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Talk:Mode (statistics)
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Contents
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1 If the highest frequency is one...
2 Commute with
3 Sample from a continuous distribution
4 I have a similar question to the first one above
5 Computing the mode
6 Infinite mode
7 Comparison of mean, median and mode: move to "average"?
8 Better Data Set for the Comparison of common averages table
9 Criticism of the mode
10 Probably an error
11 Unimodal distributions
12 Assessment comment
13 Proposed removal of section
14 Different treatment of discrete and continuous?
15 External links modified (February 2018)
If the highest frequency is one...[edit]
Question: Let's say the set is 1,2,3,4.
Is the mode all of them, since the highest frequency is one, or is there no mode whatsoever?
I brought this to question since the definition uses "frequency," and it would appear that they are all equally frequent and thus all modes? I'm not sure of the official accepted interpretation.
DOES ANYONE KNOW THE ANSWER?
The mode cannot be calculated in such a varied set of data and thus should not be used. - Ferret 03:27, 23 March 2006 (UTC)
This was a multiple choice question on a standardized test, so would I say 4 modes or no modes?--Dch111 21:02, 16 May 2006 (UTC)
No modes, according to my programming teacher who worked for NASA. —The preceding unsigned comment was added by 75.8.96.7 (talk) 04:42, 11 December 2006 (UTC).
I got a proublem from my math class which implies there is a mode for a set of 5 numbers. (It's a hard proublem to explain so I won't right it out.) Lophoole 01:00, 26 January 2007 (UTC)Lophoole
Commute with[edit]
Do we really need to say commute with when discussing the linearity of mean, mode, and median? It does mean what we want to say, but it will convey it only to readers who don't need this article. Septentrionalis 21:23, 5 May 2006 (UTC)
In both places where it is used, it is immediately explained. Doesn't that work? I can't judge, because I don't need this article. LambiamTalk 01:28, 6 May 2006 (UTC)
That's the problem. Nobody who is editing it does. (And linear or independent might be just as bad.) Septentrionalis 01:51, 6 May 2006 (UTC)
I have reverted the addition of a wikilink to Commutative operation because it does not really cover the concept meant here. Commutative diagram is more to the point, but even I :) can see that that is a bridge too far. LambiamTalk 02:13, 6 May 2006 (UTC)
Well it would be nice if commutative operation really brought out the idea of commutative diagrams; it is in there, it's just not going to be obvious to the audience here.... Septentrionalis 03:25, 6 May 2006 (UTC)
The way I see it is as follows. For "commutative" in the sense of commutative diagram, reducing everything to the simplest case, operations f and g commute if:
f o g = g o f.
There is no universal quantification here, and the commuting operations are a pair of unary operations (although generalizations are possible and usual).
For "commutative" in the sense of commutative operation, operation ⊕ is commutative if,
x ⊕ y = y ⊕ x for all x and y in the domain.
Here the universal quantification is essential, and there is one operation which is binary.
I expect that attempts to unify the two notions will produced strained and non-natural results. LambiamTalk 18:51, 6 May 2006 (UTC)
Rather than try to explain the usage, I kept the idea. See what you think; I wouldn't put Commutative diagram under the See also, but go ahead if you like. Perhaps a general article on Commutativity? Septentrionalis 01:58, 7 May 2006 (UTC)
Sample from a continuous distribution[edit]
I don't understand why in "a sample from a continuous distribution [...] each value will occur precisely once." I'm not particularly familiar with statistics (I didn't know what a 'mode' was before reading this article), so this may just reflect confusion on my part, but I don't see why sample from a continuous distribution will contain every element only once (although it seems to me that it would do so with probabililty 1, i.e., almost certainly). Could someone explain? Benja 13:23, 15 May 2006 (UTC)
Indeed, more precisely formulated, the probability of any duplicate elements is 0. So if you were to repeat this for the rest of the life of the universe, you'd never expect a duplicate. Isn't that good enough for "will occur precisely once"? More strongly, though, the values have to be represented somehow, and one imaginable way is a random-value producing apparatus that keeps producing a stream of digits making up the decimal expansion of the values for the random variables. So imagine that after several eons of steady production we have that one stream is "0.7785[seventeen umptyzillion and four digits omitted]459(to be continued)" and so is another. Are the two values duplicates of each other? Perhaps, but most likely not. They may diverge at every next digit. But even if they are, we'll never know, since the streams will never be complete. So actually, the probability of any duplicate elements is 0, and, moreover, even if duplicacy (is that an English word?) occurs, it remains forever unknowable. Might as well say: can't happen. --LambiamTalk 13:50, 15 May 2006 (UTC)
Huh. Seems like I understood alright, but you guys have a seriously different way of looking at these things. :-) (I was assuming that we were using the language of theory, since continuous distributions are a theoretical/modelling tool rather than something you'd deal with in practice.) Although, if we look at this from a practical point of view, wouldn't it be more helpful to talk about the practical situation where you get, say, temperature readings with +/- .01 degrees accuracy and it's quite possible, but unlikely, that you'll get the same reading twice, rendering the mode meaningless and requiring the technique described? -- In any case, thanks for explaining! Benja 14:30, 15 May 2006 (UTC)
"You guys"? I'm just one editor with one viewpoint (who happened to have written the relevant paragraph). The above was all very theoretical and only thought experimental; actual random-value producing apparatus doesn't last that long. For practical application, the article tells you to discretize the data, as for a histogram, so that you will actually get (still in practice) frequencies greather than 1. --LambiamTalk
