are no gaps between bars like in a bar chart.
- Histogram is used to show the shape of the distribution of a continuous variable.
- However, the histogram is also used for discrete variables when the data are grouped in to class intervals.
- In a histogram, the area of a bar should be proportional to the frequency of the corresponding class.
- If all the bars have the same width, then the height of a bar can represent the frequency.
- The bar corresponding to a class interval should be drawn from the lower class boundary to the upper class boundary. In this way there will be no gaps between the bars.
Example: The marks(out of 50) of a group of students are recorded in the accompanying table. Draw a histogram for the data
| Marks | Number of students |
|---|---|
| 10 - 14 | 4 |
| 15 - 19 | 5 |
| 20 - 24 | 11 |
| 25 - 29 | 9 |
| 30 - 34 | 6 |
| 35 - 39 | 3 |
| 40 - 44 | 2 |
| Total | 40 |
Example 2
III Frequency polygon
- If the mid-point of the top of each block in a histogram is joined by a straight line, a frequency polygon is produced.
- This is done under the assumption that the frequencies in a class-interval are evenly distributed throughout the class
Example: The marks(out of 50) of a group of students are recorded in the accompanying table. Draw a frequency polygon for the data
| Marks | Number of students |
|---|---|
| 10 - 14 | 4 |
| 15 - 19 | 5 |
| 20 - 24 | 11 |
| 25 - 29 | 9 |
| 30 - 34 | 6 |
| 35 - 39 | 3 |
| 40 - 44 | 2 |
| Total | 40 |
IV Frequency curve
- A frequency curve is drawn by smoothing the frequency polygon.
- It is smooth in such a way that the sharp turns are avoided
Example: The marks(out of 50) of a group of students are recorded in the accompanying table. Draw a frequency curve for the data
| Marks | Number of students |
|---|---|
| 10 - 14 | 4 |
| 15 - 19 | 5 |
| 20 - 24 | 11 |
| 25 - 29 | 9 |
| 30 - 34 | 6 |
| 35 - 39 | 3 |
| 40 - 44 | 2 |
| Total | 40 |
frequency curves arising in practice take on certain characteristics shapes as shown bellow
- The symmetrical or bell shaped frequency curves are characterized by the fact that observations equidistant from the central maximum have the same frequency. An important example is the normal curve.
- In the moderately asymmetrical or skewed frequency curves the tail of the curve to one side of the central maximum is longer than that to the other. If the longer tail occurs to the right the curve is said to be skewed to the right or to have positive skewness.While if the reverse is true the curve is said to be skewed to the left or to have negative skewness.
- In a J shaped or reverse J shaped curve a maximum occurs at one end.
- A U shaped frequency curve has maxima at both ends.
- A bimodal frequency curve has two maxima. These appear as two distinct peaks (local maxima) in the frequency curve.When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode.
- A multimodal frequency curve has more than two maxima.
V Dot Plot
- A dot plot is a method of presenting data which gives a rough but rapid visual appreciation of the way in which the data are distributed
- It consists of a horizontal line marked out with divisions of the scale on which the variable is being measured -This graph can be used to represent only the numerical data.
VI Box plot (Box and whisker plot)
- Box plot is also a useful method of representing the behavior of a data set or comparing two or more data sets.
- Box plot is constructed by identifying five statistics from the data set as largest value, smallest values, median, Q1 and Q3.
Example:
Construct a box plot for the following data set (Marks of students)
\[\text{52, 88, 56, 79, 72, 91, 85, 88, 68, 63, 76, 73, 86, 95, 12, 69}\]
Xmin = 12 Xmax = 95 Q1 =64.25 Q2 = Median = 74.5 Q3 = 87.5
VII Violin plot
- A violin plot is a method of plotting quantitative data.
- It is similar to a box plot, with the addition of a rotated kernel density plot on each side.
- Violin plots are similar to box plots, except that they also show the probability density of the data at different values, usually smoothed by a kernel density estimator.
6.3 Summary Measures
Although frequency distribution serves useful purpose, there are many situations that require other types of data summarization.
What we need in many instances is the ability to summarize the data by means of a single number called a descriptive measure.
Descriptive measures may be computed from the data of a sample or the data of a population. To distinguish between them we have the following definitions.
Definitions
A descriptive measure computed from the data of a sample is called a statistic.
A descriptive measure computed from the data of a population is called a parameter.
6.3.1 Measures of Central Tendency
Measure of central tendency yield information about the center, or middle part, of a group of numbers.
Eg: Mode, Median, Arithmetic Mean, Geometric mean, Harmonic Mean, Quadratic Mean, Quartiles, Deciles, and Percentiles
<IP_ADDRESS> Mode
The Mode is the most frequently occuring value in a set of data
Organizing the data into an ordered array (an ordering of the numbers from smallest to largest) helps to locate the mode.
A series having only one mode is called as uni-modal
In the case of a tie for the most frequent occuring value, two modes are listed. Then the data are set to be bimodal
If a set of data is not exactly bimodal but contains two values that are more dominant than others, some researchers take the liberty of referring to the data set as bimodal even without an exact tie for the mode.
Data sets with more than two modes are referred to as multimodal.
The mode is an appropriate measure of central tendency for nominal-level data.
The mode can be used to determine which category occurs most frequently.
