Unit variance. Types of dispersions

Let's calculate inMSEXCELvariance and standard deviation samples. Let's also calculate the variance random variable, if its distribution is known.

Let's first consider dispersion, then standard deviation.

Sample variance

Sample variance (sample variance,samplevariance) characterizes the spread of values ​​in the array relative to .

All 3 formulas are mathematically equivalent.

From the first formula it is clear that sample variance is the sum of the squared deviations of each value in the array from average, divided by sample size minus 1.

variances samples the DISP() function is used, English. the name VAR, i.e. VARiance. From version MS EXCEL 2010, it is recommended to use its analogue DISP.V(), English. the name VARS, i.e. Sample VARiance. In addition, starting from the version of MS EXCEL 2010, there is a function DISP.Г(), English. name VARP, i.e. Population VARiance, which calculates dispersion For population. The whole difference comes down to the denominator: instead of n-1 like DISP.V(), DISP.G() has just n in the denominator. Before MS EXCEL 2010, the VAR() function was used to calculate the variance of the population.

Sample variance
=QUADROTCL(Sample)/(COUNT(Sample)-1)
=(SUM(Sample)-COUNT(Sample)*AVERAGE(Sample)^2)/ (COUNT(Sample)-1)– usual formula
=SUM((Sample -AVERAGE(Sample))^2)/ (COUNT(Sample)-1) –

Sample variance is equal to 0, only if all values ​​are equal to each other and, accordingly, equal average value. Usually, the larger the value variances, the greater the spread of values ​​in the array.

Sample variance is a point estimate variances distribution of the random variable from which it was made sample. About construction confidence intervals when assessing variances can be read in the article.

Variance of a random variable

To calculate dispersion random variable, you need to know it.

For variances random variable X is often denoted Var(X). Dispersion equal to the square of the deviation from the mean E(X): Var(X)=E[(X-E(X)) 2 ]

dispersion calculated by the formula:

where x i is the value that a random variable can take, and μ is the average value (), p(x) is the probability that the random variable will take the value x.

If a random variable has , then dispersion calculated by the formula:

Dimension variances corresponds to the square of the unit of measurement of the original values. For example, if the values ​​in the sample represent part weight measurements (in kg), then the variance dimension would be kg 2 . This can be difficult to interpret, so to characterize the spread of values, a value equal to the square root of variances – standard deviation.

Some properties variances:

Var(X+a)=Var(X), where X is a random variable and a is a constant.

Var(aХ)=a 2 Var(X)

Var(X)=E[(X-E(X)) 2 ]=E=E(X 2)-E(2*X*E(X))+(E(X)) 2 =E(X 2)- 2*E(X)*E(X)+(E(X)) 2 =E(X 2)-(E(X)) 2

This dispersion property is used in article about linear regression.

Var(X+Y)=Var(X) + Var(Y) + 2*Cov(X;Y), where X and Y are random variables, Cov(X;Y) is the covariance of these random variables.

If random variables are independent, then they covariance is equal to 0, and therefore Var(X+Y)=Var(X)+Var(Y). This property of dispersion is used in derivation.

Let us show that for independent quantities Var(X-Y)=Var(X+Y). Indeed, Var(X-Y)= Var(X-Y)= Var(X+(-Y))= Var(X)+Var(-Y)= Var(X)+Var(-Y)= Var( X)+(-1) 2 Var(Y)= Var(X)+Var(Y)= Var(X+Y). This dispersion property is used to construct .

Sample standard deviation

Sample standard deviation is a measure of how widely scattered the values ​​in a sample are relative to their .

A-priory, standard deviation equal to the square root of variances:

Standard deviation does not take into account the magnitude of the values ​​in sample, but only the degree of dispersion of values ​​around them average. To illustrate this, let's give an example.

Let's calculate the standard deviation for 2 samples: (1; 5; 9) and (1001; 1005; 1009). In both cases, s=4. It is obvious that the ratio of the standard deviation to the array values ​​differs significantly between samples. For such cases it is used The coefficient of variation(Coefficient of Variation, CV) - ratio Standard Deviation to the average arithmetic, expressed as a percentage.

In MS EXCEL 2007 and earlier versions for calculation Sample standard deviation the function =STDEVAL() is used, English. name STDEV, i.e. STandard DEViation. From the version of MS EXCEL 2010, it is recommended to use its analogue =STDEV.B() , English. name STDEV.S, i.e. Sample STandard DEViation.