observe the following data 1,3,4,5,7,9,12,13,15,15,15,15,15,16,17,19,22,24,24,25,25,26,26,27,30,30,31,32,32,33,33,34,35,35,36,36,37,,39,40,40,45,45,46,47,48,49,49. class intervals freq. 0-10 6 10-20 10 20-30 8 30-40 14 40-50 9 according to the defination of the mode, 30-40 is the modal class. and the mode will lie in that interval. But if we look at the raw data then 15 is the mode. how do we resolve this contradiction ? Reeded 15:11, 28 March 2007 (UTC)reeded
If the population distribution is continuous, the chance of observing two or more identical values is 0. If you discretize the data, then the result may (of course) depend on the details. If you have a distribution whose density function has a thin spike, it gets diluted when the intervals are much wider. The sample data show a thin spike at 15 and a robust mound around 35. The choice of intervals does actually not conform to the advice given in the article, but it is also possible to construct an example that does conform but still displays this phenomenon. --LambiamTalk 23:01, 28 March 2007 (UTC)
Thanks for that! another hiccup! what happens if the data is given as a discontinuous data. do we make the intervals continuous, if so why? 59.180.85.146 06:30, 8 April 2007 (UTC)reeded13
Do you mean contiguous intervals? And isn't the data (if finite in size) always "discontinuous"? If the data comes from a discrete probability distribution, such as the Poisson distribution, then usually you don't use intervals but just tally the values: 3× a 0, 3× a 1, 4× a 2, 9× a 3, 5× a 4, 1× a 5, 2× a 6. If the frequencies are very low, you could lump groups of adjacent values together, making sure the groups have equal sizes. If the underlying distribution is continuous, like for the gamma distribution, then you must use intervals, and these intervals should partition the sample space into equi-sized parts. These intervals must be contiguous: together they must cover the whole space without gaps. --LambiamTalk 12:27, 8 April 2007 (UTC)
thanks for that!59.180.55.151 15:32, 11 April 2007 (UTC)reeded
I have a similar question to the first one above[edit]
My question is I have a set of 10 numbers with 44 and 93 appearing twice. Is the mode "44 and 93" or is it the average of the two, as calculated with "median" involving even numbers?
Thanks! LandOfIsrael 18:26, 15 August 2006 (UTC)
The article gives the example of a data sample [1, 1, 2, 4, 4] and states: "the mode is not unique". That is about all that can be said about it. Depending on your needs, predilections, and local customs, you can pick your choice between: (a) for this sample the mode is undefined; (b) this sample actually has two modes: 1 and 4; and (c) the mode is indeterminate (whatever that means). I would not take the average except when you have a histogram peaking in two adjacent slots, because that could be severely misleading. Now here is a question: what is the mode of an empty data sample? --LambiamTalk 01:22, 16 August 2006 (UTC)
Thank you so much!!! I appreciate your taking the time to help me. I didn't see that example in the article even though it was staring me in the face. Oops! :) In this case I think I'll write that the mode is "44 and 93" and see how that works. Again, thank you!!! (Thanks as well to whoever moved my question to the bottom -- I wasn't aware of the wiki custom of oldest on the top/newest at the bottom!) LandOfIsrael 10:52, 16 August 2006 (UTC)
Computing the mode[edit]
Hi. I would like to see a section on how to effectively compute the mode of a list of numbers. That would be useful. --Spoon! 02:03, 9 February 2007 (UTC)
Compute the mode by three successive passes through the data list thus:
Pass 1: Count the number of different sample values
Start a separate count variable = 0 for each sample value and do
Pass 2: Increment the appropriate count for each sample
Start a frequency variable = 0 and do
Pass 3: For each value, if its occurence is greater than the frequency variable, then update the frequency variable and remember which value did it.
At the end of Pass 3 the last remembered (i.e. stored) value is the mode. (With a little more thought, Pass 1 and 2 can be achieved in a single pass.) Cuddlyable3 07:48, 8 May 2007 (UTC)
This requires that you maintain a data structure for counting, for each possible sample value, its frequency. In general, that has to be an associative array (lookup table). For example, if the data is something like {2404030, 712, 20574, 2404030, 657032180, 20574, 712, 2404030, 712, 20574, 2404030, 657032180, 657032180, 2404030, 31710510, 657032180, 31710510, 20574, 712, 712}, you don't want to allocate a normal array for this (and you cannot know the required size in advance). Unless you have a set of library functions for this, that is not an easy programming exercise. If all values are different, the size grows to be as large as the data sample. It is almost always easier and more efficient to sort the values, after which you can find the mode in a single pass. Here is a way of doing that, presented in pseudocode:
  (maxCnt, loMode, hiMode) := (0, NAN, NAN);

  (cnt, last) := (0, NAN);
  for x ← sorted(data):
     if x ≠ last:
        if cnt = 0:
           skip;
        else if cnt > maxCnt:
           (maxCnt, loMode, hiMode) := (cnt, last, last);
        else if cnt = maxCnt:
           if last < loMode:
                 loMode := last;
           end if
           if last > hiMode:
                 hiMode := last;
           end if
        else:
           skip;
        end if
        (cnt, last) := (1, x);
     else:
        cnt := cnt+1;
     end if
  end for

  if cnt = 0:
     skip;
  else if cnt > maxCnt:
     (maxCnt, loMode, hiMode) := (cnt, last, last);
  else if cnt = maxCnt:
     if last < loMode:
        loMode := last;
     end if
     if last > hiMode:
        hiMode := last;
     end if
  else:
     skip;
  end if

The constant NAN is supposed to be a value that is different from all possible data values (and always compares as different). At the end, the variable maxCnt contains the highest frequency observed. The result is delivered in a pair of program variables: loMode for the lowest data value having maxCnt occurrences, and hiMode for the highest data value having maxCnt occurrences. If the data is unimodal, these two will be the same. If it is possible to append an "infinite" value to the data (or any value exceeding all original data values), the last block can (and should) be omitted.
