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Meaning of the word dispersion

variance in the crossword dictionary

Economic dictionary of terms

dispersion

a value characterizing the degree of dispersion of quantitative measurements of individual participants in a statistical sample (random variables) relative to the average value for this sample.

Explanatory dictionary of the Russian language. D.N. Ushakov

dispersion

variances, plural no, w. (Latin dispersio).

The divergence of light rays of different colors when passing through a refractive medium (opt.).

A state of greater or lesser fragmentation of a substance (ed.).

New explanatory and word-formative dictionary of the Russian language, T. F. Efremova.

dispersion

and. Decomposition, dispersion, separation.

Encyclopedic Dictionary, 1998

dispersion

DISPERSION (from Latin dispersio - dispersion) in mathematical statistics and probability theory is a measure of dispersion (deviation from the average). In statistics, variance is the arithmetic mean of the squared deviations of the observed values ​​(x1, x2,...,xn) of a random variable from their arithmetic mean. In probability theory, the variance of a random variable is the mathematical expectation of the squared deviation of a random variable from its mathematical expectation.

Dispersion

(from Latin dispersio ≈ dispersion), in mathematical statistics and probability theory, the most common measure of dispersion, i.e. deviation from the average. In the statistical sense D.

is the arithmetic mean of the squared deviations of values ​​xi from their arithmetic mean

In probability theory, the probability of a random variable X is called the mathematical expectation E (X ≈ mx)2 of the squared deviation of X from its mathematical expectation mx = E (X). The dimension of a random variable X is denoted by D (X) or by s2X. The square root of D. (that is, s, if D. is s2) is called the standard deviation (see Square deviation).

For a random variable X with a continuous probability distribution, characterized by the probability density p (x), D. is calculated using the formula

For D.'s assessment based on observation results, see Statistical assessments.

In probability theory, the theorem is of great importance: the sum of independent terms is equal to the sum of their sums. No less important is the Chebyshev inequality, which allows one to estimate the probability of large deviations of a random variable X from its mathematical expectation.

Lit.: Gnedenko B.V., Course of Probability Theory, 5th ed., M., 1969.

Wikipedia

Dispersion

Dispersion depending on the context may mean:

• Wave dispersion - in physics, the dependence of the phase velocity of a wave on its frequency is distinguished:
  • Light dispersion
  • Sound dispersion
• The dispersion law is a law in physics that expresses the dependence of the phase velocity of a wave on its frequency.
• The variance of a random variable is one of the averaged characteristics of a random variable.
• Dispersion - formations of two or more phases that are completely or practically immiscible and do not react chemically with each other
• Dispersion is a term that refers to the diversity of traits in a population.
• Dispersion
• Second viscosity dispersion

Dispersion (biology)

Dispersion- a term denoting the diversity of traits in a population.

One of the quantitative characteristics of a population. For description asexual And hermaphrodite population except variances for each trait ( σ ) you also need to know the number of individuals ( N) and average values ​​of features ( Δx).

IN dioecious In a population, each sex has its own dispersion value - . Other parameters are the number of individuals ( N), sex ratio and sexual dimorphism.

Examples of the use of the word dispersion in the literature.

These include Wood's almost innumerable results on diffraction, interference, polarization, anomalous variances, absorption.

After all the calculations made during the journey, after countless corrections and checks of the calculations, Erwin could easily calculate the mathematical expectation and dispersion the time of the appearance of another lucky survivor on the Happy Islands - and could not bring himself to begin the calculation, foreseeing the result.

Normal for thinking is dispersion, sleep, daydream, illogicality, simultaneous action of different thought centers without central control.

Absorption, fluorescence, magnetic rotation and anomalous dispersion mercury vapor.

Julius, a Dutch astronomer who put forward a bold theory that the chromosphere flare spectrum was caused by an anomalous dispersion white light emitted by the liquid surface of the sun.

While lecturing in Madison, I came to the topic of anomalous variances, caused by highly absorbing media.

Then I took out my long gas burner and after half an hour I had set up a demonstration with an anomalous dispersion in a long tube containing sodium vapor.

About cyanine prisms and a new method for demonstrating anomalous variances.

About the anomalous variances, absorption and surface color of nitrosodimethylaniline with notes on variances toluin.

Quantitative determination of abnormal variances sodium vapor in the visible and ultraviolet regions.

I use high frequency matrices with fast dispersion and bipolar amplifiers.

The variance of a random variable is a measure of the spread of the values ​​of this variable. Low variance means that the values ​​are clustered close together. Large dispersion indicates a strong spread of values. The concept of variance of a random variable is used in statistics. For example, if you compare the variance of two values ​​(such as between male and female patients), you can test the significance of a variable. Variance is also used when building statistical models, since low variance can be a sign that you are overfitting the values.