In addition, starting from the version of MS EXCEL 2010, there is a function STANDARDEV.G(), English. name STDEV.P, i.e. Population STandard DEViation, which calculates standard deviation For population. The whole difference comes down to the denominator: instead of n-1 as in STANDARDEV.V(), STANDARDEVAL.G() has just n in the denominator.

Standard deviation can also be calculated directly using the formulas below (see example file)
=ROOT(QUADROTCL(Sample)/(COUNT(Sample)-1))
=ROOT((SUM(Sample)-COUNT(Sample)*AVERAGE(Sample)^2)/(COUNT(Sample)-1))

Other measures of scatter

The SQUADROTCL() function calculates with a sum of squared deviations of values ​​from their average. This function will return the same result as the formula =DISP.G( Sample)*CHECK( Sample) , Where Sample- a reference to a range containing an array of sample values ​​(). Calculations in the QUADROCL() function are made according to the formula:

The SROTCL() function is also a measure of the spread of a data set. The function SROTCL() calculates the average of the absolute values ​​of deviations of values ​​from average. This function will return the same result as the formula =SUMPRODUCT(ABS(Sample-AVERAGE(Sample)))/COUNT(Sample), Where Sample- a link to a range containing an array of sample values.

Calculations in the function SROTCL () are made according to the formula:

Probability theory - special section mathematics, which is studied only by students of higher educational institutions. Do you like calculations and formulas? Aren't you scared by the prospects of getting acquainted with the normal distribution, ensemble entropy, mathematical expectation and dispersion of a discrete random variable? Then this subject will be very interesting to you. Let's get acquainted with several of the most important basic concepts of this branch of science.

Let's remember the basics

Even if you remember the most simple concepts theory of probability, do not neglect the first paragraphs of the article. The point is that without a clear understanding of the basics, you will not be able to work with the formulas discussed below.

So, some random event occurs, some experiment. As a result of the actions we take, we can get several outcomes - some of them occur more often, others less often. The probability of an event is the ratio of the number of actually obtained outcomes of one type to the total number of possible ones. Only knowing the classic definition of this concept can you begin to study mathematical expectation and variances of continuous random variables.

Average

Back in school, during math lessons, you started working with the arithmetic mean. This concept is widely used in probability theory, and therefore cannot be ignored. The main thing for us is this moment is that we will encounter it in the formulas for the mathematical expectation and dispersion of a random variable.

We have a sequence of numbers and want to find the arithmetic mean. All that is required of us is to sum up everything available and divide by the number of elements in the sequence. Let us have numbers from 1 to 9. The sum of the elements will be equal to 45, and we will divide this value by 9. Answer: - 5.

Dispersion

In scientific terms, dispersion is the average square of deviations of the obtained values ​​of a characteristic from the arithmetic mean. It is denoted by one capital Latin letter D. What is needed to calculate it? For each element of the sequence, we calculate the difference between the existing number and the arithmetic mean and square it. There will be exactly as many values ​​as there can be outcomes for the event we are considering. Next, we sum up everything received and divide by the number of elements in the sequence. If we have five possible outcomes, then divide by five.

Dispersion also has properties that need to be remembered in order to be used when solving problems. For example, when increasing a random variable by X times, the variance increases by X squared times (i.e. X*X). It is never less than zero and does not depend on shifting values ​​up or down by equal amounts. Additionally, for independent trials, the variance of the sum is equal to the sum of the variances.

Now we definitely need to consider examples of the variance of a discrete random variable and the mathematical expectation.

Let's say we ran 21 experiments and got 7 different outcomes. We observed each of them 1, 2, 2, 3, 4, 4 and 5 times, respectively. What will the variance be equal to?

First, let's calculate the arithmetic mean: the sum of the elements, of course, is 21. Divide it by 7, getting 3. Now subtract 3 from each number in the original sequence, square each value, and add the results together. The result is 12. Now all we have to do is divide the number by the number of elements, and, it would seem, that’s all. But there's a catch! Let's discuss it.

Dependence on the number of experiments

It turns out that when calculating variance, the denominator can contain one of two numbers: either N or N-1. Here N is the number of experiments performed or the number of elements in the sequence (which is essentially the same thing). What does this depend on?