For ungrouped data
Example 01: Find mode of the following datasets
Dataset 1: 12, 14, 10, 8, 6, 8, 15, 8
Dataset 2: 40, 44, 57, 48, 78
Dataset 3: 42, 45, 55, 50, 45, 40, 55, 45, 52, 55, 54
For grouped frequency data
Example 02: Find mode of the following data
| Marks | Number of students |
|---|---|
| 20 | 8 |
| 30 | 10 |
| 40 | 16 |
| 50 | 8 |
| 60 | 5 |
| 70 | 3 |
- Advantages and disadvantages of mode
Advantages
- Easy to understand
- Easy to calculate
- Not affected by extreme values in the dataset
- Good for qualitative data
Disadvantages
- Not suitable for further mathematical calculations
- There may be more than one mode for a given dataset
- It is not based upon all the observations
- In some cases, we may not be able to find a mode for a given dataset
<IP_ADDRESS> Median
The median is the middle value in an ordered array of numbers.
Median divides the series into equal parts
The following steps are used to determine the median.
STEP 1: Arrange the observations in an ordered data array.
STEP 2: For an odd number of terms, find the middle term of the ordered array. It is the median.
STEP 3: For an even number of terms, find the arithmetic mean of the middle two terms. This arithmetic mean is the median.
\[Median = \text{the }(\frac{n+1}{2})\text{th item in the data array} \]
- The level of data measurement must be at least ordinal for a median to be meaningful.
Example 1: Find the median of the dataset 1, 8, 6, 3, 2
Example 2: Find the median of the dataset 8, 9, 1, 2, 14, 12
- Advantages and disadvantages of median
Advantages
- Simple to understand
- Easy to calculate
- Not affected by extreme values in the dataset
- Can be calculated even for qualitative variables (ordinal scale data)
Disadvantages
- It is not based upon all the observations
<IP_ADDRESS> Arithmetic Mean
The arithmetic mean (usually called mean) is the sum of all observations divided by the total number of observations.
Population Mean
The poupulation mean us represented by the Greek letter \(mu\) (\(\mu\)).
Let, \(N\) is the number of terms in the population.
\[\mu = \frac{\sum{x}}{N}=\frac{x_1+x_2+x_3+...+x_N}{N}\]
Sample Mean
- The sample mean is represented by \(\bar{x}\)
- Let, \(n\) is the number of terms in the sample \[\bar{x} = \frac{\sum{x}}{n}=\frac{x_1+x_2+x_3+...+x_n}{n}\]
It is inappropriate to use the mean to analyse data that are not at least interval level in measurement.
Example 1: Calculate the mean from the following data
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Marks | 40 | 50 | 53 | 78 | 58 | 60 | 73 | 35 | 43 | 48 |
- Advantages and disadvantages of arithmetic mean
Advantages
- Simple to understand
- Easy to calculate
- Based on all the observations
- Well defined
- Unique
- Can be used in further calculation
Disadvantages
- Can be affected by extreme values in the dataset
- May lead to false conclusions
- Only applicable to quantitative data (not applicable to qualitative data)
Empirical relationship between mean, mode, median
- In case of symmetrical distribution, mean, median and mode coincide \((mean = meadian= mode)\)
- For a moderately asymmetrical distribution, the following relationship exists \(Mean – Mode = 3(Mean - Median)\)
Choice between mean and median
- Mean is very sensitive to outliers.Median is not sensitive to outliers
- When there are outliers in a data set, median is more appropriate than mean
<IP_ADDRESS> Quartiles, Deciles and Percentiles
- Median divides the data set into two equal parts.
- There are other values which divide the data set into a number of equal parts
- Those are Quartiles, Deciles and Percentiles
(a) Quartiles (Q) – Quartiles divide an array into four equal parts
\[Q_i = \text{the }\frac{i}{4}(n+1)\text{th item in the data array} \]
(b) Deciles (D) – Deciles divide an array into ten equal parts
\[D_i = \text{the }\frac{i}{10}(n+1)\text{th item in the data array} \]
(c) Percentiles (P) – Percentiles divide an array into 100 equal parts
\[P_i = \text{the }\frac{i}{100}(n+1)\text{th item in the data array} \]
6.3.2 Measures of Variability
Measure of central tendency yield information about particular points of a data set.
However, business researchers can use another group of analytic tools to describe a set of data.
These tools are measures of variability, which describe the spread or the dispersion of a set of data.
Using measures of variability in conjunction with measures of central tendency makes possible a more complete numerical description of the data.
This section focuses on seven measures of variability for ungrouped data: range, interquartile range, variance, standard deviation, z score and coefficient of variation.
<IP_ADDRESS> Range
- The range is the difference between the largest value of a data set and the smallest value.
\[Range = Maximum - Minimum\]
One important use of the range is in quality assurance, where the range is used to construct control charts
Advantages and disadvantages of range
Advantages
- Easy to understand and calculate
Disadvantages
- Consider only the highest and lowest values of the data and fails to take account of any other observations in the dataset
- Heavily influenced by extreme values
<IP_ADDRESS> Interquartile Range (IQR)
We use the interquartile range (IQR) to measure the spread of a data around the median (M).
The interquartile range is the range of values between the first and third quartile.
Essentially it is the range of the middle 50% of the data and is determined by computing the value of \(Q_3 - Q_1\).
The interquartile range is especially useful in situations where data users are more interested in values towards the middle and less interested in extremes.
The interquartile range is used in the construction of box and whisker plots.
By eliminating the lowest 25% and the highest 25% of the items in a series, we are left with the central 50% , which are ordinarily free of extreme values.
Advantages
- Easy to understand and calculate
- Not influenced by extreme values
Disadvantages
- Ignore the first \(25\%\) and the last \(25\%\) in the dataset
<IP_ADDRESS> Variance and Standard Deviation
- To measure the spread of data around the mean, we use the standard deviation (S).
- The variance and standard deviation are two very popular measures of dispersion.