A problem with including this algorithm in the text of the article is that this is "original research". --LambiamTalk 15:38, 8 May 2007 (UTC)
Before computing the mode of a series of values it is probable wise to see if the distribution is unimodal. The dip test can be used for this purpose. If it is not unimodal this may complicate the programming somewhat. Determining the number of modes in an arbitrary list of numbers if it is not unimodal is difficult. It may be impossible but I am not sure if this is a known fact. DrMicro (talk) 13:35, 7 July 2012 (UTC)
Infinite mode[edit]
Can someone give an example of "Furthermore, like the mean, the mode of a probability distribution can be (plus or minus) infinity"? --Rumping 00:07, 7 September 2007 (UTC)
I've replaced the sentence by one that seems more true. --Lambiam 12:28, 8 September 2007 (UTC)
Thought not. Thanks --Rumping 11:56, 10 September 2007 (UTC)
Are you trying to say that the distribution associated with the Cantor function does have a defined mode? --Lambiam 14:29, 10 September 2007 (UTC)
Not at all - just that I originally doubted whether there was a dististibution on the reals with an infinite mode.
As to your question, it is an interesting point; I think it might be possible to say that the modes of the Cantor distribution are the members of the Cantor set. See what you think of the following singular distribution: imagine a random variable X on [0,1] with p(x)=2x. X obviously has a mode at x=1. Then define a random variable Y determined by X by (a) writing X as a binary fraction [with 1 as 0.1111...], (b) changing all the '1' digits to '2' digits, and (c) reading the result as a ternary fraction. This is a monotonic transformation so the mode of Y should be at 0.2222...(base 3), i.e. at 1. I have not convinced myself and I won't press the point. --Rumping 16:10, 10 September 2007 (UTC)
It is indeed an interesting point, and it crossed my mind that the Cantor set could be said to be the mode. On the other hand, the somewhat loose definition given for mode ("the most frequent value assumed by a random variable") is not designed for such pathological cases, and it would be stressing its applicability beyond its scope. I see that the anonymous editor who set up the "Probability distribution" box at Cantor distribution agreed, giving "Mode: n/a". For the Rumping distribution, the answer 1 seems rather reasonable; in an experimental process, the experimentator would definitely be led to hypothesize this, and in the limit, using appropriately finer and finer discretizations as the number of trials grows, the modes of the discretized distributions will tend to 1. --Lambiam 23:02, 10 September 2007 (UTC)
Comparison of mean, median and mode: move to "average"?[edit]
Does anyone else think this section would fit better in average ? I propose moving it there with a link to it from here. Let me know what you think. Qwfp (talk) 14:50, 22 February 2008 (UTC)
In my opinion, Average should really be rewritten to cover mainly the three well-known measures of central tendency, arithmetic mean, median and mode (with "See also"s for Running average aka Moving average and Weighted average aka Weighted mean), and then this section can be moved to the rewritten article Average. The current content of Average – inasmuch as it is not OR – should be merged into Mean. Unfortunately, we did not reach consensus on this over at Talk:Average, where it was even argued that mentioning the concept of statistical mode might harm laypeople who would not know how to use it in a safe way. --Lambiam 11:13, 23 February 2008 (UTC)
I think it belongs in a section on a central tenancy article. Central tendency currently redirects to average, which I think is a mistake because average is just one example of a measure of central tendency. Comparison logically belongs there. Jason Quinn (talk) 01:34, 21 July 2008 (UTC)
Also, I should mention that anybody searching for "central tendency" is likely a technical person not interested in most of the average article. They are interested in what all the different measures are to apply in some application they are building. I will in the next week or so break the redirect and start a new article if nobody persuades me otherwise. Jason Quinn (talk) 01:38, 21 July 2008 (UTC)
I don't like the current average article at all. I think it should focus much more on just the arithmetic mean and the mean and central tendency pages should discuss the rest. Jason Quinn (talk) 01:42, 21 July 2008 (UTC)
The article Central tendency redirects to Average because it was merged with that article. As used in statistics, average and measure of central tendency are synonyms. The mean, the median and the mode are all different kinds of averages. As I wrote on the talk page of Average:[1]
Just do a Google search on ["measures of central tendency"]. The first hit: "This section defines the three most common measures of central tendency: the mean, the median, and the mode."[2] The next: "Measures of central tendency—mean, median, and mode—can help you capture, with a single number, what is typical of the data."[3] And so on. The search term ["measures of average"] gives similar results: --Lambiam 06:26, 27 November 2007 (UTC)
P.S. And here is a quote from the intro paragraph of our own article Mean: "It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, median, and mode. For instance, average house prices almost always use the median value for the average." --Lambiam 06:31, 27 November 2007 (UTC)
It is fine if arithmetic mean gets more attention than other kinds of mean and than median and mode, because it is the most common kind of average, but it would be a mistake to leave the others out. But since each already has an article, the treatment in Average can be relatively brief. --Lambiam 21:24, 26 July 2008 (UTC)
On a more practical bent, elementary school texts are discussing the three concepts, "mode," "median," and "mean" together without even mentioning the "average" concept. What drew me to the mode article (and to the section on mode, mean, and median) was a basic exercise from my daughter's sixth grade math text that elicits answers regarding these three particular concepts. The exercise in her homework does not discuss the concept of "average" or use "mean" interchangeably with the former term. Additionally, from a layman's perspective, the concepts of "mean" or "average", "mode," and "median" are distinct concepts, not just different flavors of "average," and the most common use of "average" does not encompass either "mode" or "median". Otherwise, there would be confusion as to what, for example, the "average age" of a given population group was (i.e., is it the "mode," "median," or "mean"?). A more in-depth treatment of the subtle differences between various "tendencies" is best addressed elsewhere; the topic's breath is somewhat greater than what most people will be searching for. —Preceding unsigned comment added by 71.34.165.94 (talk) 02:44, 26 August 2008 (UTC)