Steps

Calculating sample variance

1. Record the sample values. In most cases, statisticians only have access to samples of specific populations. For example, as a rule, statisticians do not analyze the cost of maintaining the totality of all cars in Russia - they analyze a random sample of several thousand cars. Such a sample will help determine the average cost of a car, but, most likely, the resulting value will be far from the real one.

   • For example, let's analyze the number of buns sold in a cafe over 6 days, taken in random order. The sample looks like this: 17, 15, 23, 7, 9, 13. This is a sample, not a population, because we do not have data on buns sold for each day the cafe is open.
   • If you are given a population rather than a sample of values, continue to the next section.

2. Write down a formula to calculate sample variance. Dispersion is a measure of the spread of values ​​of a certain quantity. The closer the variance value is to zero, the closer the values ​​are grouped together. When working with a sample of values, use the following formula to calculate variance:

   • s 2 (\displaystyle s^(2)) = ∑[(x i (\displaystyle x_(i))- x̅) 2 (\displaystyle ^(2))] / (n - 1)
   • s 2 (\displaystyle s^(2))– this is dispersion. Dispersion is measured in square units.
   • x i (\displaystyle x_(i))– each value in the sample.
   • x i (\displaystyle x_(i)) you need to subtract x̅, square it, and then add the results.
   • x̅ – sample mean (sample mean).
   • n – number of values ​​in the sample.

3. Calculate the sample mean. It is denoted as x̅. The sample mean is calculated as a simple arithmetic mean: add up all the values ​​in the sample, and then divide the result by the number of values ​​in the sample.

   • In our example, add the values ​​in the sample: 15 + 17 + 23 + 7 + 9 + 13 = 84
     Now divide the result by the number of values ​​in the sample (in our example there are 6): 84 ÷ 6 = 14.
     Sample mean x̅ = 14.
   • The sample mean is the central value around which the values ​​in the sample are distributed. If the values ​​in the sample cluster around the sample mean, then the variance is small; otherwise the variance is large.

4. Subtract the sample mean from each value in the sample. Now calculate the difference x i (\displaystyle x_(i))- x̅, where x i (\displaystyle x_(i))– each value in the sample. Each result obtained indicates the degree of deviation of a particular value from the sample mean, that is, how far this value is from the sample mean.

   • In our example:
     x 1 (\displaystyle x_(1))- x = 17 - 14 = 3
     x 2 (\displaystyle x_(2))- x̅ = 15 - 14 = 1
     x 3 (\displaystyle x_(3))- x = 23 - 14 = 9
     x 4 (\displaystyle x_(4))- x̅ = 7 - 14 = -7
     x 5 (\displaystyle x_(5))- x̅ = 9 - 14 = -5
     x 6 (\displaystyle x_(6))- x̅ = 13 - 14 = -1
   • The correctness of the results obtained is easy to check, since their sum should be equal to zero. This is related to the definition of the average, since negative values ​​(distances from the average to smaller values) are completely offset by positive values ​​(distances from the average to larger values).

5. As noted above, the sum of the differences x i (\displaystyle x_(i))- x̅ must be equal to zero. This means that the average variance is always zero, which does not give any idea about the spread of values ​​of a certain quantity. To solve this problem, square each difference x i (\displaystyle x_(i))- x̅. This will result in you only getting positive numbers, which will never add up to 0.

   • In our example:
     (x 1 (\displaystyle x_(1))- x̅) 2 = 3 2 = 9 (\displaystyle ^(2)=3^(2)=9)
     (x 2 (\displaystyle (x_(2))- x̅) 2 = 1 2 = 1 (\displaystyle ^(2)=1^(2)=1)
     9 2 = 81
     (-7) 2 = 49
     (-5) 2 = 25
     (-1) 2 = 1
   • You found the square of the difference - x̅) 2 (\displaystyle ^(2)) for each value in the sample.

6. Calculate the sum of the squares of the differences. That is, find that part of the formula that is written like this: ∑[( x i (\displaystyle x_(i))- x̅) 2 (\displaystyle ^(2))]. Here the sign Σ means the sum of squared differences for each value x i (\displaystyle x_(i)) in the sample. You have already found the squared differences (x i (\displaystyle (x_(i))- x̅) 2 (\displaystyle ^(2)) for each value x i (\displaystyle x_(i)) in the sample; now just add these squares.

   • In our example: 9 + 1 + 81 + 49 + 25 + 1 = 166 .

7. Divide the result by n - 1, where n is the number of values ​​in the sample. Some time ago, to calculate sample variance, statisticians simply divided the result by n; in this case you will get the mean of the squared variance, which is ideal for describing the variance of a given sample. But remember that any sample is only a small part of the population of values. If you take another sample and perform the same calculations, you will get a different result. As it turns out, dividing by n - 1 (rather than just n) gives a more accurate estimate of the population variance, which is what you're interested in. Division by n – 1 has become common, so it is included in the formula for calculating sample variance.

   • In our example, the sample includes 6 values, that is, n = 6.
     Sample variance = s 2 = 166 6 − 1 = (\displaystyle s^(2)=(\frac (166)(6-1))=) 33,2

8. The difference between variance and standard deviation. Note that the formula contains an exponent, so the dispersion is measured in square units of the value being analyzed. Sometimes such a magnitude is quite difficult to operate; in such cases, use the standard deviation, which is equal to the square root of the variance. That is why the sample variance is denoted as s 2 (\displaystyle s^(2)), and the standard deviation of the sample is as s (\displaystyle s).

   • In our example, the standard deviation of the sample is: s = √33.2 = 5.76.

   Calculating Population Variance

   1. Analyze some set of values. The set includes all values ​​of the quantity under consideration. For example, if you are studying the age of residents of the Leningrad region, then the totality includes the age of all residents of this region. When working with a population, it is recommended to create a table and enter the population values ​​into it. Consider the following example:

      • In a certain room there are 6 aquariums. Each aquarium contains the following number of fish:
        x 1 = 5 (\displaystyle x_(1)=5)
        x 2 = 5 (\displaystyle x_(2)=5)
        x 3 = 8 (\displaystyle x_(3)=8)
        x 4 = 12 (\displaystyle x_(4)=12)
        x 5 = 15 (\displaystyle x_(5)=15)
        x 6 = 18 (\displaystyle x_(6)=18)

   2. Write down a formula to calculate the population variance. Since the population includes all values ​​of a certain quantity, the formula below allows you to obtain the exact value of the population variance. To distinguish population variance from sample variance (which is only an estimate), statisticians use various variables:

      • σ 2 (\displaystyle ^(2)) = (∑(x i (\displaystyle x_(i)) - μ) 2 (\displaystyle ^(2)))/n
      • σ 2 (\displaystyle ^(2))– population dispersion (read as “sigma squared”). Dispersion is measured in square units.
      • x i (\displaystyle x_(i))– each value in its entirety.
      • Σ – sum sign. That is, from each value x i (\displaystyle x_(i)) you need to subtract μ, square it, and then add the results.
      • μ – population mean.
      • n – number of values ​​in the population.

   3. Calculate the population mean. When working with a population, its mean is denoted as μ (mu). The population mean is calculated as a simple arithmetic mean: add up all the values ​​in the population, and then divide the result by the number of values ​​in the population.

      • Keep in mind that averages are not always calculated as the arithmetic mean.
      • In our example, the population mean: μ = 5 + 5 + 8 + 12 + 15 + 18 6 (\displaystyle (\frac (5+5+8+12+15+18)(6))) = 10,5

   4. Subtract the population mean from each value in the population. The closer the difference value is to zero, the closer the specific value is to the population mean. Find the difference between each value in the population and its mean, and you will get a first idea of ​​the distribution of values.

      • In our example:
        x 1 (\displaystyle x_(1))- μ = 5 - 10.5 = -5.5
        x 2 (\displaystyle x_(2))- μ = 5 - 10.5 = -5.5
        x 3 (\displaystyle x_(3))- μ = 8 - 10.5 = -2.5
        x 4 (\displaystyle x_(4))- μ = 12 - 10.5 = 1.5
        x 5 (\displaystyle x_(5))- μ = 15 - 10.5 = 4.5
        x 6 (\displaystyle x_(6))- μ = 18 - 10.5 = 7.5

   5. Square each result obtained. The difference values ​​will be both positive and negative; If these values ​​are plotted on a number line, they will lie to the right and left of the population mean. This is not good for calculating variance because positive and negative numbers cancel each other out. So square each difference to get exclusively positive numbers.

      • In our example:
        (x i (\displaystyle x_(i)) - μ) 2 (\displaystyle ^(2)) for each population value (from i = 1 to i = 6):
        (-5,5)2 (\displaystyle ^(2)) = 30,25
        (-5,5)2 (\displaystyle ^(2)), Where x n (\displaystyle x_(n))– the last value in the population.
      • To calculate the average value of the results obtained, you need to find their sum and divide it by n:(( x 1 (\displaystyle x_(1)) - μ) 2 (\displaystyle ^(2)) + (x 2 (\displaystyle x_(2)) - μ) 2 (\displaystyle ^(2)) + ... + (x n (\displaystyle x_(n)) - μ) 2 (\displaystyle ^(2)))/n
      • Now let's write down the above explanation using variables: (∑( x i (\displaystyle x_(i)) - μ) 2 (\displaystyle ^(2))) / n and get a formula for calculating the population variance.

Dispersion

An indicator of data dispersion corresponding to the mean square deviation of these data from the arithmetic mean. Equal to the square of the standard deviation.

Dictionary of a practical psychologist. - M.: AST, Harvest. S. Yu. Golovin. 1998.

Dispersion

The degree of dispersion in a series of results. giving a clear idea of ​​the variability of these results. The higher the variance, the more results are scattered around the mean (rather than clustered around one central result).

Psychology. AND I. Dictionary reference / Transl. from English K. S. Tkachenko. - M.: FAIR PRESS. Mike Cordwell. 2000.

Synonyms:

See what “variance” is in other dictionaries:

dispersion- Scattering something. In mathematics, dispersion defines the deviation of quantities from the average value. The dispersion of white light leads to its decomposition into components. Sound dispersion causes it to spread out. Scattering of stored data across... ... Technical Translator's Guide

DISPERSION Modern encyclopedia

DISPERSION- (variance) A measure of data dispersion. The variance of a set of N members is found by adding the squares of their deviations from the mean and dividing by N. Therefore, if the members are xi for i = 1, 2,..., N, and their mean is m, the variance... ... Economic dictionary

Dispersion- (from the Latin dispersio scattering) of waves, the dependence of the speed of propagation of waves in a substance on the wavelength (frequency). Dispersion is determined by the physical properties of the medium in which the waves propagate. For example, in a vacuum... ... Illustrated Encyclopedic Dictionary

DISPERSION- (from Latin dispersio scattering) in mathematical statistics and probability theory, a measure of dispersion (deviation from the average). In statistics, dispersion is the arithmetic mean of the squared deviations of the observed values ​​(x1, x2,...,xn) of a random... ... Big Encyclopedic Dictionary

Dispersion- in probability theory, the most commonly used measure of deviation from the mean (dispersion measure). In English: Dispersion Synonyms: Statistical dispersion English synonyms: Statistical dispersion See also: Sample populations Financial... ... Financial Dictionary

DISPERSION- [lat. dispersus scattered, scattered] 1) scattering; 2) chemistry, physics. breaking up a substance into very small particles. D. light decomposition of white light into a spectrum using a prism; 3) mat. deviation from the average. Dictionary of foreign words. Komlev N.G.,... ... Dictionary of foreign words of the Russian language

dispersion- scattering, scattering Dictionary of Russian synonyms. dispersion noun, number of synonyms: 6 nanodispersion (1) ... Synonym dictionary

Dispersion- characteristic of the dispersion of the values ​​of a random variable, measured by the square of their deviations from the average value (denoted by d2). D. differs between theoretical (continuous or discrete) and empirical (also continuous and... ... Economic and mathematical dictionary

Dispersion- * dispersion * dispersion 1. Dispersion; scatter; variation (see). 2. A theoretical probability concept that characterizes the measure of deviation of a random variable from its mathematical expectation. In biometric practice, sample variance s2 ... Genetics. encyclopedic Dictionary

Books

• Anomalous dispersion in broad absorption bands, D.S. Christmas. Reproduced in the original author's spelling of the 1934 edition (publishing house 'Izvestia of the USSR Academy of Sciences'). IN…

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

and perform a check, namely, find the expectation:

Dispersion in statistics is found as the individual values ​​of the characteristic squared from . Depending on the initial data, it is determined using the simple and weighted variance formulas:

1. (for ungrouped data) is calculated using the formula:

2. Weighted variance (for variation series):

where n is frequency (repeatability of factor X)

An example of finding variance

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

Example 1. 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 is 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 dispersion formula can be transformed as follows:

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

(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.

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 average of the within-group variances reflects random, i.e., that part of the variation that occurred under the influence of all other factors, with the exception of the grouping factor. It is calculated using the formula:

Characterizes the systematic variation of the resulting characteristic, which is due to the influence of the factor-sign that forms the basis of the group. It is equal to the mean square of the deviations of the group means from the overall mean. Intergroup variance is calculated using the formula:

The rule for adding variance in statistics

According to rule of adding variances the total variance is equal to the sum of the average of the within-group and between-group variances:

The meaning of this rule is that the total variance that arises under the influence of all factors is equal to the sum of the variances that arise under the influence of all other factors and the variance that arises due to the grouping factor.

Using the formula for adding variances, you can determine the third unknown variance from two known variances, and also judge the strength of the influence of the grouping characteristic.

Dispersion properties

1. If all values ​​of a characteristic are reduced (increased) by the same constant amount, then the dispersion will not change.
2. If all values ​​of a characteristic are reduced (increased) by the same number of times n, then the variance will correspondingly decrease (increase) by n^2 times.