- These measures are not meaningful unless the data are at least interval-level data.
- Their formulations are categorized into whether to evaluate from a population or from a sample.
NOTE
Sum of deviations from the arithmetic mean is always zero. \[{\sum{(x-\mu)} = 0}\]
This property requires considering the alternative ways to obtain measure of variability.
<IP_ADDRESS>.1 Variance
The variance is the average of the squared deviations about the mean for a set of numbers.
The population variance is denoted by \(\sigma^2\)
\[\sigma^2 = \frac{\sum(x-\mu)^2}{N}\]
The sum of the squared deviations about the mean of a set of values - called the sum of squares of \(x\) and sometimes abbreviated as \(SS_x\)
Because the variance is computed from squared deviations, the final result is expressed in terms of squared units of measurements.
Statistics measured in squared units are problematic to interpret.
<IP_ADDRESS>.2 Standard Deviation
The standard deviation is the square root of the variance.
The population standard deviation is denoted by \(\sigma\)
\[\sigma = \sqrt{\frac{\sum(x-\mu)^2}{N}}=\sqrt{\sigma^2}\]
- One feature of standard deviation that distinguishes it from a variance is that the standard deviation is expressed in the same units as the raw data, whereas the variance is expressed in those units squared.
Advantages
- Based on all the observations
- Since this is based on arithmetic mean, it has all the merits of it
- The most important and widely used measure of dispersion
Disadvantages
- Not easy to understand and difficult to calculate
- Gives more weight to extreme values, because the values are squared up
<IP_ADDRESS> Empirical Rule
The empirical rule is an important rule of thumb that is used to state the approximate percentage of values that lie within a given number of standard deviations from the mean of a set of data if the data are normally distributed
The empirical rule is used only for three numbers of standard deviations: \(1\sigma\), \(2\sigma\), \(3\sigma\)
| Distance from the mean | Values within distance |
|---|---|
| \(\mu\pm1\sigma\) | \(68\%\) |
| \(\mu\pm2\sigma\) | \(95\%\) |
| \(\mu\pm3\sigma\) | \(99.7\%\) |
- If a set of data is normally distributed, or bell shaped, approximately \(68\%\) of the data values are within one standard deviation of the mean, \(95\%\) are within two standard deviations, and almost \(100\%\) are within three standard deviations.
<IP_ADDRESS> Population versus sample variance and standard deviations
The sample variance is denoted by \(s^2\) and the sample standard deviation by \(s\).
The main use for sample variances and standard deviations is as estimators of population variances and standard deviations.
Thus, computation of the sample variance and standard deviation differs slightly from computation of the population variance and standard deviation.
Both the sample variance and sample standard deviation use \(n-1\)int he denominator instead of \(n\) because using \(n\) in the denominator of a sample variance results in a statistic that tends to underestimate the population variance.
While discussion of the properties of good estimator is beyond the scope of this course, one of the properties of a good estimator is being unbiased.
Whereas, using \(n\) in the denominator of the sample variance makes it a biased estimator, using \(n-1\) allows it to be an unbiased estimator, which is a desirable property in inferential statistics.
Sample variance
\[s^2 = \frac{\sum{(x-\bar{x})^2}}{n-1}\]
Sample standard deviation
\[s = \sqrt{\frac{\sum{(x-\bar{x})^2}}{n-1}}= \sqrt{s^2}\]
<IP_ADDRESS> Computational formulas for variance and standard deviation
An alternative method of computing variance and standard deviation, sometimes referred to as the computational method or shortcut method, is available.
Algebraically,
\[\sum{(x-\mu)^2}= \sum{x^2}- \frac{(\sum x)^2}{N}\]
and
\[\sum{(x-\bar{x})^2}= \sum{x^2}- \frac{(\sum x)^2}{n}\]
- Substituting these equivalent expressions into the original formulas for variance and standard deviation yields the following computational formulas.
Computational formula for population variance and standard deviation
\[\sigma^2 = \frac{\sum{x^2}- \frac{(\sum x)^2}{N}}{N}\]
\[\sigma = \sqrt{\sigma^2}\]
Computational formula for sample variance and standard deviation
\[s^2 = \frac{\sum{x^2}- \frac{(\sum x)^2}{n}}{n-1}\]
\[s = \sqrt{s^2}\]
- For situations in which the mean is already computed or is given, alternative forms of these formulas are:
\[\sigma^2 = \frac{\sum{x^2}-N\mu^2}{N}\]
\[s^2 = \frac{\sum{x^2}-n(\bar{x})^2}{n-1}\]
<IP_ADDRESS> Coefficient of variation
In general, for two variables measured with he same units (eg: two groups of people both weighed in kg), the group with the larger variance and standard deviation has n=more variability among their scores.
The unit of measure affects the size od the variance.
The same population weights, expressed in ‘grams’ rather than kg would have a larger variance and standard deviation.
The coefficient of variation, a measure of relative variability gets around this difficulty and makes it possible to compare variability across variables measured in different units.
The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage and is denoted CV.
\[CV = \frac{\sigma}{\mu}(100)\]
Using median and quartile deviation
\[CV = \frac{\frac{Q_3 - Q_1}{2}}{Median}(100)\]
The coefficient of variation essentially is a relative comparison of a standard deviation to its mean.
The coefficient of variation can be useful in comparing standard deviations that have been computed from data with different means.
The choice of whether to use a coefficient of variation or raw standard deviations to compare multiple standard deviations is a matter of preference
The coefficient of variation also provides and optional method of interpreting the value of a standard deviation.
6.3.3 Measures of Shape
Measures of shape are tools that can be used to describe the shape of a distribution of data.
