To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Median

From Wikipedia, the free encyclopedia

Finding the median in sets of data with an odd and even number of values

The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution). For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.

The median is a commonly used measure of the properties of a data set in statistics and probability theory. The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a "typical" value. For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a "typical" income is.

Because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.

YouTube Encyclopedic

• 1/5
Views:
1 050 906
17 970
238 134
2 059 690
52 928
• ✪ Math Antics - Mean, Median and Mode
• ✪ How do we Find the Median?
• ✪ Statistics - Find the median
• ✪ Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy
• ✪ Median of a Triangle Formula, Example Problems, Properties, Definition, Geometry, Midpoint & Centroi

Transcription

Hi, this is Rob. Welcome to Math Antics! In this lesson, we’re gonna learn about three important math concepts called the Mean, the Median and the Mode. Math often deals with data sets, and data sets are often just collections (or groups) of numbers. These numbers may be the results of scientific measurements or surveys or other data collection methods. For example, you might record the ages of each member of you family into a data set. Or you might measure the weight of each of your pets and list them in a data set. Those data sets are fairly small and easy to understand. But you could have much bigger data sets. A really big data set might contain the cost of every item in a store, or the top speed of every land mammal, or the brightness of all the stars in our galaxy! Those data sets would contain a lot of different numbers! And if you had to look at a big data set all at one time… it would be pretty hard to make sense of it or say much about it besides “well that’s a lot of numbers”! But that’s where Mean, Median and Mode can really help us out. They’re three different properties of data sets that can give us useful, easy to understand information about a data set so that we can see the big picture and understand what the data means about the world we live in. That sounds pretty useful, huh? So let’s learn what each property really is and find out how to calculate them for any particular data set. Let’s start with the Mean. You may not have ever heard of something called “the mean” before, but I’ll bet you’ve heard of “the average”. If so, then I’ve got good news! Mean means average! “Mean” and “average” are just two different terms for the exact same property of a data set. The mean (or average) is an extremely useful property. To understand what it is, let’s look at a simple data set that contains 5 numbers. As a visual aid, let’s also represent those numbers with stacks of blocks who’s heights correspond to their values: one, eight, three, two, six Right now, since each of the 5 numbers is different, the stacks of blocks are all different heights. But what if we rearrange the blocks with the goal of making the stacks the same height? In other words, if each stack could have the exact same amount, what would that amount be? Well, with a bit of trial and error, you’ll see that we have enough blocks for each stack to have a total of 4. That means that the Mean (or average) for our original data set would be 4. Some of the numbers are greater than 4 and some are less, but if the amounts could all be made the same, they would all become 4. So that’s the concept of Mean; it’s the value you’d get if you could smooth out or flatten all of the different data values into one consistent value. But, is there a way we can use math to calculate the mean of a data set? After all, it would be very inconvenient if we always had to use stacks of blocks to do it! There’s got to be an easier way!! [crash] To learn the mathematical procedure for calculating the Mean, lets start with blocks again. But this time, instead of using trial and error, let’s use a more systematic way to make the stacks all the same height. This way involves a clever combination of addition and division. We know that we want to end up with 5 stacks that all have the same number of blocks, right? So first, let’s add up all of the numbers, which is like putting all of the blocks we have into one big stack. Adding up all of the numbers (or counting all the blocks) shows us that we have a total of 20. Next, we divide that number (or stack) into 5 equal parts. Since the stack has a total of 20 blocks, dividing it into 5 equal stacks means that we’ll have 4 in each, since 20 divided by 5 equals 4. So that’s the math procedure you use to find the mean of a data set. It’s just two simple steps. First, you add up all the numbers in the set. And then you divide the total you get by how many numbers you added up. The answer you get is the Mean of the data set. Let’s use that procedure to find the mean age of the members of this fine looking family here. If we add them all up using a calculator (or by hand if you’d like) the total of the ages is 222 years. But then, we need to divide that total by the number of ages we added which is 6. 222 divided by 6 is 37. So that’s the mean age of all the members in this family. Alright, that’s the Mean. Now what about the Median? The Median is the middle of a data set. It’s the number that splits the data set into two equally sized group or halves. One half contains members that are greater than or equal to the Median, and the other half contains members that are less than or equal to the Median. Sometimes finding the Median of a data set is easy, and sometimes it’s hard. That’s because finding the middle value of a data set requires that its members be in order from the least to the greatest (or vice versa). And if the data set has a lot of numbers, it might take a lot of work to put them in the right order if they aren’t already that way. So to make things easier, let’s start with a really basic data set that isn’t in order. It’s pretty easy to see that we can put this data set in order from the least to the greatest value just by switching the 2 and the 1. There, now we have the data set {1, 2, 3} and finding the Median (or middle) of this data set is easy! It’s just 2 because the 2 is located exactly in the middle. That almost seems too easy, doesn’t it? But don’t worry… it gets harder! But before we try a harder problem, I want to point out that sometimes the Mean and the Median of a data set are the same number, and sometimes they’re not. In the case of our simple data set {1, 2, 3}, the Median is 2 and the Mean is also 2, as you can see if we rearrange the amounts or follow the procedure we learned to calculate the Mean. But what about the first data set that we found the mean of? We determined than the Mean of this data set is 4. But what about the Median? Well, the Median is the middle, and since this data set is already in order from least to greatest, it’s easy to see that the 3 is located in the middle since it splits the other members into two equal groups. So for this data set, the Mean is 4 but the Median is 3. So to find the Median of a set of numbers, first you need to make sure that all the numbers are in order and then you can identify the member that’s exactly in the middle by making sure there’s an equal number of members on either side of it. Okay, ...so far so good. But some of you may be wondering, “What if a data set doesn’t have an obvious middle member?” All of the sets we’ve found the Median of so far have an odd number of members. But, what if it has an even number of members? …like the data set {1, 2, 3, 4} There isn’t a member in the middle that splits the set into two equally sized groups. If that’s the case, we can actually use what we learned about the Mean to help us out. If the data set has an even number of members, then to find the Median, we need to take the middle TWO numbers and calculate the Mean (or average) of those two. By doing that, we’re basically figuring out what number WOULD be exactly half way between the two middle numbers, and that number will be our Median. For example, in the set {1, 2, 3, 4} we need to take the middle TWO numbers (2 and 3) and find the Mean of those numbers. We can do that by adding 2 and 3 and then dividing by 2. 2 plus 3 equals 5 and 5 divided by 2 is 2.5 So the Median of the data set is 2.5 Even though the number 2.5 isn’t actually a member of the data set, it’s the Median because it represents the middle of the data set and it splits the members into two equally sized groups. Okay, so now you know the difference between Mean and Median. But what about the Mode of a data set? What in the world does that mean? Well, “Mode” is just a technical word for the value in a data set that occurs most often. In the data sets we’ve seen so far, there hasn’t even been a Mode because none of the data values were ever repeated. But what if you had this data set? This set has 6 members, but some of the value are repeated. If we rearrange them, you can see that there’s one ‘1’, two ‘2’s and three ‘3’s The Mode of this data set is the value that occurs most often (or most frequently) so that would be 3 since there’s three ‘3’s. Now don’t get confused just because the number 3 was repeater 3 times. The mode is the number that’s repeated most often, NOT how many times it was repeated. As I mentioned, if each member in a data set occurs only once, it had no mode, but it’s also possible for a data set to have more than one mode. Here’s an example of a data set like that: In this set, the number 7 is repeated twice but so is the number 15. That means they tie for the title of Mode. This set has two modes: 7 and 15. Okay, so now that you know what the Mean, Median and Mode of a data set are. Let’s put all that new information to use on one final real-world example. Suppose there’s this guy who makes and sells custom electric guitars. Here’s a table showing how many guitars he sold during each month of the year. Let’s find the Mean, Median and Mode of this data set. First, to find the Mean we need to add up the number of guitars sold in each month. You can do the addition by hand or you can use a calculator if you want to. Either way, be careful since that’s a lot of numbers to add up and we don’t want to make a mistake. The answer I get is 108. So that’s the total he sold for the whole year, but to get the Mean sold each month, we need to divide that total by the number of months which is 12. 108 divided by 12 is 9, so the Mean (or average) is 9. Next, to find the Median of the data set, we’re going to have to rearrange the 12 data points in order from smallest to largest so we can figure out what the middle value is. There, that’s better. Since there’s an even number of members in this set, we can’t just choose the middle number, so we’re going to have to pick the middle two numbers and then find the Mean of them. 9 and 10 are in the middle since there’s an equal number of data values on either side of them. So we need to take the Mean of 9 and 10. That’s easy, 9 plus 10 equals 19 and then 19 divided by 2 is 9.5 So, the Median number of guitars sold is 9.5. That means that in half of the months, he sold more than 9.5, and in half of the months, he sold less than 9.5. Last of all, let’s identify the Mode of this data set (if there is one). We let’s see… there’s two ‘8’s in the data set… Oh… but there’s three ’10’s. That looks like the most frequent number, so 10 is the Mode of this data set. It’s the result that occurred most often. Alright, so that’s the basics of Mean, Median, and Mode. They are three really useful properties of data sets and now you know how to find them. But sometimes, the hardest part about Mean, Median and Mode is just remembering which is which. So remember that “Mean means average”, Median is in the middle, and Mode starts with ‘M’ ‘O’ which can remind you that it’s the number that occurs “Most Often”. Remember, to get good at math, you need to do more than just watch videos about it. You need to Practice! So be sure to try finding the Mean, Median and Mode on your own. As always, thank for watching Math Antics, and I’ll see ya next time. Learn more at www.mathantics.com