If the number of tests is measured in hundreds, then we must put N in the denominator. If in units, then N-1. Scientists decided to draw the border quite symbolically: today it passes through the number 30. If we conducted less than 30 experiments, then we will divide the amount by N-1, and if more, then by N.

Task

Let's return to our example of solving the problem of variance and mathematical expectation. We got an intermediate number 12, which needed to be divided by N or N-1. Since we conducted 21 experiments, which is less than 30, we will choose the second option. So the answer is: the variance is 12 / 2 = 2.

Expected value

Let's move on to the second concept, which we must consider in this article. The mathematical expectation is the result of adding all possible outcomes multiplied by the corresponding probabilities. It is important to understand that the obtained value, as well as the result of calculating the variance, is obtained only once for the whole task, no matter how many outcomes are considered.

The formula for mathematical expectation is quite simple: we take the outcome, multiply it by its probability, add the same for the second, third result, etc. Everything related to this concept is not difficult to calculate. For example, the sum of the expected values ​​is equal to the expected value of the sum. The same is true for the work. Not every quantity in probability theory allows you to perform such simple operations. Let's take the problem and calculate the meaning of two concepts we have studied at once. Besides, we were distracted by theory - it's time to practice.

One more example

We ran 50 trials and got 10 types of outcomes - numbers from 0 to 9 - appearing in different percentages. These are, respectively: 2%, 10%, 4%, 14%, 2%,18%, 6%, 16%, 10%, 18%. Recall that to obtain probabilities, you need to divide the percentage values ​​by 100. Thus, we get 0.02; 0.1, etc. Let us present an example of solving the problem for the variance of a random variable and the mathematical expectation.

We calculate the arithmetic mean using the formula that we remember from elementary school: 50/10 = 5.

Now let’s convert the probabilities into the number of outcomes “in pieces” to make it easier to count. We get 1, 5, 2, 7, 1, 9, 3, 8, 5 and 9. From each value obtained, we subtract the arithmetic mean, after which we square each of the results obtained. See how to do this using the first element as an example: 1 - 5 = (-4). Next: (-4) * (-4) = 16. For other values, do these operations yourself. If you did everything correctly, then after adding them all up you will get 90.

Let's continue calculating the variance and expected value by dividing 90 by N. Why do we choose N rather than N-1? Correct, because the number of experiments performed exceeds 30. So: 90/10 = 9. We got the variance. If you get a different number, don't despair. Most likely, you made a simple mistake in the calculations. Double-check what you wrote, and everything will probably fall into place.

Finally, remember the formula for mathematical expectation. We will not give all the calculations, we will only write an answer that you can check with after completing all the required procedures. The expected value will be 5.48. Let us only recall how to carry out operations, using the first elements as an example: 0*0.02 + 1*0.1... and so on. As you can see, we simply multiply the outcome value by its probability.

Deviation

Another concept closely related to dispersion and mathematical expectation is standard deviation. It is denoted either by the Latin letters sd, or by the Greek lowercase “sigma”. This concept shows how much on average the values ​​deviate from the central feature. To find its value, you need to calculate Square root from dispersion.

If you plot a normal distribution graph and want to see the squared deviation directly on it, this can be done in several stages. Take half of the image to the left or right of the mode (central value), draw a perpendicular to the horizontal axis so that the areas of the resulting figures are equal. The size of the segment between the middle of the distribution and the resulting projection onto the horizontal axis will represent the standard deviation.

Software

As can be seen from the descriptions of the formulas and the examples presented, calculating variance and mathematical expectation is not the simplest procedure from an arithmetic point of view. In order not to waste time, it makes sense to use the program used in higher education educational institutions- it's called "R". It has functions that allow you to calculate values ​​for many concepts from statistics and probability theory.

For example, you specify a vector of values. This is done as follows: vector<-c(1,5,2…). Теперь, когда вам потребуется посчитать какие-либо значения для этого вектора, вы пишете функцию и задаете его в качестве аргумента. Для нахождения дисперсии вам нужно будет использовать функцию var. Пример её использования: var(vector). Далее вы просто нажимаете «ввод» и получаете результат.