In this section, we examine two measures of shapes: skewness and Kurtosis.
<IP_ADDRESS> Skewness
A distribution of data in which the right half is a mirror image of the left half is said to be symmetrical.
One example of a symmetrical distribution is the normal distribution, or bell curve.
Skewness is a measure of symmetry, or more precisely, the lack of symmetry
The measures of asymmetry are called as measures of skewness.
The skewed portion is the long, thin part of the curve
Skewness and the relationship of the mean, median and mode
The concept of skewness helps us to understand the relationship between the mean, median and mode.
In a unimodal distribution (distribution with a single peak or mode) that is skewed, the mode is the apex (high point) of the curve and the median is the middle value.
The mean tends to be located toward the tail of the distribution, because the mean is affected by all values, including the extreme ones.
A bell-shaped or normal distribution with the mean, median and mode all at the centre of the distribution has no skewness.
<IP_ADDRESS>.1 Pearsonian coefficient of skewness
- This coefficient compares the mean and median in light of the magnitude of the standard deviation
\[S_k = \frac{3(\mu-M_d)}{\sigma}\]
where \(S_k\) = coefficient of skewness, \(M_d\) = median
Note that if the distribution is symmetrical, the mean and median are the same value and hence the coefficient of skewness is equal to zero.
If the value of \(S_k\) is positive, the distribution is positively skewed.
If the value of \(S_k\) is negative, the distribution is negatively skewed.
The greater the magnitude of \(S_k\), the more skewed is the distribution.
<IP_ADDRESS> Kurtosis
Kurtosis describes the amount of peakedness of a distribution.
Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution
Distributions that are high and thin are referred to as leptokurtic distributions.
Distributions that are flat and spread out are referred to as platykurtic distributions.
Between the above two types are distributions that are more ‘normal’ in shape, referred to as mesokurtic distributions
6.3.4 Measures of Association
- Many times in business it is important to explore the relationship between two numerical variables
- Measures of association are statistics that yeild information about the relatedness of numerical variables.
- In this section, we discuss only one measure of association, correlation, and do so only for two numerical variables.
<IP_ADDRESS> Scatter plot
- A scatter plot is a two dimensional graph of pairs of points from two numerical variables
- In a quantitative bi-variate dataset, we have a \((x,y)\) pair for each sampling unit, where \(x\) denotes the independent variable and \(y\) denotes the dependent variable.
- Each \((x,y)\) pair can be considered as a point on the Cartesian plan.
- Scatter plot is a plot of all the \((x,y)\) pairs in the dataset.
- The purpose of scatter plot is to illustrate any relationship between two quantitative variables.
- If the variables are related, what kind of relationship it is, linear or nonlinear?
- If the relationship is linear, the scatter plot will show whether it is negative or positive.
Example: Edgar Anderson’s Iris Data
This famous (Fisher’s or Anderson’s) iris data set gives the measurements in centimeters of the variables sepal length and width and petal length and width, respectively, for 50 flowers from each of 3 species of iris. The species are Iris setosa, versicolor, and virginica.
<IP_ADDRESS> Correlation
- Correlation is a measure of the degree of relatedness of two or more variables.
- Several measures of correlation are available , the selection of which depends mostly on the level of data being analysed.
- Ideally, researchers would like to calculate \(\rho\), the population coefficient of correlation.
- However, because researchers virtually always deal with sample data, this section introduces a widely used sample coefficient of correlation, \(r\).
- This measure is applicable only if both variables being analysed have at least an interval level of data
Pearson product-moment correlation coefficient (\(r\))
- The statistic \(r\) is the Pearson product-moment correlation coefficient, named after Karl Pearson (1857 - 1936).
- The tern \(r\) is a measure of the linear correlation of two variables.
- It is a number that ranges from -1 to 0 to +1, representing th strength of the linear relationship between the variables.
- An \(r\) value of \(+1\) denotes a perfect linear positive relationship between two variables.
- An \(r\) value of \(-1\) denotes a perfect linear negative relationship between two variables, which indicates an inverse relationship between two variables: as one variable gets larger, the other gets smaller.
- An \(r\) value of 0 means no linear relationship is present between the two variables.
\[ r = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2\sum(y-\bar{y})^2}}\]
\[ r = \frac{\sum{xy} - \frac{(\sum x\sum y)}{n}}{\sqrt{[\sum{x^2- \frac{(\sum{x})^2}{n}}][\sum y^2-\frac{(\sum y)^2}{n}]}}\]
- Examples: Following figure shows five different degrees of correlation:
NOTE
- When \(r=0\), it signifies there is no linear relationship between the two variables. (There can be a non-linear relationship, Figure (e))
- Figure (e): There is a very strong curvilinear relationship. But there is no linear relationship.
6.4 Tutorial
Chapter 6: Descriptive Statistics
- A power company in Sri Lanka designs and manufactures power distribution switchboards for hospitals, bridges, airports, highways and water treatment plants. Power company marketing director wants to determine client satisfaction with their products and services. He developed a questionnaire that yields a satisfaction score between 0 and 100 for participant responses. A random sample of 50 of the company’s 1000 clients is asked to complete a satisfaction survey. The satisfaction scores for the 50 participants are averaged to produce a mean satisfaction score.
- What is the objective of this study?
- What is the population for this study?
- What is the sample for this study?
- What is the statistic for this study?
- What would be a parameter for this study?