Finite set of numbers

The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest.

If there is an odd number of numbers, the middle one is picked. For example, consider the list of numbers

1, 3, 3, 6, 7, 8, 9

This list contains seven numbers. The median is the fourth of them, which is 6.

If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values.[1][2] For example, in the data set

1, 2, 3, 4, 5, 6, 8, 9

the median is the mean of the middle two numbers: this is ${\displaystyle (4+5)/2}$, which is ${\displaystyle 4.5}$. (In more technical terms, this interprets the median as the fully trimmed mid-range).

The formula used to find the index of the middle number of a data set of n numerically ordered numbers is ${\displaystyle (n+1)/2}$. This either gives the middle number (for an odd number of values) or the halfway point between the two middle values. For example, with 14 values, the formula will give an index of 7.5, and the median will be taken by averaging the seventh (the floor of this index) and eighth (the ceiling of this index) values. So the median can be represented by the following formula:

${\displaystyle \mathrm {median} (a)={\frac {a_{\lfloor (\#x+1)\div 2\rfloor }+a_{\lceil (\#x+1)\div 2\rceil }}{2}}}$
Comparison of common averages of values { 1, 2, 2, 3, 4, 7, 9 }
Type Description Example Result
Arithmetic mean Sum of values of a data set divided by number of values: ${\displaystyle \scriptstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$ (1+2+2+3+4+7+9) / 7 4
Median Middle value separating the greater and lesser halves of a data set 1, 2, 2, 3, 4, 7, 9 3
Mode Most frequent value in a data set 1, 2, 2, 3, 4, 7, 9 2

One can find the median using the Stem-and-Leaf Plot.

There is no widely accepted standard notation for the median, but some authors represent the median of a variable x either as or as μ1/2[1] sometimes also M.[3][4] In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced.