Better Data Set for the Comparison of common averages table[edit]
I find it unfortunate that this table uses a different data set for each example, and that the clarity and usefulness would be greatly improved if the same data set was used for each entry. I would like to edit it with the following data set: 1, 2, 2, 3, 4, 7, 9. This would give a mean of 4, a median of 3, and a mode of 2. --Pdcurry (talk) 23:31, 22 December 2009 (UTC)
Criticism of the mode[edit]
Why isn't there a criticism section??? —Preceding unsigned comment added by 129.98.192.217 (talk) 03:01, 25 October 2010 (UTC)
Probably an error[edit]
In the Wikipedia article we read "For samples, if it is known that they are drawn from a symmetric distribution, the sample mean can be used as an estimate of the population mode." I propose to change to: "For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode." Otherwise, it fails for example for values 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, having mean 3.5 which is not a good estimate of the two modes 2 and 5. — Preceding unsigned comment added by 140.105.52.132 (talk) 22:22, 21 March 2012 (UTC)
Unimodal distributions[edit]
I texified the "Unimodal distributions" section under median, and found this similar section on this article. However, the two contradict each other. The media article claims that the difference between the mean and median are bounded by σ ( 3 / 5 ) 1 / 2 {\displaystyle \sigma (3/5)^{1/2}} , whereas this article claims that it is σ 3 1 / 2 {\displaystyle \sigma 3^{1/2}} .
Which one is correct?
Tebello TheWHAT!!?? 21:34, 18 March 2012 (UTC)
The correct statement is that the difference between the mean and the MEDIAN is σ ( 3 / 5 ) 1 / 2 {\displaystyle \sigma (3/5)^{1/2}} and that the difference between the mean and the MODE is σ 3 1 / 2 {\displaystyle \sigma 3^{1/2}} . This is what is written in the articles.DrMicro (talk) 13:31, 7 July 2012 (UTC)
Assessment comment[edit]
The comment(s) below were originally left at Talk:Mode (statistics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Quite a reasonable article, though the language could perhaps be made simpler and flow better in places (i've just had a go at the first sentence). This isn't a complex concept and not as important as the mean or median in practice so i don't think the article needs to be any longer – if anything, it should be more concise. Qwfp (talk) 15:00, 22 February 2008 (UTC)
Last edited at 15:00, 22 February 2008 (UTC). Substituted at 20:06, 1 May 2016 (UTC)
Proposed removal of section[edit]
I plan to remove the section Confidence interval for the mode with a single data point unless I hear an objection. There are at least three problems with this section:
Undue weight. It comprises well over 1/3 of the article, despite being about a very esoteric point (samples with only a single data point).
At least part of it doesn't make sense (the first part of the subsection "Statistical tests").
It assumes at the beginning that we know the mode, in order to form a confidence interval for the mode. Likewise, later in discussing symmetric distributions, the confidence interval for the mean/median/mode depends on the true value of the mean/median/mode. This renders it not useful in practice even if we did have a single-value sample.
Let me know if there is any objection to removing the entire section. Loraof (talk) 15:46, 24 June 2017 (UTC)
Different treatment of discrete and continuous?[edit]
From the lead:
"The mode of a continuous probability distribution is the value x at which its probability density function has a locally maximum value";
"The mode is not necessarily unique to a given discrete distribution, since the probability mass function may take the same maximum value at several points".
Rather strange. A multimodal continuous case: several local maxima, possibly of different "heights". But multimodal discrete case: several local maxima of the same "height". Really so? Boris Tsirelson (talk) 18:55, 26 June 2017 (UTC)
Strange, yes. But this seems to be how statisticians use these terms. References would be worthwhile. I'll add one for the discrete case. Loraof (talk) 14:39, 27 June 2017 (UTC)
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Root mean square
From Wikipedia, the free encyclopedia
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In statistics and its applications, the root mean square (abbreviated RMS or rms) is defined as the square root of mean square (the arithmetic mean of the squares of a set of numbers).[1] The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle.
For a cyclically alternating electric current, RMS is equal to the value of the direct current that would produce the same average power dissipation in a resistive load.[1] In Estimation theory, the root mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data.
Contents
[hide]
1 Definition
2 In common waveforms
2.1 In waveform combinations
3 Uses
3.1 In electrical engineering
3.1.1 Voltage
3.1.2 Average electrical power
3.2 Speed
3.3 Error
4 In frequency domain
5 Relationship to other statistics
6 See also
7 References
8 External links
Definition[edit]
The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In Physics, the RMS current is the "value of the direct current that dissipates power in a resistor."
In the case of a set of n values { x 1 , x 2 , … , x n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}} , the RMS
x r m s = 1 n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) . {\displaystyle x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}.}
The corresponding formula for a continuous function (or waveform) f(t) defined over the interval T 1 ≤ t ≤ T 2 {\displaystyle T_{1}\leq t\leq T_{2}} is
f r m s = 1 T 2 − T 1 ∫ T 1 T 2 [ f ( t ) ] 2 d t , {\displaystyle f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},}
and the RMS for a function over all time is
f r m s = lim T → ∞ 1 T ∫ 0 T [ f ( t ) ] 2 d t . {\displaystyle f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}.}
The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sequence of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.[2]
In the case of the RMS statistic of a random process, the expected value is used instead of the mean.