- Determine the type and the scale of measurement of the following variables
- The time required to produce an item on an assembly line
- the number of litres of milk a family drinks in a week
- The ranking of 50 students in your class after their overall performance have been designated as excellent, good, satisfactory or poor
- The telephone area code of clients in Australia
- The age of each of your batch mates
- The sales at the local pizza restaurant each month
- A student index number
- The response time of emergency services
- The number of tickets sold at a ticket counter on any given day
- Monthly maximum air temperature
The grades of 30 students for Statistics are as follows:
B, C, B, D, B, C, C, A , B, C,
C, B, E, B, B, D, D, F, B, D,
D, A, B, A, B, C, E, A, A, E
- Construct a frequency table with suitable values (eg: absolute frequency, relative frequency, cumulative frequency, cumulative relative frequency)
The gender type of a group of 10 students are as follows:
Male, Female, Female, Male, Female Female, Male, Male, Female, Female
- Construct a frequency table with suitable values (eg: absolute frequency, relative frequency, cumulative frequency, cumulative relative frequency )
- In a study conducted to investigate the relationship between delivery time and computer-assisted ordering, the following data were gathered from a sample of 40 firms, that 16 use computer-assisted ordering, while 24 do not. Furthermore, past data are used to categorize each firm’s delivery times below the industry average, equal to the industry average, or above the industry average. The results obtained are given in the table below.
| Computer Assisted Ordering | Delivery Time Below Industry Average | Delivery Time Equal to Industry Average | Delivery Time Above industry Average | Row Total |
|---|---|---|---|---|
| No | 4 | 12 | 8 | 24 |
| Yes | 10 | 4 | 2 | 16 |
| Total | 14 | 16 | 10 | 40 |
- For each row and column total, calculate the corresponding row or column percentage.
- For each cell, calculate the corresponding cell, row, and column percentages.
- Carry out graphical analysis to investigate the relationship between delivery-time performance and computer-assisted ordering.
- What conclusions can be made about the nature of the relationship?
- Giving suitable examples distinguish between multiple bar chart and a component bar chart.
- Write down advantages and disadvantages of each chart and when they should be used
- In a study conducted to investigate the effect of wearing helmets in riding motorcycles to reduce the head injuries, the following data were gathered by investigating 151 motorcycle accidents reported during last 10 month period. Out of the total number of cases (151) investigated, only 102 people were wearing helmets properly at the time of the accidents. Out of them, only 9 got severe head injuries. The rest got either minor head injuries or no head injuries at all. Among 49 who were not wearing helmets at the time of the accident, 15 got server head injuries and the rest got minor head injuries or no head injuries at all.
Use an appropriate table to represent the above data and state what conclusions can be drawn from it
Carry out a suitable graphical analysis to investigate the effect of wearing helmets in riding motorcycles to reduce the head injuries.
- The table below shows a frequency distribution of the weekly wages of 65 employees at the ABC Company. With reference to this table,
| Wages (in $) | Number of Employees |
|---|---|
| 250.00 - 259.99 | 8 |
| 260.00 - 269.99 | 10 |
| 270.00 - 279.99 | 16 |
| 280.00 - 289.99 | 14 |
| 290.00 - 299.99 | 10 |
| 300.00 - 309.99 | 5 |
| 310.00 - 319.99 | 2 |
| Total | 65 |
- Construct a histogram
- Construct a frequency polygon
- Construct a frequency curve
- State what conclusions can be drawn from the above graphical analysis
- The contribution of the agriculture, industrial, and service sector to the Gross Domestic Product (GDP) for each province in Sri Lanka is given in the table below. It is required to investigate whether the contributions are varying from province to province. What kind of graph you would suggest for representing data to serve the purpose. Justify your answer. Sketch the proposed graph.
- Marks of 16 students are given below.
\[\text{52, 88, 56, 79, 72, 91, 85, 88, 68, 63, 76, 73, 86, 95, 12, 69}\]
- Find the quartile of the distribution and interpret the values
- Construct a box plot for the data set
- Are there outliers in the data set?
- Listed below, ordered from smallest to largest, is the time in days the customers take to pay their invoices.
\[13, 13, 13, 20, 26, 27, 31, 34, 34, 34, 35, 35, 36, 37, 38, 41, 41, 41, 45, 47, 47, 47, 50, 51, 53, 54, 56, 62, 67, 82\]
- Determine the median
- Determine the first and third quartiles
- Determine the 2nd decile and the 8th decile
- Determine the 67th percentile
- What are outliers?
- Which of the following is/are unaffected by outliers? Underline the correct answer/ answers.
- Mean
- Median
- Mode
- Range
- Standard deviation
- Inter-quartile range
- Some summary measures of a variable is given below. Descriptive Statistics:
| N | Mean | StDev | Minimum | Q1 | Median | Q3 | Maximum |
|---|---|---|---|---|---|---|---|
| 50 | 5.389 | 0.997 | 3.150 | 4.658 | 5.350 | 6.020 | 8.600 |
- Are there outliers in this data set? YES/NO Justify your answer using a box plot
The mean and the standard deviation of 250 observations of a variable X are 62.1 and 4.3 respectively. However, in a re-scrutinizing process, it was found that the observations 72 and 81 were incorrectly recorded as 92 and 87. Find the correct mean and standard deviation of the data set.
The weekly sales from a sample of AB company were organized into a frequency distribution. The mean of weekly sales was computed to be $105,900, the median $105,000, and the mode $104,500
- Sketch the sales in the form of a smoothed frequency polygon. Note the location of the mean, median and mode on the X-axis
- Is the distribution symmetrical, positively skewed, or negatively skewed? Explain.