The median is used primarily for skewed distributions, which it summarizes differently from the arithmetic mean. Consider the multiset { 1, 2, 2, 2, 3, 14 }. The median is 2 in this case, (as is the mode), and it might be seen as a better indication of central tendency (less susceptible to the exceptionally large value in data) than the arithmetic mean of 4.

The median is a popular summary statistic used in descriptive statistics, since it is simple to understand and easy to calculate, while also giving a measure that is more robust in the presence of outlier values than is the mean. The widely cited empirical relationship between the relative locations of the mean and the median for skewed distributions is, however, not generally true.[5] There are, however, various relationships for the absolute difference between them; see below.

With an even number of observations (as shown above) no value need be exactly at the value of the median. Nonetheless, the value of the median is uniquely determined with the usual definition. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid.

In a population, at most half have values strictly less than the median and at most half have values strictly greater than it. If each group contains less than half the population, then some of the population is exactly equal to the median. For example, if a < b < c, then the median of the list {abc} is b, and, if a < b < c < d, then the median of the list {abcd} is the mean of b and c; i.e., it is (b + c)/2. Indeed, as it is based on the middle data in a group, it is not necessary to even know the value of extreme results in order to calculate a median. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated.[6]

The median can be used as a measure of location when a distribution is skewed, when end-values are not known, or when one requires reduced importance to be attached to outliers, e.g., because they may be measurement errors.

A median is only defined on ordered one-dimensional data, and is independent of any distance metric. A geometric median, on the other hand, is defined in any number of dimensions.

The median is one of a number of ways of summarising the typical values associated with members of a statistical population; thus, it is a possible location parameter. The median is the 2nd quartile, 5th decile, and 50th percentile. Since the median is the same as the second quartile, its calculation is illustrated in the article on quartiles. A median can be worked out for ranked but not numerical classes (e.g. working out a median grade when students are graded from A to F), although the result might be halfway between grades if there is an even number of cases.

When the median is used as a location parameter in descriptive statistics, there are several choices for a measure of variability: the range, the interquartile range, the mean absolute deviation, and the median absolute deviation.

For practical purposes, different measures of location and dispersion are often compared on the basis of how well the corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient than the sample median when data are uncontaminated by data from heavy-tailed distributions or from mixtures of distributions, but less efficient otherwise, and that the efficiency of the sample median is higher than that for a wide range of distributions. More specifically, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean—see asymptotic efficiency and references therein.

Probability distributions

Geometric visualisation of the mode, median and mean of an arbitrary probability density function.[7]

For any probability distribution on the real line R with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (which has a probability density function), or a discrete probability distribution, a median is by definition any real number m that satisfies the inequalities

${\displaystyle \operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,\!}$

or, equivalently, the inequalities

${\displaystyle \int _{(-\infty ,m]}dF(x)\geq {\frac {1}{2}}{\text{ and }}\int _{[m,\infty )}dF(x)\geq {\frac {1}{2}}\,\!}$

in which a Lebesgue–Stieltjes integral is used. For an absolutely continuous probability distribution with probability density function ƒ, the median satisfies

${\displaystyle \operatorname {P} (X\leq m)=\operatorname {P} (X\geq m)=\int _{-\infty }^{m}f(x)\,dx={\frac {1}{2}}.\,\!}$

Any probability distribution on R has at least one median, but in specific cases there may be more than one median. Specifically, if a probability density is zero on an interval [ab], and the cumulative distribution function at a is 1/2, any value between a and b will also be a median.

Medians of particular distributions

The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking a well-defined mean, such as the Cauchy distribution:

Populations

Optimality property

The mean absolute error of a real variable c with respect to the random variable X is

${\displaystyle E(\left|X-c\right|)\,}$

Provided that the probability distribution of X is such that the above expectation exists, then m is a median of X if and only if m is a minimizer of the mean absolute error with respect to X.[9] In particular, m is a sample median if and only if m minimizes the arithmetic mean of the absolute deviations.

More generally, a median is defined as a minimum of

${\displaystyle E(|X-c|-|X|),}$

as discussed below in the section on multivariate medians (specifically, the spatial median).

This optimization-based definition of the median is useful in statistical data-analysis, for example, in k-medians clustering.

Unimodal distributions

Comparison of mean, median and mode of two log-normal distributions with different skewness.

It can be shown for a unimodal distribution that the median ${\displaystyle {\tilde {X}}}$ and the mean ${\displaystyle {\bar {X}}}$ lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other.[10] In symbols,

${\displaystyle {\frac {\left|{\tilde {X}}-{\bar {X}}\right|}{\sigma }}\leq \left({\frac {3}{5}}\right)^{\frac {1}{2}}}$

where |·| is the absolute value.

A similar relation holds between the median and the mode: they lie within 31/2 ≈ 1.732 standard deviations of each other:

${\displaystyle {\frac {|{\tilde {X}}-\mathrm {mode} |}{\sigma }}\leq 3^{\frac {1}{2}}.}$

Inequality relating means and medians

If the distribution has finite variance, then the distance between the median and the mean is bounded by one standard deviation.

This bound was proved by Mallows,[11] who used Jensen's inequality twice, as follows. We have

{\displaystyle {\begin{aligned}|\mu -m|=|\operatorname {E} (X-m)|&\leq \operatorname {E} (|X-m|)\\&\leq \operatorname {E} (|X-\mu |)\\&\leq {\sqrt {\operatorname {E} \left((X-\mu )^{2}\right)}}=\sigma .\end{aligned}}}

The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes the absolute deviation function

${\displaystyle a\mapsto \operatorname {E} (|X-a|).\,}$

This proof also follows directly from Cantelli's inequality.[12] The result can be generalized to obtain a multivariate version of the inequality,[13] as follows:

{\displaystyle {\begin{aligned}\|\mu -m\|=\|\operatorname {E} (X-m)\|&\leq \operatorname {E} \|X-m\|\\&\leq \operatorname {E} (\|X-\mu \|)\\&\leq {\sqrt {\operatorname {E} \left(\|X-\mu \|^{2}\right)}}={\sqrt {\operatorname {trace} \left(\operatorname {var} (X)\right)}}\end{aligned}}}

where m is a spatial median, that is, a minimizer of the function ${\displaystyle a\mapsto \operatorname {E} (\|X-a\|).\,}$ The spatial median is unique when the data-set's dimension is two or more.[14][15] An alternative proof uses the one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters.