Finally

Dispersion and mathematical expectation are without which it is difficult to calculate anything in the future. In the main course of lectures at universities, they are discussed already in the first months of studying the subject. It is precisely because of the lack of understanding of these simple concepts and the inability to calculate them that many students immediately begin to fall behind in the program and later receive bad grades at the end of the session, which deprives them of scholarships.

Practice for at least one week, half an hour a day, solving tasks similar to those presented in this article. Then, on any test in probability theory, you will be able to cope with the examples without extraneous tips and cheat sheets.

This page describes a standard example of finding variance, you can also look at other problems for finding it

Example 1. Determination of group, group average, intergroup and total variance

Example 2. Finding the variance and coefficient of variation in a grouping table

Example 3. Finding variance in a discrete series

Example 4. The following data is available for a group of 20 correspondence students. It is necessary to construct an interval series of the distribution of the characteristic, calculate the average value of the characteristic and study its dispersion

Let's build an interval grouping. Let's determine the range of the interval using the formula:

where X max is the maximum value of the grouping characteristic;
X min – minimum value of the grouping characteristic;
n – number of intervals:

We accept n=5. The step is: h = (192 - 159)/ 5 = 6.6

Let's create an interval grouping

For further calculations, we will build an auxiliary table:

X"i – the middle of the interval. (for example, the middle of the interval 159 – 165.6 = 162.3)

We determine the average height of students using the weighted arithmetic average formula:

Let's determine the variance using the formula:

The formula can be transformed like this:

From this formula it follows that variance is equal to the difference between the average of the squares of the options and the square and the average.

Dispersion in variation series with equal intervals using the method of moments can be calculated in the following way using the second property of dispersion (dividing all options by the value of the interval). Determining variance, calculated using the method of moments, using the following formula is less laborious:

where i is the value of the interval;
A is a conventional zero, for which it is convenient to use the middle of the interval with the highest frequency;
m1 is the square of the first order moment;
m2 - moment of second order

Alternative trait variance (if in a statistical population a characteristic changes in such a way that there are only two mutually exclusive options, then such variability is called alternative) can be calculated using the formula:

Substituting q = 1- p into this dispersion formula, we get:

Types of variance

Total variance measures the variation of a characteristic across the entire population as a whole under the influence of all factors that cause this variation. It is equal to the mean square of the deviations of individual values ​​of a characteristic x from the overall mean value of x and can be defined as simple variance or weighted variance.

Within-group variance characterizes random variation, i.e. part of the variation that is due to the influence of unaccounted factors and does not depend on the factor-attribute that forms the basis of the group. Such dispersion is equal to the mean square of the deviations of individual values ​​of the attribute within group X from the arithmetic mean of the group and can be calculated as simple dispersion or as weighted dispersion.

Thus, within-group variance measures variation of a trait within a group and is determined by the formula:

where xi is the group average;
ni is the number of units in the group.

For example, intragroup variances that need to be determined in the task of studying the influence of workers’ qualifications on the level of labor productivity in a workshop show variations in output in each group caused by all possible factors (technical condition of equipment, availability of tools and materials, age of workers, labor intensity, etc. .), except for differences in qualification category (within a group all workers have the same qualifications).

The main generalizing indicators of variation in statistics are dispersions and standard deviations.

Dispersion this arithmetic mean squared deviations of each characteristic value from the overall average. The variance is usually called the mean square of deviations and is denoted by  2. Depending on the source data, the variance can be calculated using the simple or weighted arithmetic mean:

 unweighted (simple) variance;

 variance weighted.

Standard deviation this is a generalizing characteristic of absolute sizes variations signs in the aggregate. It is expressed in the same units of measurement as the attribute (in meters, tons, percentage, hectares, etc.).

The standard deviation is the square root of the variance and is denoted by :

 standard deviation unweighted;

 weighted standard deviation.

The standard deviation is a measure of the reliability of the mean. The smaller the standard deviation, the better the arithmetic mean reflects the entire represented population.

The calculation of the standard deviation is preceded by the calculation of the variance.

The procedure for calculating the weighted variance is as follows:

1) determine the weighted arithmetic mean:

2) calculate the deviations of the options from the average:

3) square the deviation of each option from the average:

4) multiply the squares of deviations by weights (frequencies):

5) summarize the resulting products:

6) the resulting amount is divided by the sum of the weights:

Example 2.1

Let's calculate the weighted arithmetic mean:

The values ​​of deviations from the mean and their squares are presented in the table. Let's define the variance:

The standard deviation will be equal to:

If the source data is presented in the form of interval distribution series , then you first need to determine the discrete value of the attribute, and then apply the described method.