In common waveforms[edit]
Further information: RMS amplitude
Sine, square, triangle, and sawtooth waveforms.
A rectangular pulse wave of duty cycle D, the ratio between the pulse duration ( τ {\displaystyle \tau } ) and the period (T); illustrated here with a = 1.
Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.
If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may or may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is:
Peak-to-peak = 2 2 × R M S ≈ 2.8 × R M S . {\displaystyle =2{\sqrt {2}}\times {RMS}\approx 2.8\times {RMS}.}
For other waveforms the relationships are not the same as they are for sine waves.
Waveform
Equation
RMS
DC, constant y = A 0 {\displaystyle y=A_{0}\,} A 0 {\displaystyle A_{0}\,}
Sine wave y = A 1 sin  ( 2 π f t ) {\displaystyle y=A_{1}\sin(2\pi ft)\,} A 1 2 {\displaystyle {\frac {A_{1}}{\sqrt {2}}}}
Square wave y = { A 1 frac  ( f t ) < 0.5 − A 1 frac  ( f t ) > 0.5 {\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}} A 1 {\displaystyle A_{1}\,}
DC-shifted square wave y = A 0 + { A 1 frac  ( f t ) < 0.5 − A 1 frac  ( f t ) > 0.5 {\displaystyle y=A_{0}+{\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}} A 0 2 + A 1 2 {\displaystyle {\sqrt {A_{0}^{2}+A_{1}^{2}}}\,}
Modified sine wave y = { 0 frac  ( f t ) < 0.25 A 1 0.25 < frac  ( f t ) < 0.5 0 0.5 < frac  ( f t ) < 0.75 − A 1 frac  ( f t ) > 0.75 {\displaystyle y={\begin{cases}0&\operatorname {frac} (ft)<0.25\\A_{1}&0.25<\operatorname {frac} (ft)<0.5\\0&0.5<\operatorname {frac} (ft)<0.75\\-A_{1}&\operatorname {frac} (ft)>0.75\end{cases}}} A 1 2 {\displaystyle {\frac {A_{1}}{\sqrt {2}}}}
Triangle wave y = | 2 A 1 frac  ( f t ) − A 1 | {\displaystyle y=\left|2A_{1}\operatorname {frac} (ft)-A_{1}\right|} A 1 3 {\displaystyle A_{1} \over {\sqrt {3}}}
Sawtooth wave y = 2 A 1 frac  ( f t ) − A 1 {\displaystyle y=2A_{1}\operatorname {frac} (ft)-A_{1}\,} A 1 3 {\displaystyle A_{1} \over {\sqrt {3}}}
Pulse train y = { A 1 frac  ( f t ) < D 0 frac  ( f t ) > D {\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<D\\0&\operatorname {frac} (ft)>D\end{cases}}} A 1 D {\displaystyle A_{1}{\sqrt {D}}}
Phase-to-phase voltage y = A 1 sin  ( t ) − A 1 sin  ( t − 2 π 3 ) {\displaystyle y=A_{1}\sin(t)-A_{1}\sin \left(t-{\frac {2\pi }{3}}\right)\,} A 1 3 2 {\displaystyle A_{1}{\sqrt {\frac {3}{2}}}}
Notes:
t is time
f is frequency
Ai is amplitude (peak value)
D is the duty cycle or the proportion of the time period (1/f) spent high.
frac(r) is the fractional part of r
In waveform combinations[edit]
Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).[3]
R M S Total = R M S 1 2 + R M S 2 2 + ⋯ + R M S n 2 {\displaystyle RMS_{\text{Total}}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}}
A special case of this, particularly helpful in electrical engineering, is given #In electrical engineering below. Another special case, useful in statistics, is given in #Relationship to other statistics.
(If the waveforms are in phase, then their RMS amplitudes sum directly.)
Uses[edit]
In electrical engineering[edit]
Voltage[edit]
Further information: Root mean square AC voltage
In electrical engineering, a special case of #RMS of waveform combinations (see also #Relationship to other statistics) is:
R M S Total = R M S DC 2 + R M S AC 2 {\displaystyle RMS_{\text{Total}}={\sqrt {{RMS_{\text{DC}}}^{2}+{RMS_{\text{AC}}}^{2}}}}
where R M S DC {\displaystyle RMS_{\text{DC}}} refers to the direct current component of the signal and R M S AC {\displaystyle RMS_{\text{AC}}} is the alternating current component of the signal.
Average electrical power[edit]
Further information: AC power
Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:
P = I 2 R . {\displaystyle P=I^{2}R.}
However, if the current is a time-varying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation:
P avg {\displaystyle P_{\text{avg}}\,\!} = ⟨ I ( t ) 2 R ⟩ {\displaystyle {}=\left\langle I(t)^{2}R\right\rangle \,\!} (where ⟨ … ⟩ {\displaystyle \langle \ldots \rangle } denotes the mean of a function)
= R ⟨ I ( t ) 2 ⟩ {\displaystyle {}=R\left\langle I(t)^{2}\right\rangle \,\!} (as R does not vary over time, it can be factored out)
= I RMS 2 R {\displaystyle {}=I_{\text{RMS}}^{2}R\,\!} (by definition of RMS)
So, the RMS value, IRMS, of the function I(t) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I(t).
Average power can also be found using the same method that in the case of a time-varying voltage, V(t), with RMS value VRMS,
P Avg = V RMS 2 R . {\displaystyle P_{\text{Avg}}={V_{\text{RMS}}^{2} \over R}.}
This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.
By taking the square root of both these equations and multiplying them together, the power is found to be:
P Avg = V RMS I RMS . {\displaystyle P_{\text{Avg}}=V_{\text{RMS}}I_{\text{RMS}}.}
Both derivations depend on voltage and current being proportional (i.e., the load, R, is purely resistive). Reactive loads (i.e., loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.
In the common case of alternating current when I(t) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If Ip is defined to be the peak current, then:
I RMS = 1 T 2 − T 1 ∫ T 1 T 2 [ I p sin  ( ω t ) ] 2 d t . {\displaystyle I_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}\left[I_{\text{p}}\sin(\omega t)\right]^{2}dt}}.\,\!}
where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave).
Since Ip is a positive constant:
I RMS = I p 1 T 2 − T 1 ∫ T 1 T 2 sin 2  ( ω t ) d t . {\displaystyle I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{\sin ^{2}(\omega t)}\,dt}}}.}
Using a trigonometric identity to eliminate squaring of trig function:
I RMS = I p 1 T 2 − T 1 ∫ T 1 T 2 1 − cos  ( 2 ω t ) 2 d t I RMS = I p 1 T 2 − T 1 [ t 2 − sin  ( 2 ω t ) 4 ω ] T 1 T 2 {\displaystyle {\begin{aligned}I_{\text{RMS}}&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{1-\cos(2\omega t) \over 2}\,dt}}}\\I_{\text{RMS}}&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}-{\sin(2\omega t) \over 4\omega }\right]_{T_{1}}^{T_{2}}}}\end{aligned}}}
but since the interval is a whole number of complete cycles (per definition of RMS), the sin terms will cancel out, leaving:
I RMS = I p 1 T 2 − T 1 [ t 2 ] T 1 T 2 = I p 1 T 2 − T 1 T 2 − T 1 2 = I p 2 . {\displaystyle I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}\right]_{T_{1}}^{T_{2}}}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{{T_{2}-T_{1}} \over 2}}}={I_{\text{p}} \over {\sqrt {2}}}.}
A similar analysis leads to the analogous equation for sinusoidal voltage:
V RMS = V p 2 . {\displaystyle V_{\text{RMS}}={V_{\text{p}} \over {\sqrt {2}}}.}
Where IP represents the peak current and VP represents the peak voltage.
Because of their usefulness in carrying out power calculations, listed voltages for power outlets (e.g., 120 V in the USA, or 230 V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies VP = VRMS × √2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × √2, or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and the peak-to-peak mains voltage, about 650 volts.
RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each day represents an RMS current of 5 amps in the long term.
The term "RMS power" is sometimes erroneously used in the audio industry as a synonym for "mean power" or "average power" (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power.
Speed[edit]
Main article: Root-mean-square speed
In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation:
v RMS = 3 R T M {\displaystyle {v_{\text{RMS}}}={\sqrt {3RT \over {M}}}}
where R represents the ideal gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms. The generally accepted terminology for speed as compared to velocity is that the former is the scalar magnitude of the latter. Therefore, although the average speed is between zero and the RMS speed, the average velocity for a stationary gas is zero.
Error[edit]
Main article: Root-mean-square error
When two data sets—one set from theoretical prediction and the other from actual measurement of some physical variable, for instance—are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0. The mean of the pairwise differences does not measure the variability of the difference, and the variability as indicated by the standard deviation is around the mean instead of 0. Therefore, the RMS of the differences is a meaningful measure of the error.
In frequency domain[edit]
The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal x [ n ] = x ( t = n T ) {\displaystyle x[n]=x(t=nT)} , where T {\displaystyle T} is the sampling period,
∑ n = 1 N x 2 [ n ] = 1 N ∑ m = 1 N | X [ m ] | 2 {\displaystyle \sum _{n=1}^{N}{x^{2}[n]}={\frac {1}{N}}\sum _{m=1}^{N}{\bigl |}X[m]{\bigr |}^{2}} ,
where X [ m ] = FFT  { x [ n ] } {\displaystyle X[m]=\operatorname {FFT} \{x[n]\}} and N is number of samples and FFT coefficients.
In this case, the RMS computed in the time domain is the same as in the frequency domain:
R M S { x [ n ] } = 1 N ∑ n x 2 [ n ] = 1 N 2 ∑ m | X [ m ] | 2 = ∑ m | X [ m ] N | 2 . {\displaystyle \mathrm {RMS} \{x[n]\}={\sqrt {{\frac {1}{N}}\sum _{n}{x^{2}[n]}}}={\sqrt {{\frac {1}{N^{2}}}\sum _{m}{{\bigl |}X[m]{\bigr |}}^{2}}}={\sqrt {\sum _{m}{\left|{\frac {X[m]}{N}}\right|^{2}}}}.}
Relationship to other statistics[edit]
If x ¯ {\displaystyle {\bar {x}}} is the arithmetic mean and σ x {\displaystyle \sigma _{x}} is the standard deviation of a population or a waveform then:[4]
x r m s 2 = x ¯ 2 + σ x 2 = x 2 ¯ . {\displaystyle x_{\mathrm {rms} }^{2}={\overline {x}}^{2}+\sigma _{x}^{2}={\overline {x^{2}}}.}
From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.
Physical scientists often use the term "root mean square" as a synonym for standard deviation when it can be assumed the input signal has zero mean, i.e., referring to the square root of the mean squared deviation of a signal from a given baseline or fit.[5][6] This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the root mean square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. RMS(signal) = Stdev(signal) if the mean signal is 0). A special case of this, particularly helpful in electrical engineering, is given above.