Jensen's inequality for medians

Jensen's inequality states that for any random variable x with a finite expectation E(x) and for any convex function f

${\displaystyle f[E(x)]\leq E[f(x)]}$

It has been shown[16] that if x is a real variable with a unique median m and f is a C function then

${\displaystyle f(m)\leq \operatorname {Median} [f(x)]}$

A C function is a real valued function, defined on the set of real numbers R, with the property that for any real t

${\displaystyle f^{-1}\left(\,(-\infty ,t]\,\right)=\{x\in R\mid f(x)\leq t\}}$

is a closed interval, a singleton or an empty set.

Medians for samples

The sample median

Efficient computation of the sample median

Even though comparison-sorting n items requires Ω(n log n) operations, selection algorithms can compute the k'th-smallest of n items with only Θ(n) operations. This includes the median, which is the n/2'th order statistic (or for an even number of samples, the arithmetic mean of the two middle order statistics).

Selection algorithms still have the downside of requiring Ω(n) memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates the median as the median of a three-element subsample; this is commonly used as a subroutine in the quicksort sorting algorithm, which uses an estimate of its input's median. A more robust estimator is Tukey's ninther, which is the median of three rule applied with limited recursion:[17] if A is the sample laid out as an array, and

med3(A) = median(A[1], A[n/2], A[n]),

then

ninther(A) = med3(med3(A[1 ... 1/3n]), med3(A[1/3n ... 2/3n]), med3(A[2/3n ... n]))

The remedian is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.[18]

Easy explanation of the sample median

In individual series (if number of observation is very low) first one must arrange all the observations in order. Then count(n) is the total number of observation in given data.

If n is odd then Median (M) = value of ((n + 1)/2)th item term.

If n is even then Median (M) = value of [(n/2)th item term + (n/2 + 1)th item term]/2

For an odd number of values

As an example, we will calculate the sample median for the following set of observations: 1, 5, 2, 8, 7.

Start by sorting the values: 1, 2, 5, 7, 8.

In this case, the median is 5 since it is the middle observation in the ordered list.

The median is the ((n + 1)/2)th item, where n is the number of values. For example, for the list {1, 2, 5, 7, 8}, we have n = 5, so the median is the ((5 + 1)/2)th item.

median = (6/2)th item
median = 3rd item
median = 5
For an even number of values

As an example, we will calculate the sample median for the following set of observations: 1, 6, 2, 8, 7, 2.

Start by sorting the values: 1, 2, 2, 6, 7, 8.

In this case, the arithmetic mean of the two middlemost terms is (2 + 6)/2 = 4. Therefore, the median is 4 since it is the arithmetic mean of the middle observations in the ordered list.

Sampling distribution

The distributions of both the sample mean and the sample median were determined by Laplace.[19] The distribution of the sample median from a population with a density function ${\displaystyle f(x)}$ is asymptotically normal with mean ${\displaystyle m}$ and variance[20]

${\displaystyle {\frac {1}{4nf(m)^{2}}}}$

where ${\displaystyle m}$ is the median of ${\displaystyle f(x)}$ and ${\displaystyle n}$ is the sample size.

These results have also been extended.[21] It is now known for the ${\displaystyle p}$-th quantile that the distribution of the sample ${\displaystyle p}$-th quantile is asymptotically normal around the ${\displaystyle p}$-th quantile with variance equal to

${\displaystyle {\frac {p(1-p)}{nf(x_{p})^{2}}}}$

where ${\displaystyle f(x_{p})}$ is the value of the distribution density at the ${\displaystyle p}$-th quantile.

In the case of a discrete variable, the sampling distribution of the median for small-samples can be investigated as follows. We take the sample size to be an odd number ${\displaystyle N=2n+1}$. If a given value ${\displaystyle v}$ is to be the median of the sample then two conditions must be satisfied. The first is that at most ${\displaystyle n}$ observations can have a value of ${\displaystyle v-1}$ or less. The second is that at most ${\displaystyle n}$ observations can have a value of ${\displaystyle v+1}$ or more. Let ${\displaystyle i}$ be the number of observations which have a value of ${\displaystyle v-1}$ or less and let ${\displaystyle k}$ be the number of observations which have a value of ${\displaystyle v+1}$ or more. Then ${\displaystyle i}$ and ${\displaystyle k}$ both have a minimum value of 0 and a maximum of ${\displaystyle n}$. If an observation has a value below ${\displaystyle v}$, it is not relevant how far below ${\displaystyle v}$ it is and conversely, if an observation has a value above ${\displaystyle v}$, it is not relevant how far above ${\displaystyle v}$ it is. We can therefore represent the observations as following a trinomial distribution with probabilities ${\displaystyle F(v-1)}$, ${\displaystyle f(v)}$ and ${\displaystyle 1-F(v)}$. The probability that the median ${\displaystyle m}$ will have a value ${\displaystyle v}$ is then given by

${\displaystyle \Pr(m=v)=\sum _{i=0}^{n}\sum _{k=0}^{n}{\frac {N!}{i!(N-i-k)!k!}}[F(v-1)]^{i}[f(v)]^{N-i-k}[1-F(v)]^{k}.}$

Summing this over all values of ${\displaystyle v}$ defines a proper distribution and gives a unit sum. In practice, the function ${\displaystyle f(v)}$ will often not be known but it can be estimated from an observed frequency distribution. An example is given in the following table where the actual distribution is not known but a sample of 3,800 observations allows a sufficiently accurate assessment of ${\displaystyle f(v)}$.

v 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
f(v) 0.000 0.008 0.010 0.013 0.083 0.108 0.328 0.220 0.202 0.023 0.005
F(v) 0.000 0.008 0.018 0.031 0.114 0.222 0.550 0.770 0.972 0.995 1.000

Using these data it is possible to investigate the effect of sample size on the standard errors of the mean and median. The observed mean is 3.16, the observed raw median is 3 and the observed interpolated median is 3.174. The following table gives some comparison statistics. The standard error of the median is given both from the above expression for ${\displaystyle pr(m=v)}$ and from the asymptotic approximation given earlier.