- Compare the variation of the annual incomes of executives with the variation of the incomes of unskilled employees. The sample information is given below.
| Type | \(\bar{x}\) | \(SD\) |
|---|---|---|
| Executives | 500,000 | 50,000 |
| Unskilled employees | 22,000 | 2,200 |
- Marks for the course module AB and the corresponding lecture attendance of 10 students are given below
| Student Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Marks in course module AB | 84 | 51 | 91 | 60 | 68 | 89 | 98 | 58 | 53 | 47 |
| Lecture attendance | 13 | 6 | 15 | 4 | 12 | 14 | 15 | 10 | 11 | 6 |
- Draw a scatter plot for the above data
- Calculate the coefficient of correlation
- Comment on the relationship between the marks and the lecture attendance
References
Black, K., Asafu-Adjaye, J., Khan, N., Perera, N., Edwards, P., & Harris, M. (2007). Australasian business statistics. John Wiley & Sons.
Spiegel, M. R., & Stephens, L. J. (2017). Schaum’s outline of statistics. McGraw Hill Professional.


[Teleport to the hunting grounds]
You can move to any of these locations. Requires an ancient adena.
Necropolis of Sacrifice (20-30) - 5500 AA


The best way to understand energy is to learn about its different forms.
Energy changes forms without loss or gain. The law of conservation of energy states that the total energy of an isolated system remains constant—it is said to be conserved over time.
Energy cannot be created or destroyed, but it can change forms.
- Chemical energy is converted to kinetic energy in the explosion of a stick of dynamite.
- A microwave converts electrical energy into light energy and then into the heat energy.
- As an object falls the gravitational potential energy of position is converted to kinetic energy.
- Chemical energy stored in methane is burned to form heat energy.
Work
Work measures a change in energy caused by a force.
Work is only done when a force causes an object to change position.
- An elevator lifting an person is work.
- Pedaling a moving bike is work.
- Pushing a box across the floor is work.
- A car skidding to a stop is work.
- Pushing a stationary wall is not work.
- A ball rolling on flat frictionless ground is not work.
$$W = F \Delta x \cos\theta$$
\(W\) = work [J, Joules, kg m²/s²]\(F\) = Force [N, Newtons, kg m/s²] vector
\(\Delta x\) = displacement [m] vector
\(\theta\) = angle between applied force and displacement
F cos θ returns the component of the force vector in the same direction as the displacement vector. vector.
solution
$$W = F\Delta x\cos\theta$$ $$W = (100)(200)\cos(30)$$ $$W = (100)(200)(0.866)$$ $$W = 17320\, \mathrm{J}$$solution
$$W = F\Delta x\cos\theta$$ $$W = (10)(2)\cos(0)$$ $$W = (10)(2)(1)$$ $$W = 20\, \mathrm{J}$$solution
The force to lift must be slightly larger than the force of gravity.
$$F_{g} = mg$$$$W = F\Delta x\cos\theta$$ $$W = (mg)\Delta x\cos\theta$$ $$W = (2085)(9.8)(2.5)\cos(0)$$ $$W = 51082.5\, \mathrm{J}$$
Kinetic Energy
Kinetic energy is the energy objects have from motion.
derivation of kinetic energy
$$v^2 = u^2 +2a\Delta x$$
Kinetic energy is defined as relative to zero velocity, so set the initial velocity to zero.
$$v^2 = 2a\Delta x$$ $$a\Delta x =\frac{v^2}{2}$$ $$ma\Delta x =\tfrac{1}{2}mv^2 $$ $$F\Delta x =\tfrac{1}{2}mv^2 $$ $$W = \tfrac{1}{2}mv^2$$Kinetic energy is the same as the work required to bring a mass to a speed.
$$K =\tfrac{1}{2}mv^2$$$$K = \tfrac{1}{2} mv^{2}$$
\(K\) = kinetic energy [J, Joules]\(m\) = mass [kg]
\(v\) = velocity [m/s]
Energy has units of Joules in the metric system. 1 Joule is the same as a 2 kg mass moving at 1 m/s.
Energy is a scalar. It has no direction. Not having to work with vectors can make solving 2-D and 3-D problems much easier.
Zoe has a mass of 20kg. She is 0.56 meters tall with reddish fur.
solution
$$KE = \tfrac{1}{2}mv^{2}$$ $$K = \tfrac{1}{2}(20)(5)^{2}$$ $$K = 250 \, \mathrm{J}$$solution
$$K = \tfrac{1}{2}mv^{2}$$ $$K = \tfrac{1}{2}(20)(-5)^{2}$$Energy is a scalar so direction doesn't matter. The negative sign on the velocity goes away because it is squared, and we get the same answer when she runs left or right.
$$K = 250 \, \mathrm{J}$$solution
$$K = \tfrac{1}{2}mv^{2}$$ $$100 = \tfrac{1}{2}(80)v^{2}$$ $$100 = (40)v^{2}$$ $$2.5 = v^{2}$$ $$1.58 \mathrm{\tfrac{m}{s}} = v$$That's about walking speed. It takes about 100 J to accelerate a person from rest to walking speed.
Potential Energy
Potential energy is stored energy from the position of an object.
Examples of potential energy:
How could each of these potential energies convert into kinetic energy?
Gravitational Potential Energy
If you drop a massive object, it will gain kinetic energy as it falls. We call the potential for the energy of its position to turn into kinetic energy gravitational potential energy
derivation of gravitational potential energy
Gravitational energy can be calculated with the work equation. $$W = F\Delta x\cos\theta$$ As an object falls the force of gravity pulls the object vertically down. $$F_{g} = mg$$ $$W = mg\Delta x\cos\theta$$ The displacement is vertical so lets call it height: h. $$W = mgh\cos\theta$$ The force of gravity and the height are pointed in the same direction, so cos(0)=1. $$W = mgh$$ We now have a nice equation for the energy that comes from the work of lifting an object.
$$U_{g} = mgh$$
\(U_g\) = gravitational potential energy [J, Joules]\(m\) = mass [kg]
\(g\) = acceleration from gravity, 9.8 on Earth [m/s²]
\(h\) = height [m]
Height is the tricky part of this equation, because where is zero height? The ground floor can be the zero point, or the floor of the basement or the top of a table, or any point.
Which of these zero points are correct?
We get 3 different answers because of the different zero points. Unlike kinetic energy, gravitational potential energy isn't consistent on it's own, but we can still use it to find a change in gravitational potential energy.
solution
$$\text{6th story - 1st story = 5 stories} $$ $$ \left(5\,\mathrm{story} \right) \left(\frac{3 \, \mathrm{m}} {1 \, \mathrm{story}}\right) = 15 \, \mathrm{m}$$$$U_{g} = mgh$$ $$U_{g} = (100)(9.8)(15)$$ $$U_{g} = 14700 \, \mathrm{J}$$
convert ft to m
solution
$$-400\,\mathrm{ft}\left(\frac{0.3048\,\mathrm{m}}{1\,\mathrm{ft}}\right) = -122\,\mathrm{m}$$$$U_{g} = mgh$$ $$U_{g} = (80.0)(9.8)(-122)$$ $$U_{g} = -95648\,\mathrm{J}$$
Hooke's Law
Hooke's law is an approximation the force from a spring. It is an accurate approximation for solids under small deformations.
$$F = -k x$$
\(F\) = Restoring elastic force [N, Newtons, kg m/s²] Vector\(k\) = spring constant [kg/s²]
\(x\) = displacement from equilibrium [m] vector
The spring constant k is different for every spring. Sometimes when you buy a spring the constant will be listed, but we can also measure k directly.
The Spring force is always directed back towards the equilibrium point. This works for compressing the spring or expanding the spring.
answer
The red/orange one is correctly drawn. Displacement is defined as the the distance from equilibrium for the spring only.
solution
$$ x= \mathrm{80\,(m)m = 80\left(\frac{1}{1000}\right)m =0.08 \, m}$$$$F= -k x$$ $$F= -(0.4 \, \mathrm{\tfrac{kg}{s^2}})(0.08 \, \mathrm{m})$$ $$F= -0.032 \, \mathrm{N}$$
hmmm, why is the force negative. We know the force should point right. It needs to be positive. Oh, the displacement points left. It should be negative.
$$F= -(-0.4 \, \mathrm{\tfrac{kg}{s^2}})(0.08 \, \mathrm{m})$$ $$F= 0.032 \, \mathrm{N}$$Elastic Potential Energy
We can convert our equation for the elastic force into energy.
derivation of elastic potential energy
The energy stored in an elastic material can be calculated with the equation for work and some calculus.
$$W = F x\cos\theta$$The elastic force and the displacement are always at 0 degrees.
$$W = F x$$ The force is the elastic force. $$F = -k x$$We can't directly replace F with -kx in the work equation because the force changes over the displacement.
F = 0 at x = 0
F = -k (10) at x = 10
We need to use an operation from calculus called an integral. The integral will find the area of F times x, which equals work.
$$W = \int_{0}^{x} k x \ dx$$ $$W = \tfrac{1}{2} kx^2 $$ $$U_s = \tfrac{1}{2} k x^2 $$$$U_{s} = \tfrac{1}{2}kx^{2}$$
\(U_s\) = elastic/spring potential energy [J, Joules, kg m²/s²]\(k\) = spring constant [kg/s²]
\(x\) = displacement [m]
Elastic potential energy is equal to the work required to bring a spring to a position. Hooke's law is an good approximation until the spring get stretched too much and deformed.
k = kg/s² m = kg
How does the spring constant affect the behavior of the spring?
How does mass affect the behavior of the spring?
solution
$$200 \, \mathrm{(c)m} = \mathrm{200\left(\frac{1}{100}\right)m = 2\,m}$$$$U_{s} = \tfrac{1}{2}kx^{2}$$ $$U_{s} = \tfrac{1}{2} (0.3 \,\mathrm{\tfrac{kg}{s^2}})(2\,\mathrm{m})^{2}$$ $$U_{s} = 0.6 \, \mathrm{J}$$
solution
$$x= \mathrm{40\,(c)m = 40\left(\frac{1}{100}\right)m =0.4 \, m}$$$$F= -k x$$ $$-\frac{F}{ x} = k$$ $$-\frac{-100 \,\mathrm{N} }{0.4\,\mathrm{m}} = k$$ $$250 \mathrm{\tfrac{kg}{s^2}} = k$$


This is my first go at creating a starter template/css-framework for quick website builds. It uses
almost no external sources, except for fonts and social media icons. I have
separated the .css and .js files for each of the effects or segments of the
page so that you may find what you need more easily.
Furthermore, I've tried
naming these files and their contents in such a way to make them as understandable
as possible, but since I made them, and therefore know them like the back of my hand, I realise that this most certainly
won't be the case for others. I'm only human after all. Some of the stuff is covered on this page, but
you will find out the most by reading the code. Hope you find this template useful!
First, we have the whole page, which contains the main menu, the page, and footer. Inside the page, there are rows that contain a couple of columns, each with a different width, stacked next to each other.
The column width is adjusted using the col00 - col10 classes, used in combination with the column01 class. The usage of both is so that the user can define a specific width for each column on a larger screen, but when the screen size is smaller, like a phone or tablet, all the columns will receive the full width as specified in the column01 class at the 768px @media query.