See also[edit]
Central moment
Geometric mean
L2 norm
Least squares
Mean squared displacement
Table of mathematical symbols
True RMS converter
Average rectified value (ARV)
References[edit]
^ Jump up to: a b A Dictionary of Physics (6 ed.). Oxford University Press. 2009. ISBN 9780199233991.
Jump up ^ Cartwright, Kenneth V (Fall 2007). "Determining the Effective or RMS Voltage of Various Waveforms without Calculus" (PDF). Technology Interface. 8 (1): 20 pages.
Jump up ^ Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". MasteringElectronicsDesign.com. Retrieved 21 January 2015.
Jump up ^ Chris C. Bissell and David A. Chapman (1992). Digital signal transmission (2nd ed.). Cambridge University Press. p. 64. ISBN 978-0-521-42557-5.
Jump up ^ "Root-Mean-Square".
Jump up ^ "ROOT, TH1:GetRMS".
External links[edit]
A case for why RMS is a misnomer when applied to audio power
A Java applet on learning RMS
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Gunsmoke: To the Last Man
From Wikipedia, the free encyclopedia
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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (January 2012) (Learn how and when to remove this template message)
Gunsmoke: To the Last Man
Genre
Western
Written by
Earl W. Wallace
Directed by
Jerry Jameson
Starring
James Arness
Pat Hingle
Amy Stoch
Theme music composer
Artie Kane
Country of origin
United States
Original language(s)
English
Production
Producer(s)
Larry Rapaport (coordinating producer)
Location(s)
Tucson, Arizona
Editor(s)
Scott Powell
Running time
91 min.
Production company(s)
CBS Entertainment Production
Distributor
CBS
Release
Original network
CBS
Original release
January 10, 1992
Gunsmoke: To the Last Man is a 1992 American TV-movie starring James Arness as retired Marshal Matt Dillon and featuring Pat Hingle. Hapless rustlers make the fatal mistake of stealing Matt Dillon's cattle and Dillon blunders into the gory Pleasant Valley War during the process of hunting them down and killing two of them. The film, set in the 1880s, was directed by Jerry Jameson and based upon the long-running American television series Gunsmoke (1955 to 1975). The supporting cast features Matt Mulhern as Will McCall, Joseph Bottoms as villainous Tommy Graham, and Morgan Woodward as Sheriff Abel Rose.
Cast[edit]
James Arness ... Matt Dillon
Pat Hingle ... Col. Tucker
Amy Stock-Poynton ... Beth Dillon
Matt Mulhern ... Will McCall
Jason Lively ... Rusty Dover
Joseph Bottoms ... Tommy Graham
Morgan Woodward ... Sheriff Abel Rose
Mills Watson ... Horse Trader
James Booth ... Zack the Preacher
Amanda Wyss ... Lizzie Tewksbury
Jim Beaver ... Deputy Willie Rudd
Herman Poppe ... John Tewksbury
Ken Swofford ... Charlie Tewksbury
Don Collier ... Sheriff Tom, Tombstone
Ed Adams ... Billy Wilson
Kathleen Erickson ... Mrs. Claire Oliver (billed as Kathleen Todd Erickson)
Loy Burns ... Kirby Tewksbury (billed as Loy W. Burns)
Andy Sherman ... Virgil Tucker
Clark A. Ray ... Rowe Blevin
Michael F. Woodson ... Bartender at Payson House
Erol Landis ... Cole Tucker
William J. Fisher ... Undertaker
Stephen C. Foster ... Luther
Ric San Nicholas ... Tink
Jimmy Don Cox ... Onlooker
Richard Glover ... David Henry
External links[edit]
Gunsmoke: To the Last Man in the Internet Movie Database
[hide]
v
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Films directed by Jerry Jameson
The Bat People (1974)
The Elevator (1974)
The Deadly Tower (1975)
Airport '77 (1977)
Superdome (1978)
L'arma (1978)
A Fire in the Sky (1978)
Raise the Titanic (1980)
High Noon, Part II: The Return of Will Kane (1980)
Hotline (1982)
Starflight: The Plane That Couldn't Land (1983)
Fire and Rain (1989)
Gunsmoke: To the Last Man (1992)
Gunsmoke IV: The Long Ride (1993)
Bonanza: The Return (1993)
Land of the Free (1998)
Safe Harbor (2009)
Captive (2015)
This 1990s Western film–related article is a stub. You can help Wikipedia by expanding it.
v
t
e
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gunsmoke:_To_the_Last_Man&oldid=748740599"
Categories:
1992 television films
English-language films
1990s Western (genre) films
American Western (genre) films
American films
Films directed by Jerry Jameson
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text:
List of bones of the human skeleton
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Part of a series of lists about
Human anatomy
General[show]
features
regions
Variations
Movements
Systems
Structures[show]
Arteries
Bones
Eponymous
Foramina
Glands
endocrine
Lymphatic vessels
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The skeleton of an adult human consists of 206 bones. It is composed of 301 bones at birth, which decreases to 94 bones in the axial skeleton (28 in the skull and 52 in the torso) and 126 bones in the appendicular skeleton (32 × 2 in the upper extremities including both arms and 31 × 2 in the lower extremities including both legs). Many small and variable supernumerary bones, such as some sesamoid bones, are not included in this count.
Contents
[hide]
1 Introduction
2 Bones
2.1 Spine (vertebral column)
2.2 Chest (thorax)
2.3 Skull (cranium and mandible)
2.4 Arm
2.5 Pelvis (pelvic girdle)
2.6 Leg
2.7 Sesamoid bones
3 References
Introduction[edit]
The bones in a human skeleton will be more as the age will be more, as multiple bones fuse, a process which typically reaches completion in the third decade of life. In addition, the bones of the skull and face are counted as separate bones, despite being fused naturally. Some reliable sesamoid bones such as the pisiform are counted, while others, such as the hallux sesamoids, are not.
Individuals may have more or fewer bones than this owing to anatomical variations. The most common variations include sutural (wormian) bones, which are located along the sutural lines on the back of the skull, and sesamoid bones which develop within some tendons, mainly in the hands and feet. Some individuals may also have additional (i.e. supernumerary) cervical ribs or lumbar vertebrae.
Bones[edit]
Spine (vertebral column)[edit]
A fully grown adult features 30 bones in the spine, whereas a child can have 33.
The cervical vertebrae (7)
The thoracic vertebrae (12)
The lumbar vertebrae (5)
The sacral vertebrae (5 at birth, later fused into one)
The coccygeal vertebrae (5 at birth, some or all of the bones fuse together but there seems to be a disagreement between researchers as to what the most common number should be. Some say the most common is 1, others say 2 or 3, with 4 being the least likely. It is counted as 1 in this article.[citation needed])
Chest (thorax)[edit]
There are usually 25 bones in the chest but sometimes there can be additional cervical ribs in humans. Cervical ribs occur naturally in other animals such as reptiles.
The sternum (1)
The ribs (24, in 12 pairs)
It is important to note that three pairs (the 8th, 9th and 10th), also known as false ribs, are attached to each other. They are also attached to the 7th rib by cartilage and synovial joints. Also two pairs of floating ribs (the 11th and 12th), have no anterior attachment.
Cervical ribs are extra ribs that occur in some humans.
Skull (cranium and mandible)[edit]
There are 22 bones in the skull. Including the hyoid and the bones of the middle ear, the head contains 29 bones.
The cranial bones (8)
The occipital bone
The parietal bones (2)
The frontal bone
The temporal bones (2)
The sphenoid bone (sometimes counted as facial)
The ethmoid bone (sometimes counted as facial)
The facial bones (14)
The nasal bones (2)
The maxillae (upper jaw) (2)
The lacrimal bone (2)
The zygomatic bone or cheekbone (2);
The palatine bone (2)
The inferior nasal concha (2)
The vomer
The mandible (lower jaw)
The hyoid bone (not connected to any other bone)
In the middle ears (6)
malleus (2)
incus (2)
stapes (2)
Arm[edit]
There are a total of 64 bones in the arm.
The upper arm bones (6 bones, 3 on each side)
The humerus
The shoulder (pectoral girdle)
The scapula
The clavicles
The lower arm bones (4 bones, 2 on each side)
The ulna
The radius
The hand (54 bones, 27 in each hand)
The carpals
scaphoid bone (2)
lunate bone (2)
triquetral bone (2)
pisiform bone (2)
trapezium (2)
trapezoid bone (2)
capitate bone (2)
hamate bone (2)
The metacarpals (5 × 2 = 10)
The phalanges of the hand
proximal phalanges (5 × 2 = 10)
intermediate phalanges (4 × 2 = 8)
distal phalanges (5 × 2 = 10)
Pelvis (pelvic girdle)[edit]
The hip bone has three regions: ilium, ischium, and pubis (2)
The sacrum and the coccyx attach to the two hip bones to form the pelvis, but are more important to the spinal column. For this reason it is omitted from the pelvic girdle.
Leg[edit]
The femur (2)
The patella or kneecap (2)
The tibia (2)
The fibula (2)
The foot (52 bones in total, 26 per foot)
The tarsus
calcaneus or heel bone (2)
talus (2)
navicular bone (2)
medial cuneiform bone (2)
intermediate cuneiform bone (2)
lateral cuneiform bone (2)
cuboid bone (2)
The metatarsals (10)
The phalanges of the foot
proximal phalanges (5 × 2 = 10)
intermediate phalanges (4 x 2 = 8)
distal phalanges (5 x 2 = 10)
Sesamoid bones[edit]
Patella
Pisiform bone
Fabella
Sesamoids in the first and second metacarpal bones
Sesamoids in the first metatarsal bone
Lenticular process of the incus
For inconsistent bones, see Accessory bone.
References[edit]
The 206 Bones of the Human Body
[hide]
v
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Bones in the human skeleton
Axial skeleton
Skull
Neurocranium
occipital
parietal
frontal
temporal
sphenoid
ethmoid
Face
nasal
maxilla
lacrimal
zygomatic
palatine
inferior nasal conchae
vomer
mandible
hyoid
Ear
ossicles
malleus
incus
stapes
Thorax
sternum
rib cage
rib
Vertebral column
vertebrae
cervical
thoracic
lumbar
sacrum
coccyx
Appendicular
Shoulder
clavicle
scapula
Arm
humerus
ulna
radius
Hand
carpals (scaphoid, lunate bone, triquetral, pisiform, trapezium, trapezoid, capitate, hamate)
metacarpals
phalanges (prox, int, dist)
Pelvis
ilium
ischium
pubis
Leg
femur
patella
fibula
tibia
Foot
tarsals
heelbone
anklebone
navicular
cuneiform
cuboid
metatarsals
phalanges
proximal
intermediate
distal
accessory
accessory navicular
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_bones_of_the_human_skeleton&oldid=818782589"
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value: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-List_of_bones_of_the_human_skeleton rootpage-List_of_bones_of_the_human_skeleton skin-vector action-view