Sample size
Statistic
3 9 15 21
Expected value of median 3.198 3.191 3.174 3.161
Standard error of median (above formula) 0.482 0.305 0.257 0.239
Standard error of median (asymptotic approximation) 0.879 0.508 0.393 0.332
Standard error of mean 0.421 0.243 0.188 0.159

The expected value of the median falls slightly as sample size increases while, as would be expected, the standard errors of both the median and the mean are proportionate to the inverse square root of the sample size. The asymptotic approximation errs on the side of caution by overestimating the standard error.

In the case of a continuous variable, the following argument can be used. If a given value ${\displaystyle v}$ is to be the median, then one observation must take the value ${\displaystyle v}$. The elemental probability of this is ${\displaystyle f(v)\,dv}$. Then, of the remaining ${\displaystyle 2n}$ observations, exactly ${\displaystyle n}$ of them must be above ${\displaystyle v}$ and the remaining ${\displaystyle n}$ below. The probability of this is the ${\displaystyle n}$th term of a binomial distribution with parameters ${\displaystyle F(v)}$ and ${\displaystyle 2n}$. Finally we multiply by ${\displaystyle 2n+1}$ since any of the observations in the sample can be the median observation. Hence the elemental probability of the median at the point ${\displaystyle v}$ is given by

${\displaystyle f(v){\frac {(2n)!}{n!n!}}[F(v)]^{n}[1-F(v)]^{n}(2n+1)\,dv.}$

Now we introduce the beta function. For integer arguments ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, this can be expressed as ${\displaystyle \mathrm {B} (\alpha ,\beta )=(\alpha -1)!(\beta -1)!/(\alpha +\beta -1)!}$. Also, we note that ${\displaystyle f(v)=dF(v)/dv}$. Using these relationships and setting both ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ equal to ${\displaystyle (n+1)}$ allows the last expression to be written as

${\displaystyle {\frac {[F(v)]^{n}[1-F(v)]^{n}}{\mathrm {B} (n+1,n+1)}}\,dF(v)}$

Hence the density function of the median is a symmetric beta distribution over the unit interval which supports ${\displaystyle F(v)}$. Its mean, as we would expect, is 0.5 and its variance is ${\displaystyle 1/(4(N+2))}$. The corresponding variance of the sample median is

${\displaystyle {\frac {1}{4(N+2)f(m)^{2}}}.}$

However this finding can only be used if the density function ${\displaystyle f(v)}$ is known or can be assumed. As this will not always be the case, the median variance has to be estimated sometimes from the sample data.

Estimation of variance from sample data

The value of ${\displaystyle (2f(x))^{-2}}$—the asymptotic value of ${\displaystyle n^{-{\frac {1}{2}}}(\nu -m)}$ where ${\displaystyle \nu }$ is the population median—has been studied by several authors. The standard "delete one" jackknife method produces inconsistent results.[22] An alternative—the "delete k" method—where ${\displaystyle k}$ grows with the sample size has been shown to be asymptotically consistent.[23] This method may be computationally expensive for large data sets. A bootstrap estimate is known to be consistent,[24] but converges very slowly (order of ${\displaystyle n^{-{\frac {1}{4}}}}$).[25] Other methods have been proposed but their behavior may differ between large and small samples.[26]

Efficiency

The efficiency of the sample median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size and on the underlying population distribution. For a sample of size ${\displaystyle N=2n+1}$ from the normal distribution, the efficiency for large N is

${\displaystyle {\frac {2}{\pi }}{\frac {N+2}{N}}}$

The efficiency tends to ${\displaystyle {\frac {2}{\pi }}}$ as ${\displaystyle N}$ tends to infinity.

Other estimators

For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median.[27]

If data are represented by a statistical model specifying a particular family of probability distributions, then estimates of the median can be obtained by fitting that family of probability distributions to the data and calculating the theoretical median of the fitted distribution.[citation needed] Pareto interpolation is an application of this when the population is assumed to have a Pareto distribution.

Coefficient of dispersion

The coefficient of dispersion (CD) is defined as the ratio of the average absolute deviation from the median to the median of the data.[28] It is a statistical measure used by the states of Iowa, New York and South Dakota in estimating dues taxes.[29][30][31] In symbols

${\displaystyle CD={\frac {1}{n}}{\frac {\sum |m-x|}{m}}}$

where n is the sample size, m is the sample median and x is a variate. The sum is taken over the whole sample.

Confidence intervals for a two-sample test in which the sample sizes are large have been derived by Bonett and Seier[28] This test assumes that both samples have the same median but differ in the dispersion around it. The confidence interval (CI) is bounded inferiorly by

${\displaystyle \exp \left[\log \left({\frac {t_{a}}{t_{b}}}\right)-z_{\alpha }\left(\operatorname {var} \left[\log \left({\frac {t_{a}}{t_{b}}}\right)\right]\right)^{\frac {1}{2}}\right]}$

where tj is the mean absolute deviation of the jth sample, var() is the variance and zα is the value from the normal distribution for the chosen value of α: for α = 0.05, zα = 1.96. The following formulae are used in the derivation of these confidence intervals

${\displaystyle \operatorname {var} [\log(t_{a})]={\frac {1}{n}}\left[{\frac {s_{a}^{2}}{t_{a}^{2}}}+\left({\frac {x_{a}-{\bar {x}}}{t_{a}}}\right)^{2}-1\right]}$
${\displaystyle \operatorname {var} \left[\log \left({\frac {t_{a}}{t_{b}}}\right)\right]=\operatorname {var} [\log(t_{a})]+\operatorname {var} [\log(t_{b})]-2r(\operatorname {var} [\log(t_{a})]\operatorname {var} [\log(t_{b})])^{\frac {1}{2}}}$

where r is the Pearson correlation coefficient between the squared deviation scores

${\displaystyle d_{ia}=|x_{ia}-{\bar {x}}_{a}|}$ and ${\displaystyle d_{ib}=|x_{ib}-{\bar {x}}_{b}|}$

a and b here are constants equal to 1 and 2, x is a variate and s is the standard deviation of the sample.

Multivariate median

Previously, this article discussed the univariate median, when the sample or population had one-dimension. When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one.[27][32][33][34]

Marginal median

The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.[27][35]

Centerpoint

An alternative generalization of the median in higher dimensions is the centerpoint.

Other median-related concepts

Interpolated median

When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median ${\displaystyle m}$ is 3 since the median is the smallest value of ${\displaystyle x}$ for which ${\displaystyle F(x)}$ is greater than a half. But the interpolated median is somewhere between 2.50 and 3.50. First we add half of the interval width ${\displaystyle w}$ to the median to get the upper bound of the median interval. Then we subtract that proportion of the interval width which equals the proportion of the 33% which lies above the 50% mark. In other words, we split up the interval width pro rata to the numbers of observations. In this case, the 33% is split into 28% below the median and 5% above it so we subtract 5/33 of the interval width from the upper bound of 3.50 to give an interpolated median of 3.35. More formally, if the values ${\displaystyle f(x)}$ are known, the interpolated median can be calculated from

${\displaystyle m_{\text{int}}=m+w\left[{\frac {1}{2}}-{\frac {F(m)-{\frac {1}{2}}}{f(m)}}\right].}$

Alternatively, if in an observed sample there are ${\displaystyle k}$ scores above the median category, ${\displaystyle j}$ scores in it and ${\displaystyle i}$ scores below it then the interpolated median is given by

${\displaystyle m_{\text{int}}=m+{\frac {w}{2}}\left[{\frac {k-i}{j}}\right].}$

Pseudo-median

For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population pseudo-median, which is the median of a symmetrized distribution and which is close to the population median.[36] The Hodges–Lehmann estimator has been generalized to multivariate distributions.[37]

Variants of regression

The Theil–Sen estimator is a method for robust linear regression based on finding medians of slopes.[38]

Median filter

In the context of image processing of monochrome raster images there is a type of noise, known as the salt and pepper noise, when each pixel independently becomes black (with some small probability) or white (with some small probability), and is unchanged otherwise (with the probability close to 1). An image constructed of median values of neighborhoods (like 3×3 square) can effectively reduce noise in this case.[citation needed]

Cluster analysis

In cluster analysis, the k-medians clustering algorithm provides a way of defining clusters, in which the criterion of maximising the distance between cluster-means that is used in k-means clustering, is replaced by maximising the distance between cluster-medians.

Median–median line

This is a method of robust regression. The idea dates back to Wald in 1940 who suggested dividing a set of bivariate data into two halves depending on the value of the independent parameter ${\displaystyle x}$: a left half with values less than the median and a right half with values greater than the median.[39] He suggested taking the means of the dependent ${\displaystyle y}$ and independent ${\displaystyle x}$ variables of the left and the right halves and estimating the slope of the line joining these two points. The line could then be adjusted to fit the majority of the points in the data set.

Nair and Shrivastava in 1942 suggested a similar idea but instead advocated dividing the sample into three equal parts before calculating the means of the subsamples.[40] Brown and Mood in 1951 proposed the idea of using the medians of two subsamples rather the means.[41] Tukey combined these ideas and recommended dividing the sample into three equal size subsamples and estimating the line based on the medians of the subsamples.[42]

Median-unbiased estimators

Any mean-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function, as observed by Gauss. A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace. Other loss functions are used in statistical theory, particularly in robust statistics.

The theory of median-unbiased estimators was revived by George W. Brown in 1947:[43]

An estimate of a one-dimensional parameter θ will be said to be median-unbiased if, for fixed θ, the median of the distribution of the estimate is at the value θ; i.e., the estimate underestimates just as often as it overestimates. This requirement seems for most purposes to accomplish as much as the mean-unbiased requirement and has the additional property that it is invariant under one-to-one transformation.

— page 584

Further properties of median-unbiased estimators have been reported.[44][45][46][47] Median-unbiased estimators are invariant under one-to-one transformations.

There are methods of construction median-unbiased estimators that are optimal (in a sense analogous to minimum-variance property considered for mean-unbiased estimators). Such constructions exist for probability distributions having monotone likelihood-functions.[48][49] One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao—Blackwell procedure but for a larger class of loss functions.[50]

History

The idea of the median appeared in the 13th century in the Talmud [51][52] (further[citation needed] for possible older mentions)

The idea of the median also appeared later in Edward Wright's book on navigation (Certaine Errors in Navigation) in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations.

In 1757, Roger Joseph Boscovich developed a regression method based on the L1 norm and therefore implicitly on the median.[53]

In 1774, Laplace suggested the median be used as the standard estimator of the value of a posterior pdf. The specific criterion was to minimize the expected magnitude of the error; ${\displaystyle |\alpha -\alpha ^{*}|}$ where ${\displaystyle \alpha ^{*}}$ is the estimate and ${\displaystyle \alpha }$ is the true value. Laplaces's criterion was generally rejected for 150 years in favor of the least squares method of Gauss and Legendre which minimizes ${\displaystyle (\alpha -\alpha ^{*})^{2}}$ to obtain the mean.[54] The distribution of both the sample mean and the sample median were determined by Laplace in the early 1800s.[19][55]

Antoine Augustin Cournot in 1843 was the first[56] to use the term median (valeur médiane) for the value that divides a probability distribution into two equal halves. Gustav Theodor Fechner used the median (Centralwerth) in sociological and psychological phenomena.[57] It had earlier been used only in astronomy and related fields. Gustav Fechner popularized the median into the formal analysis of data, although it had been used previously by Laplace.[57]

Francis Galton used the English term median in 1881,[58] having earlier used the terms middle-most value in 1869, and the medium in 1880.[59][60]

References

1. ^ a b Weisstein, Eric W. "Statistical Median". MathWorld.
2. ^ Simon, Laura J.; "Descriptive statistics" Archived 2010-07-30 at the Wayback Machine, Statistical Education Resource Kit, Pennsylvania State Department of Statistics
3. ^ David J. Sheskin (27 August 2003). Handbook of Parametric and Nonparametric Statistical Procedures: Third Edition. CRC Press. pp. 7–. ISBN 978-1-4200-3626-8. Retrieved 25 February 2013.
4. ^ Derek Bissell (1994). Statistical Methods for Spc and Tqm. CRC Press. pp. 26–. ISBN 978-0-412-39440-9. Retrieved 25 February 2013.
5. ^
6. ^ Robson, Colin (1994). Experiment, Design and Statistics in Psychology. Penguin. pp. 42–45. ISBN 0-14-017648-9.
7. ^ "AP Statistics Review - Density Curves and the Normal Distributions". Retrieved 16 March 2015.
8. ^ Newman, Mark EJ. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46.5 (2005): 323–351.
9. ^ Stroock, Daniel (2011). Probability Theory. Cambridge University Press. p. 43. ISBN 978-0-521-13250-3.
10. ^ Basu, S.; Dasgupta, A. (1997). "The Mean, Median, and Mode of Unimodal Distributions:A Characterization". Theory of Probability and Its Applications. 41 (2): 210–223. doi:10.1137/S0040585X97975447.
11. ^ Mallows, Colin (August 1991). "Another comment on O'Cinneide". The American Statistician. 45 (3): 257. doi:10.1080/00031305.1991.10475815.
12. ^ K.Van Steen Notes on probability and statistics
13. ^ Piché, Robert (2012). Random Vectors and Random Sequences. Lambert Academic Publishing. ISBN 978-3659211966.
14. ^ Kemperman, Johannes H. B. (1987). Dodge, Yadolah (ed.). "The median of a finite measure on a Banach space: Statistical data analysis based on the L1-norm and related methods". Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987. Amsterdam: North-Holland Publishing Co.: 217–230. MR 0949228.
15. ^ Milasevic, Philip; Ducharme, Gilles R. (1987). "Uniqueness of the spatial median". Annals of Statistics. 15 (3): 1332–1333. doi:10.1214/aos/1176350511. MR 0902264.
16. ^ Merkle, M. (2005). "Jensen's inequality for medians". Statistics & Probability Letters. 71 (3): 277–281. doi:10.1016/j.spl.2004.11.010.
17. ^ Bentley, Jon L.; McIlroy, M. Douglas (1993). "Engineering a sort function". Software—Practice and Experience. 23 (11): 1249–1265. doi:10.1002/spe.4380231105.
18. ^ Rousseeuw, Peter J.; Bassett, Gilbert W. Jr. (1990). "The remedian: a robust averaging method for large data sets" (PDF). J. Amer. Stat. Soc. 85 (409): 97–104. doi:10.1080/01621459.1990.10475311.
19. ^ a b Stigler, Stephen (December 1973). "Studies in the History of Probability and Statistics. XXXII: Laplace, Fisher and the Discovery of the Concept of Sufficiency". Biometrika. 60 (3): 439–445. doi:10.1093/biomet/60.3.439. JSTOR 2334992. MR 0326872.
20. ^ Rider, Paul R. (1960). "Variance of the median of small samples from several special populations". J. Amer. Statist. Assoc. 55 (289): 148–150. doi:10.1080/01621459.1960.10482056.
21. ^ Stuart, Alan; Ord, Keith (1994). Kendall's Advanced Theory of Statistics. London: Arnold. ISBN 0340614307.
22. ^ Efron, B. (1982). The Jackknife, the Bootstrap and other Resampling Plans. Philadelphia: SIAM. ISBN 0898711797.
23. ^ Shao, J.; Wu, C. F. (1989). "A General Theory for Jackknife Variance Estimation". Ann. Stat. 17 (3): 1176–1197. doi:10.1214/aos/1176347263. JSTOR 2241717.
24. ^ Efron, B. (1979). "Bootstrap Methods: Another Look at the Jackknife". Ann. Stat. 7 (1): 1–26. doi:10.1214/aos/1176344552. JSTOR 2958830.
25. ^ Hall, P.; Martin, M. A. (1988). "Exact Convergence Rate of Bootstrap Quantile Variance Estimator". Probab Theory Related Fields. 80 (2): 261–268. doi:10.1007/BF00356105.
26. ^ Jiménez-Gamero, M. D.; Munoz-García, J.; Pino-Mejías, R. (2004). "Reduced bootstrap for the median". Statistica Sinica. 14 (4): 1179–1198.