Example 2.2

Let us show the calculation of variance for an interval series using data on the distribution of the sown area of ​​a collective farm according to wheat yield.

The arithmetic mean is:

Let's calculate the variance:

6.3. Calculation of variance using a formula based on individual data

Calculation technique variances complex, and with large values ​​of options and frequencies it can be cumbersome. Calculations can be simplified using the properties of dispersion.

The dispersion has the following properties.

1. Reducing or increasing the weights (frequencies) of a varying characteristic by a certain number of times does not change the dispersion.

2. Decrease or increase each value of a characteristic by the same constant amount A does not change the dispersion.

3. Decrease or increase each value of a characteristic by a certain number of times k respectively reduces or increases the variance in k 2 times standard deviation  in k once.

4. The dispersion of a characteristic relative to an arbitrary value is always greater than the dispersion relative to the arithmetic mean per square of the difference between the average and arbitrary values:

If A 0, then we arrive at the following equality:

that is, the variance of the characteristic is equal to the difference between the mean square of the characteristic values ​​and the square of the mean.

Each property can be used independently or in combination with others when calculating variance.

The procedure for calculating variance is simple:

1) determine arithmetic mean :

2) square the arithmetic mean:

3) square the deviation of each variant of the series:

X i 2 .

4) find the sum of squares of the options:

5) divide the sum of the squares of the options by their number, i.e. determine the average square:

6) determine the difference between the mean square of the characteristic and the square of the mean:

Example 3.1 The following data is available on worker productivity:

Let's make the following calculations:

However, this characteristic alone is not enough to study a random variable. Let's imagine two shooters shooting at a target. One shoots accurately and hits close to the center, while the other... is just having fun and doesn’t even aim. But what's funny is that he average the result will be exactly the same as the first shooter! This situation is conventionally illustrated by the following random variables:

The “sniper” mathematical expectation is equal to , however, for the “interesting person”: – it is also zero!

Thus, there is a need to quantify how far scattered bullets (random variable values) relative to the center of the target (mathematical expectation). well and scattering translated from Latin is no other way than dispersion .

Let's see how this numerical characteristic is determined using one of the examples from the 1st part of the lesson:

There we found a disappointing mathematical expectation of this game, and now we have to calculate its variance, which denoted by through .

Let's find out how far the wins/losses are “scattered” relative to the average value. Obviously, for this we need to calculate differences between random variable values and her mathematical expectation:

–5 – (–0,5) = –4,5
2,5 – (–0,5) = 3
10 – (–0,5) = 10,5

Now it seems that you need to sum up the results, but this way is not suitable - for the reason that fluctuations to the left will cancel each other out with fluctuations to the right. So, for example, an “amateur” shooter (example above) the differences will be , and when added they will give zero, so we will not get any estimate of the dispersion of his shooting.

To get around this problem you can consider modules differences, but for technical reasons the approach has taken root when they are squared. It is more convenient to formulate the solution in a table:

And here it begs to calculate weighted average the value of the squared deviations. What is it? It's theirs expected value, which is a measure of scattering:

– definition variances. From the definition it is immediately clear that variance cannot be negative– take note for practice!

Let's remember how to find the expected value. Multiply the squared differences by the corresponding probabilities (Table continuation):
– figuratively speaking, this is “traction force”,
and summarize the results:

Don't you think that compared to the winnings, the result turned out to be too big? That's right - we squared it, and to return to the dimension of our game, we need to extract the square root. This quantity is called standard deviation and is denoted by the Greek letter “sigma”:

This value is sometimes called standard deviation .

What is its meaning? If we deviate from the mathematical expectation to the left and right by the standard deviation:

– then the most probable values ​​of the random variable will be “concentrated” on this interval. What we actually observe:

However, it so happens that when analyzing scattering one almost always operates with the concept of dispersion. Let's figure out what it means in relation to games. If in the case of arrows we are talking about the “accuracy” of hits relative to the center of the target, then here dispersion characterizes two things:

Firstly, it is obvious that as the bets increase, the dispersion also increases. So, for example, if we increase by 10 times, then the mathematical expectation will increase by 10 times, and the variance will increase by 100 times (since this is a quadratic quantity). But note that the rules of the game themselves have not changed! Only the rates have changed, roughly speaking, before we bet 10 rubles, now it’s 100.