In between this segment and the next one on the left (or above on smaller devices), there is an empty column which is used to separate them. Take note that if column sizes equal to more than 10 (or 100% width), the column which is last in that row will drop into the next row.
Furthermore, there are separator divs. These can be placed into the columns, rows, or between rows. Also, there are classes for removing content on certain screen sizes, so if you do not need some element, like separator, image, etc, apply the "not-on-X" classes.
Images can be placed in numerous ways, as seen above, and in the continuation of this page. Take a look at the classes in the style.css along with this html to understand better how to achieve these effects.
Here we have text that's being wrapped around an image that is floating on the left side. The image is placed directly into the paragraph where it should show up, and won't flow out of its bounds. By changing its class' ending letter to "-r", it will float to the right side.
Some images are used to cover the whole row or fill the column they're in.
To cover the whole row, add a column that is absolutely positioned (so that
it won't interfere with the rest of the content), add a class of col10, and full-height (so that it fills the entire row),
and place an image inside it.
Give that image the image-div-cover class, and it will fill the whole column,
and therefore the whole row.
If you wish the row to fill the whole screen, as in the top of this page, give that row
a cont-height100 class as well as max-width-override (as the default max-width for rows is 1200px), and
do the aforementioned steps. Voila! The entire screen is covered.
In the upcoming segment, the images have multiple classes as well as their container divs, all working together in order to align them on multiple screen sizes appropriately, as well as the text contained with them.
The Gallery below can be used as is, perhaps instead of the previous image segment,
or it can be used as an entire page for itself, where you would wish to put your work or products.
It is integrated with the Lightbox, just like the previous segment, but the styling for it
is separated in gallery-images.css for ease of use.
That's the basics covered! Thank you for picking up the Bare Bones template and trying it out.
Have fun!


[gecode-users] gecode autoconf setting
Christian Schulte
cschulte at kth.se
Mon Oct 22 20:13:10 CEST 2007
Hi Martin,
Yes that shouldn't be hard (actually the hard part is packing a binary
distribution). Should be there in Gecode 2.0.0.
Christian
-----Original Message-----
From: users-bounces at gecode.org [mailto:users-bounces at gecode.org] On Behalf
Of Martin Mann
Sent: Monday, October 22, 2007 3:22 PM
To: gecode user list
Subject: [gecode-users] gecode autoconf setting
Hi,
I am currently preparing an autoconf based package including
gecode-1.3.0 as subpackage (subdir) in the source distribution using the
standard autoconf macro
AC_CONFIG_SUBDIRS([gecode-1.3.0])
Everything works fine including compilation:
- my configure enters the subdir and calls the gecode configure
- my makefile enters the gecode subdir and compiles the library
- my code compiles using the previously compiled local gecode version in
the subdir
- my resulting program works and can be installed in the system using
the standard "make install" target generated by autoconf
so far everything fine!
the problem occurs, when i want to create a source distribution of my
package (include gecode of course)
calling the standard "make dist" results in
...
make[1]: Entering directory `/home/..../gecode-1.3.0'
make[1]: *** No rule to make target `distdir'. Stop.
make[1]: Leaving directory `/home/..../gecode-1.3.0'
make: *** [distdir] Error 1
a closer look at the 'gecode-1.3.0/Makefile' shows, that there are only
make targets to generate a distribution of the include-headers and
compiled library - the binary distribution. This is nice to install the
precompiled library but doesnt help source sub-supplier as I am.. ;)
Think this is also usefull for other people that want to integrate a
certain gecode version as a bundle into their source project and to
distribute it.
Is there a reason for the non-standard Makefile not providing a source
distribution generation? If not is it possible to enable this in the
comming gecode-2.0.0 version?
Would be glad about! ;)
Meanwhile I will try to write a dist make target to enable the support
locally. Do you ve got this already or are you interested to get it if I
can manage?
So far, hope to help,
Martin
--
Martin Mann, Dipl. Bioinf.
Bioinformatics - Inst. of Computer Science
Albert-Ludwigs-University Freiburg
Tel: ++49-761-203-8259
Fax: ++49-761-203-7462
http://www.bioinf.uni-freiburg.de/~mmann/
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Welcome to Jenni's documentation!
¶
Contents:
Introduction
Summary
Why use Jenni?
How does it work?
What will be next?
Security Warning
Sample Directory Structure
CLI
Seed Command
Run Command
Seed Job
Testing
Running Bandit
Implementation Details
Programmatic Job Generation
Customisation Hooks
FolderBase.add_new_job
FolderBase.add_new_folder
FolderBase.yield_items_pre_sub_item
FolderBase.yield_items_post_sub_item
FolderBase.sub_item_completed_handler
Summary
Security
Pipeline Execution Step Server
API
Folders
RootFolderBase
FolderBase
Jobs
GroovyJobBase
Job Parameters
BooleanParam
IntParam
StringParam
ValidatingStringParam
TextParam
ChoiceParam
jenni.utils
Other Classes
JobBase
PipelineJobBase
PythonPipelineJobBase
All Modules
itembase
jobbase
folderbase
rootfolderbase
pipelinejobbase
pythonpipelinejobbase
groovyjobbase
Pipeline Execution Steps
steps.archiveArtifacts
steps.build
steps.sh
steps.stashing
Examples
Basic Example
jobs/__init__.py
jobs/job1.py
jobs/job1.groovy
Indices and tables
¶
Index
Module Index
Search Page
Jenni
Navigation
Contents:
Introduction
Security
API
Examples
Basic Example
Related Topics
Documentation overview
Next:
Introduction
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