27. ^ a b c Hettmansperger, Thomas P.; McKean, Joseph W. (1998). Robust nonparametric statistical methods. Kendall's Library of Statistics. 5. London: Edward Arnold. ISBN 0-340-54937-8. MR 1604954.
28. ^ a b Bonett, DG; Seier, E (2006). "Confidence interval for a coefficient of dispersion in non-normal distributions". Biometrical Journal. 48 (1): 144–148. doi:10.1002/bimj.200410148.
29. ^ "Statistical Calculation Definitions for Mass Appraisal" (PDF). Iowa.gov. Archived from the original (PDF) on 11 November 2010. Median Ratio: The ratio located midway between the highest ratio and the lowest ratio when individual ratios for a class of realty are ranked in ascending or descending order. The median ratio is most frequently used to determine the level of assessment for a given class of real estate.
30. ^ "Assessment equity in New York: Results from the 2010 market value survey". Archived from the original on 6 November 2012.
31. ^ "Summary of the Assessment Process" (PDF). state.sd.us. South Dakota Department of Revenue - Property/Special Taxes Division. Archived from the original (PDF) on 10 May 2009.
32. ^ Small, Christopher G. "A survey of multidimensional medians." International Statistical Review/Revue Internationale de Statistique (1990): 263–277. doi:10.2307/1403809 JSTOR 1403809
33. ^ Niinimaa, A., and H. Oja. "Multivariate median." Encyclopedia of statistical sciences (1999).
34. ^ Mosler, Karl. Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach. Vol. 165. Springer Science & Business Media, 2012.
35. ^ Puri, Madan L.; Sen, Pranab K.; Nonparametric Methods in Multivariate Analysis, John Wiley & Sons, New York, NY, 197l. (Reprinted by Krieger Publishing)
36. ^ Pratt, William K.; Cooper, Ted J.; Kabir, Ihtisham (1985-07-11). "Pseudomedian Filter". Architectures and Algorithms for Digital Image Processing II. 0534: 34. doi:10.1117/12.946562.
37. ^ Oja, Hannu (2010). Multivariate nonparametric methods with R: An approach based on spatial signs and ranks. Lecture Notes in Statistics. 199. New York, NY: Springer. pp. xiv+232. doi:10.1007/978-1-4419-0468-3. ISBN 978-1-4419-0467-6. MR 2598854.
38. ^ Wilcox, Rand R. (2001), "Theil–Sen estimator", Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy, Springer-Verlag, pp. 207–210, ISBN 978-0-387-95157-7.
39. ^ Wald, A. (1940). "The Fitting of Straight Lines if Both Variables are Subject to Error". Annals of Mathematical Statistics. 11 (3): 282–300. doi:10.1214/aoms/1177731868. JSTOR 2235677.
40. ^ Nair, K. R.; Shrivastava, M. P. (1942). "On a Simple Method of Curve Fitting". Sankhyā: The Indian Journal of Statistics. 6 (2): 121–132. JSTOR 25047749.
41. ^ Brown, G. W.; Mood, A. M. (1951). "On Median Tests for Linear Hypotheses". Proc Second Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA: University of California Press. pp. 159–166. Zbl 0045.08606.
42. ^ Tukey, J. W. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley. ISBN 0201076160.
43. ^ Brown, George W. (1947). "On Small-Sample Estimation". Annals of Mathematical Statistics. 18 (4): 582–585. doi:10.1214/aoms/1177730349. JSTOR 2236236.
44. ^ Lehmann, Erich L. (1951). "A General Concept of Unbiasedness". Annals of Mathematical Statistics. 22 (4): 587–592. doi:10.1214/aoms/1177729549. JSTOR 2236928.
45. ^ Birnbaum, Allan (1961). "A Unified Theory of Estimation, I". Annals of Mathematical Statistics. 32 (1): 112–135. doi:10.1214/aoms/1177705145. JSTOR 2237612.
46. ^ van der Vaart, H. Robert (1961). "Some Extensions of the Idea of Bias". Annals of Mathematical Statistics. 32 (2): 436–447. doi:10.1214/aoms/1177705051. JSTOR 2237754. MR 0125674.
47. ^ Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393.
48. ^ Pfanzagl, Johann. "On optimal median unbiased estimators in the presence of nuisance parameters." The Annals of Statistics (1979): 187–193.
49. ^ Brown, L. D.; Cohen, Arthur; Strawderman, W. E. A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications. Ann. Statist. 4 (1976), no. 4, 712–722. doi:10.1214/aos/1176343543. http://projecteuclid.org/euclid.aos/1176343543.
50. ^ Page 713: Brown, L. D.; Cohen, Arthur; Strawderman, W. E. A Complete Class Theorem for Strict Monotone Likelihood Ratio With Applications. Ann. Statist. 4 (1976), no. 4, 712–722. doi:10.1214/aos/1176343543. http://projecteuclid.org/euclid.aos/1176343543.
51. ^ Talmud and Modern Economics
52. ^
53. ^ Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press. ISBN 0674403401.
54. ^ Jaynes, E.T. (2007). Probability theory : the logic of science (5. print. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 172. ISBN 978-0-521-59271-0.
55. ^ Laplace PS de (1818) Deuxième supplément à la Théorie Analytique des Probabilités, Paris, Courcier
56. ^ Howarth, Richard (2017). Dictionary of Mathematical Geosciences: With Historical Notes. Springer. p. 374.
57. ^ a b Keynes, J.M. (1921) A Treatise on Probability. Pt II Ch XVII §5 (p 201) (2006 reprint, Cosimo Classics, ISBN 9781596055308 : multiple other reprints)
58. ^ Galton F (1881) "Report of the Anthropometric Committee" pp 245–260. Report of the 51st Meeting of the British Association for the Advancement of Science
59. ^ encyclopediaofmath.org
60. ^ personal.psu.edu
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.