The second, more interesting point is that variance characterizes the style of play. Mentally fix the game bets at some certain level, and let's see what's what:

A low variance game is a cautious game. The player tends to choose the most reliable schemes, where he does not lose/win too much at one time. For example, the red/black system in roulette (see example 4 of the article Random variables) .

High variance game. She is often called dispersive game. This is an adventurous or aggressive style of play, where the player chooses “adrenaline” schemes. Let's at least remember "Martingale", in which the amounts at stake are orders of magnitude greater than the “quiet” game of the previous point.

The situation in poker is indicative: there are so-called tight players who tend to be cautious and “shaky” over their gaming funds (bankroll). Not surprisingly, their bankroll does not fluctuate significantly (low variance). On the contrary, if a player has high variance, then he is an aggressor. He often takes risks, makes large bets and can either break a huge bank or lose to smithereens.

The same thing happens in Forex, and so on - there are plenty of examples.

Moreover, in all cases it does not matter whether the game is played for pennies or thousands of dollars. Every level has its low- and high-dispersion players. Well, as we remember, the average winning is “responsible” expected value.

You probably noticed that finding variance is a long and painstaking process. But mathematics is generous:

Formula for finding variance

This formula is derived directly from the definition of variance, and we immediately put it into use. I’ll copy the sign with our game above:

and the found mathematical expectation.

Let's calculate the variance in the second way. First, let's find the mathematical expectation - the square of the random variable. By determination of mathematical expectation:

In this case:

Thus, according to the formula:

As they say, feel the difference. And in practice, of course, it is better to use the formula (unless the condition requires otherwise).

We master the technique of solving and designing:

Example 6

Find its mathematical expectation, variance and standard deviation.

This task is found everywhere, and, as a rule, goes without meaningful meaning.
You can imagine several light bulbs with numbers that light up in a madhouse with certain probabilities :)

Solution: It is convenient to summarize the basic calculations in a table. First, we write the initial data in the top two lines. Then we calculate the products, then and finally the sums in the right column:

Actually, almost everything is ready. The third line shows a ready-made mathematical expectation: .

We calculate the variance using the formula:

And finally, the standard deviation:
– Personally, I usually round to 2 decimal places.

All calculations can be carried out on a calculator, or even better – in Excel:

It's hard to go wrong here :)

Answer:

Those who wish can simplify their life even more and take advantage of my calculator (demo), which will not only instantly solve this problem, but also build thematic graphics (we'll get there soon). The program can be download from the library– if you have downloaded at least one educational material, or receive another way. Thanks for supporting the project!

A couple of tasks to solve on your own:

Example 7

Calculate the variance of the random variable in the previous example by definition.

And a similar example:

Example 8

A discrete random variable is specified by its distribution law:

Yes, random variable values ​​can be quite large (example from real work), and here, if possible, use Excel. As, by the way, in Example 7 - it’s faster, more reliable and more enjoyable.

Solutions and answers at the bottom of the page.

To conclude the 2nd part of the lesson, we will look at another typical problem, one might even say a small puzzle:

Example 9

A discrete random variable can take only two values: and , and . The probability, mathematical expectation and variance are known.

Solution: Let's start with an unknown probability. Since a random variable can take only two values, the sum of the probabilities of the corresponding events is:

and since , then .

All that remains is to find..., it's easy to say :) But oh well, here we go. By definition of mathematical expectation:
– substitute known quantities:

– and nothing more can be squeezed out of this equation, except that you can rewrite it in the usual direction:

or:

I think you can guess the next steps. Let's compose and solve the system:

Decimals are, of course, a complete disgrace; multiply both equations by 10:

and divide by 2:

That's better. From the 1st equation we express:
(this is the easier way)– substitute into the 2nd equation:

We are building squared and make simplifications:

Multiply by:

The result was quadratic equation, we find its discriminant:
- Great!

and we get two solutions:

1) if , That ;

2) if , That .

The condition is satisfied by the first pair of values. With a high probability everything is correct, but, nevertheless, let’s write down the distribution law:

and perform a check, namely, find the expectation: