The variance icon. Calculation of group, intergroup and total variance (according to the rule of adding variances)
Where σ 2 j is the intra-group variance of the j -th group.
For ungrouped data residual dispersion is a measure of the approximation accuracy, i.e. approximation of the regression line to the original data: 
where y(t) is the forecast according to the trend equation; y t – initial series of dynamics; n is the number of points; p is the number of coefficients of the regression equation (the number of explanatory variables).
In this example it is called unbiased estimate of variance.
Example #1. The distribution of workers of three enterprises of one association by tariff categories is characterized by the following data:
| Worker's wage category | Number of workers at the enterprise | ||
| enterprise 1 | enterprise 2 | enterprise 3 | |
| 1 | 50 | 20 | 40 |
| 2 | 100 | 80 | 60 |
| 3 | 150 | 150 | 200 |
| 4 | 350 | 300 | 400 |
| 5 | 200 | 150 | 250 |
| 6 | 150 | 100 | 150 |
Define:
1. dispersion for each enterprise (intragroup dispersion);
2. average of intragroup dispersions;
3. intergroup dispersion;
4. total variance.
Solution.
Before proceeding to solve the problem, it is necessary to find out which feature is effective and which is factorial. In the example under consideration, the effective attribute is "Tariff category", and the factor attribute is "Number (name) of the enterprise".
Then we have three groups (enterprises) for which it is necessary to calculate the group average and intragroup variances:
| Company | group average, | within-group variance, |
| 1 | 4 | 1,8 |
The average of the intragroup variances ( residual dispersion) calculated by the formula:
where you can calculate:
or:
then:
The total dispersion will be equal to: s 2 \u003d 1.6 + 0 \u003d 1.6.
The total variance can also be calculated using one of the following two formulas:
When solving practical problems, one often has to deal with a sign that takes only two alternative values. In this case, they are not talking about the weight of a particular value of a feature, but about its share in the aggregate. If the proportion of population units that have the trait under study is denoted by " R", and not possessing - through" q”, then the dispersion can be calculated by the formula:
s 2 = p×q
Example #2. Based on the data on the output of six workers of the brigade, determine the intergroup variance and evaluate the impact of the work shift on their labor productivity if the total variance is 12.2.
| No. of the working brigade | Working output, pcs. | |
| in the first shift | in 2nd shift | |
| 1 | 18 | 13 |
| 2 | 19 | 14 |
| 3 | 22 | 15 |
| 4 | 20 | 17 |
| 5 | 24 | 16 |
| 6 | 23 | 15 |
Solution. Initial data
| X | f1 | f2 | f 3 | f4 | f5 | f6 | Total |
| 1 | 18 | 19 | 22 | 20 | 24 | 23 | 126 |
| 2 | 13 | 14 | 15 | 17 | 16 | 15 | 90 |
| Total | 31 | 33 | 37 | 37 | 40 | 38 |
Then we have 6 groups for which it is necessary to calculate the group mean and intragroup variances.
1. Find the average values of each group.
2. Find the mean square of each group.
We summarize the results of the calculation in a table:
| Group number | Group average | Intragroup variance |
| 1 | 1.42 | 0.24 |
| 2 | 1.42 | 0.24 |
| 3 | 1.41 | 0.24 |
| 4 | 1.46 | 0.25 |
| 5 | 1.4 | 0.24 |
| 6 | 1.39 | 0.24 |
3. Intragroup variance characterizes the change (variation) of the studied (resulting) trait within the group under the influence of all factors, except for the factor underlying the grouping:
We calculate the average of the intragroup dispersions using the formula:
4. Intergroup variance characterizes the change (variation) of the studied (resulting) trait under the influence of a factor (factorial trait) underlying the grouping.
Intergroup dispersion is defined as:
where
Then
Total variance characterizes the change (variation) of the studied (resulting) trait under the influence of all factors (factorial traits) without exception. By the condition of the problem, it is equal to 12.2.
Empirical correlation relation measures how much of the total fluctuation of the resultant attribute is caused by the studied factor. This is the ratio of the factorial variance to total variance:
We determine the empirical correlation relation:
Relationships between features can be weak or strong (close). Their criteria are evaluated on the Chaddock scale:
0.1 0.3 0.5 0.7 0.9 In our example, the relationship between feature Y factor X is weak
Determination coefficient.
Let's define the coefficient of determination:
Thus, 0.67% of the variation is due to differences between the traits, and 99.37% is due to other factors.
Conclusion: in this case, the output of workers does not depend on work in a particular shift, i.e. the influence of the work shift on their labor productivity is not significant and is due to other factors.
Example #3. Based on average wages and squared deviations from its value for two groups of workers, find the total variance by applying the rule for adding variances:
Solution:Average of within-group variances
Intergroup dispersion is defined as:
The total variance will be: 480 + 13824 = 14304
Let's calculate inMSEXCELdispersion and standard deviation samples. We also calculate the variance of a random variable if its distribution is known.
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.
It can be seen from the first formula that sample variance is the sum of the squared deviations of each value in the array from average divided by the sample size minus 1.
dispersion samples the DISP() function is used, eng. the name of the VAR, i.e. VARIance. Since MS EXCEL 2010, it is recommended to use its analogue DISP.V() , eng. the name VARS, i.e. Sample Variance. In addition, starting from the version of MS EXCEL 2010, there is a DISP.G () function, eng. VARP name, 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. Prior to MS EXCEL 2010, the VARP() function was used to calculate the population variance.
Sample variance
=SQUARE(Sample)/(COUNT(Sample)-1)
=(SUMSQ(Sample)-COUNT(Sample)*AVERAGE(Sample)^2)/ (COUNT(Sample)-1)- the 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, are equal mean value. Usually, the larger the value dispersion, the greater the spread of values in the array.
Sample variance is a point estimate dispersion distribution of the random variable from which the sample. About building confidence intervals when evaluating dispersion can be read in the article.
Variance of a random variable
To calculate dispersion random variable, you need to know it.
For dispersion random variable X often use the notation Var(X). Dispersion is 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 the random variable can take, and μ is the average value (), р(x) is the probability that the random variable will take the value x.
If the random variable has , then dispersion calculated by the formula:
Dimension dispersion corresponds to the square of the unit of measurement of the original values. For example, if the values in the sample are measurements of the weight of the part (in kg), then the dimension of the variance would be kg 2 . This can be difficult to interpret, therefore, to characterize the spread of values, a value equal to the square root of dispersion – standard deviation.
Some properties dispersion:
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[(XE(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(Х;Y) - covariance of these random variables.
If random variables are independent, then their covariance is 0, and hence Var(X+Y)=Var(X)+Var(Y). This property of the variance is used in the output.
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 property of the variance is used to plot .
Sample standard deviation
Sample standard deviation is a measure of how widely scattered the values in the sample are relative to their .
By definition, standard deviation equals the square root of dispersion:
Standard deviation does not take into account the magnitude of the values in sampling, but only the degree of scattering of values around them middle. Let's take an example to illustrate this.
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 values of the array is significantly different for the samples. For such cases, use 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 =STDEV() is used, eng. the name STDEV, i.e. standard deviation. Since MS EXCEL 2010, it is recommended to use its analogue = STDEV.B () , eng. name STDEV.S, i.e. Sample STandard DEViation.
In addition, starting from the version of MS EXCEL 2010, there is a function STDEV.G () , eng. 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 like STDEV.V() , STDEV.G() has just n in the denominator.
Standard deviation can also be calculated directly from the formulas below (see example file)
=SQRT(SQUADROTIV(Sample)/(COUNT(Sample)-1))
=SQRT((SUMSQ(Sample)-COUNT(Sample)*AVERAGE(Sample)^2)/(COUNT(Sample)-1))
Other dispersion measures
The SQUADRIVE() function calculates with umm of squared deviations of values from their middle. This function will return the same result as the formula =VAR.G( Sample)*CHECK( Sample) , where Sample- a reference to a range containing an array of sample values (). Calculations in the QUADROTIV() function are made according to the formula:
The SROOT() function is also a measure of the scatter of a set of data. The SIROTL() function calculates the average of the absolute values of the deviations of values from middle. This function will return the same result as the formula =SUMPRODUCT(ABS(Sample-AVERAGE(Sample)))/COUNT(Sample), where Sample- a reference to a range containing an array of sample values.
Calculations in the function SROOTKL () are made according to the formula:
Variation range (or range of variation) - is the difference between the maximum and minimum values of the feature:
In our example, the range of variation in shift output of workers is: in the first brigade R=105-95=10 children, in the second brigade R=125-75=50 children. (5 times more). This suggests that the output of the 1st brigade is more “stable”, but the second brigade has more reserves for the growth of output, because. if all workers reach the maximum output for this brigade, it can produce 3 * 125 = 375 parts, and in the 1st brigade only 105 * 3 = 315 parts.
If the extreme values of the attribute are not typical for the population, then quartile or decile ranges are used. The quartile range RQ= Q3-Q1 covers 50% of the population, the first decile range RD1 = D9-D1 covers 80% of the data, the second decile range RD2= D8-D2 covers 60%.
The disadvantage of the indicator variation range is, but that its value does not reflect all the fluctuations of the attribute.
The simplest generalizing indicator that reflects all the fluctuations of a trait is mean linear deviation, which is the arithmetic mean of the absolute deviations of individual options from their average value:
,
for grouped data 
,
where хi is the value of the feature in discrete series or the middle of an interval in an interval distribution.
In the above formulas, the differences in the numerator are taken modulo, otherwise, according to the property of the arithmetic mean, the numerator will always be zero. Therefore, the average linear deviation is rarely used in statistical practice, only in those cases where summing the indicators without taking into account the sign makes economic sense. With its help, for example, the composition of employees, the profitability of production, and foreign trade turnover are analyzed.
Feature variance is the average square of the deviations of the variant from their average value:
simple variance 
,
weighted variance 
.
The formula for calculating the variance can be simplified:
Thus, the variance is equal to the difference between the mean of the squares of the variant and the square of the mean of the variant of the population: 
.
However, due to the summation of the squared deviations, the variance gives a distorted idea of the deviations, so the average is calculated from it. standard deviation
, which shows how much the specific variants of the attribute deviate on average from their average value. Calculated by taking the square root of the variance:
for ungrouped data 
,
for the variation series 
The smaller the value of the variance and the standard deviation, the more homogeneous the population, the more reliable (typical) the average value will be.
Linear Mean and Mean standard deviation- named numbers, that is, they are expressed in units of measurement of the attribute, are identical in content and close in meaning.
count absolute indicators variations are recommended using tables.
Table 3 - Calculation of the characteristics of variation (on the example of the period of data on the shift output of the work teams)
Number of workers |
The middle of the interval |
Estimated values |
|||||
Total: |
|||||||
Average shift output of workers: 
Average linear deviation: 
Output dispersion: 
The standard deviation of the output of individual workers from the average output:
.
1 Calculation of dispersion by the method of moments
The calculation of variances is associated with cumbersome calculations (especially if the average value is expressed a large number with multiple decimal places). Calculations can be simplified by using a simplified formula and dispersion properties.
The dispersion has the following properties:
- if all the values of the attribute are reduced or increased by the same value A, then the variance will not decrease from this:
,
, then or 
Using the properties of the variance and first reducing all variants of the population by the value A, and then dividing by the value of the interval h, we obtain a formula for calculating the variance in variational series with at equal intervals way of moments:
,
where is the dispersion calculated by the method of moments;
h is the value of the interval of the variation series;
– new (transformed) variant values;
A is a constant value, which is used as the middle of the interval with the highest frequency; or the variant with the highest frequency; 
is the square of the moment of the first order;
is a moment of the second order.
Let's calculate the variance by the method of moments based on the data on the shift output of the working team.
Table 4 - Calculation of dispersion by the method of moments
Groups of production workers, pcs. |
Number of workers |
The middle of the interval |
Estimated values |
||
Calculation procedure:
- calculate the variance:
2 Calculation of the variance of an alternative feature
Among the signs studied by statistics, there are those that have only two mutually exclusive meanings. These are alternative signs. They are given two quantitative values, respectively: options 1 and 0. The frequency of options 1, which is denoted by p, is the proportion of units that have this feature. The difference 1-p=q is the frequency of options 0. Thus,
xi |
|
Arithmetic mean of alternative feature 
, since p+q=1.
Feature variance
, because 1-p=q
Thus, the variance of an alternative feature is equal to the product of the proportion of units that have the given trait and the proportion of units that do not have that trait.
If the values 1 and 0 are equally frequent, i.e. p=q, the variance reaches its maximum pq=0.25.
Variance variable is used in sample surveys, for example, product quality.
3 Intergroup dispersion. Variance addition rule
Dispersion, unlike other characteristics of variation, is an additive quantity. That is, in the aggregate, which is divided into groups according to the factor criterion X , resultant variance y can be decomposed into variance within each group (within group) and variance between groups (between group). Then, along with the study of the variation of the trait throughout the population as a whole, it becomes possible to study the variation in each group, as well as between these groups.
Total variance measures the variation of a trait at over the entire population under the influence of all the factors that caused this variation (deviations). It is equal to the mean square of the deviations of the individual values of the feature at of the overall mean and can be calculated as simple or weighted variance.
Intergroup variance characterizes the variation of the effective feature at, caused by the influence of the sign-factor X underlying the grouping. It characterizes the variation of the group means and is equal to the mean square of the deviations of the group means from the total mean: 
,
where is the arithmetic mean of the i-th group;
– number of units in the i-th group (frequency of the i-th group);
is the total mean of the population.
Intragroup variance reflects random variation, i.e., that part of the variation that is caused by the influence of unaccounted for factors and does not depend on the attribute-factor underlying the grouping. It characterizes the variation of individual values relative to group averages, it is equal to the mean square of deviations of individual values of the trait at within a group from the arithmetic mean of this group (group mean) and is calculated as a simple or weighted variance for each group: 
or 
,
where is the number of units in the group.
Based on the intra-group variances for each group, it is possible to determine the overall average of the within-group variances:
.
The relationship between the three variances is called variance addition rules, according to which the total variance is equal to the sum intergroup variance and the average of the intragroup dispersions:
Example. When studying the influence tariff category(qualification) of workers on the level of productivity of their labor, the following data were obtained.
Table 5 - Distribution of workers by average hourly output.
№ p/n |
Workers of the 4th category |
Workers of the 5th category |
|||||
Working out |
Working out |
||||||
1 |
7 |
7-10=-3 |
9 |
1 |
14 |
14-15=-1 |
1 |
V this example workers are divided into two groups according to a factor criterion X- qualifications, which are characterized by their rank. The effective trait - production - varies both under its influence (intergroup variation) and due to other random factors (intragroup variation). The challenge is to measure these variations using three variances: total, between-group, and within-group. The empirical coefficient of determination shows the proportion of the variation of the resulting feature at under the influence of a factor sign X. The rest of the total variation at caused by changes in other factors.
In the example, the empirical coefficient of determination is: 
or 66.7%,
This means that 66.7% of the variation in labor productivity of workers is due to differences in qualifications, and 33.3% is due to the influence of other factors.
Empirical correlation relation shows the tightness of the relationship between the grouping and effective features. It is calculated as the square root of the empirical coefficient of determination:
The empirical correlation ratio , as well as , can take values from 0 to 1.
If there is no connection, then =0. In this case, =0, that is, the group means are equal to each other and there is no intergroup variation. This means that the grouping sign - the factor does not affect the formation of the general variation.
If the relationship is functional, then =1. In this case, the variance of the group means is equal to the total variance (), i.e., there is no intragroup variation. This means that the grouping feature completely determines the variation of the resulting feature being studied.
How closer meaning correlation relationship to the unit, the closer, closer to the functional dependence of the relationship between the features.
For a qualitative assessment of the closeness of the connection between the signs, the Chaddock relations are used.
In the example 
, which indicates a close relationship between the productivity of workers and their qualifications.
Often in statistics, when analyzing a phenomenon or process, it is necessary to take into account not only information about the average levels of the studied indicators, but also scatter or variation in the values of individual units , which is an important characteristic of the studied population.
Stock prices, volumes of supply and demand, interest rates in different periods of time and in different places are subject to the greatest variation.
The main indicators characterizing the variation , are the range, variance, standard deviation and coefficient of variation.
Span variation is the difference between the maximum and minimum values of the attribute: R = Xmax – Xmin. disadvantage this indicator is that it evaluates only the boundaries of the variation of a feature and does not reflect its fluctuation within these boundaries.
Dispersion devoid of this shortcoming. It is calculated as the average square of deviations of the attribute values from their average value:
Simplified way to calculate variance is carried out using the following formulas (simple and weighted):
Examples of the application of these formulas are presented in tasks 1 and 2.
A widely used indicator in practice is standard deviation :
The standard deviation is defined as Square root from the variance and has the same dimension as the trait under study.
The considered indicators allow to obtain absolute value variations, i.e. evaluate it in units of measure of the trait under study. Unlike them, the coefficient of variation measures fluctuation in relative terms - relative to the average level, which in many cases is preferable.
Formula for calculating the coefficient of variation.
Examples of solving problems on the topic "Indicators of variation in statistics"
Task 1 . When studying the influence of advertising on the size of the average monthly deposit in the banks of the district, 2 banks were examined. The following results are obtained:
Define:
1) for each bank: a) average monthly deposit; b) dispersion of the contribution;
2) the average monthly deposit for two banks together;
3) Dispersion of the deposit for 2 banks, depending on advertising;
4) Dispersion of the deposit for 2 banks, depending on all factors except advertising;
5) Total variance using the addition rule;
6) Coefficient of determination;
7) Correlation relation.
Solution
1) Let's make a calculation table for a bank with advertising . To determine the average monthly deposit, we find the midpoints of the intervals. In this case, the value of the open interval (the first one) is conditionally equated to the value of the interval adjacent to it (the second one).
We find the average size of the contribution using the weighted arithmetic mean formula:
29,000/50 = 580 rubles
The dispersion of the contribution is found by the formula:
23 400/50 = 468
We will perform similar actions for a bank without ads :
2) Find the average deposit for two banks together. Xav \u003d (580 × 50 + 542.8 × 50) / 100 \u003d 561.4 rubles.
3) The variance of the deposit, for two banks, depending on advertising, we will find by the formula: σ 2 =pq (formula of the variance of an alternative sign). Here p=0.5 is the proportion of factors that depend on advertising; q=1-0.5, then σ 2 =0.5*0.5=0.25.
4) Since the share of other factors is 0.5, then the variance of the deposit for two banks, which depends on all factors except advertising, is also 0.25.
5) Determine the total variance using the addition rule.
= (468*50+636,16*50)/100=552,08
= [(580-561,4)250+(542,8-561,4)250] / 100= 34 596/ 100=345,96
σ 2 \u003d σ 2 fact + σ 2 rest \u003d 552.08 + 345.96 \u003d 898.04
6) Coefficient of determination η 2 = σ 2 fact / σ 2 = 345.96/898.04 = 0.39 = 39% - the size of the contribution depends on advertising by 39%.
7) Empirical correlation ratio η = √η 2 = √0.39 = 0.62 - the relationship is quite close.
Task 2 . There is a grouping of enterprises according to the value of marketable products:
Determine: 1) the dispersion of the value of marketable products; 2) standard deviation; 3) coefficient of variation.
Solution
1) By condition, an interval distribution series is presented. It must be expressed discretely, that is, find the middle of the interval (x "). In groups of closed intervals, we find the middle by a simple arithmetic mean. In groups with an upper limit, as the difference between this upper limit and half the size of the interval following it (200-(400 -200):2=100).
In groups with a lower limit - the sum of this lower limit and half the size of the previous interval (800+(800-600):2=900).
The calculation of the average value of marketable products is done according to the formula:
Хср = k×((Σ((x"-a):k)×f):Σf)+a. Here a=500 is the size of the variant at the highest frequency, k=600-400=200 is the size of the interval at the highest frequency Let's put the result in a table:
So, the average value of marketable output for the period under study as a whole is Xav = (-5:37) × 200 + 500 = 472.97 thousand rubles.
2) We find the dispersion using the following formula:
σ 2 \u003d (33/37) * 2002-(472.97-500) 2 \u003d 35,675.67-730.62 \u003d 34,945.05
3) standard deviation: σ = ±√σ 2 = ±√34 945.05 ≈ ±186.94 thousand rubles.
4) coefficient of variation: V \u003d (σ / Xav) * 100 \u003d (186.94 / 472.97) * 100 \u003d 39.52%
According to the sample survey, depositors were grouped according to the size of the deposit in the Sberbank of the city:
Define:
1) range of variation;
2) average deposit amount;
3) average linear deviation;
4) dispersion;
5) standard deviation;
6) coefficient of variation of contributions.
Solution:
This distribution series contains open intervals. In such series, the value of the interval of the first group is conventionally assumed to be equal to the value of the interval of the next, and the value of the interval of the last group is equal to the value of the interval of the previous one.
The interval value of the second group is 200, therefore, the value of the first group is also 200. The interval value of the penultimate group is 200, which means that the last interval will also have a value equal to 200.
1) Define the range of variation as the difference between the largest and smallest value of the attribute:
The range of variation in the size of the contribution is 1000 rubles.
2) The average size of the contribution is determined by the formula of the arithmetic weighted average.
Let us preliminarily determine the discrete value of the attribute in each interval. To do this, using the simple arithmetic mean formula, we find the midpoints of the intervals.
The average value of the first interval will be equal to:
the second - 500, etc.
Let's put the results of calculations in the table:
| Deposit amount, rub. | Number of contributors, f | The middle of the interval, x | xf |
|---|---|---|---|
| 200-400 | 32 | 300 | 9600 |
| 400-600 | 56 | 500 | 28000 |
| 600-800 | 120 | 700 | 84000 |
| 800-1000 | 104 | 900 | 93600 |
| 1000-1200 | 88 | 1100 | 96800 |
| Total | 400 | - | 312000 |
The average deposit in the city's Sberbank will be 780 rubles:
3) The average linear deviation is the arithmetic average of the absolute deviations of the individual values of the attribute from the total average:
The procedure for calculating the average linear deviation in the interval distribution series is as follows:
1. The arithmetic weighted average is calculated, as shown in paragraph 2).
2. The absolute deviations of the variant from the mean are determined:
3. The obtained deviations are multiplied by the frequencies:
4. The sum of weighted deviations is found without taking into account the sign:
5. The sum of the weighted deviations is divided by the sum of the frequencies:
It is convenient to use the table of calculated data:
| Deposit amount, rub. | Number of contributors, f | The middle of the interval, x | |||
|---|---|---|---|---|---|
| 200-400 | 32 | 300 | -480 | 480 | 15360 |
| 400-600 | 56 | 500 | -280 | 280 | 15680 |
| 600-800 | 120 | 700 | -80 | 80 | 9600 |
| 800-1000 | 104 | 900 | 120 | 120 | 12480 |
| 1000-1200 | 88 | 1100 | 320 | 320 | 28160 |
| Total | 400 | - | - | - | 81280 |
The average linear deviation of the size of the deposit of Sberbank clients is 203.2 rubles.
4) Dispersion is the arithmetic mean of the squared deviations of each feature value from the arithmetic mean.
Calculation of dispersion in interval series distribution is made according to the formula:
The procedure for calculating the variance in this case is as follows:
1. Determine the arithmetic weighted average, as shown in paragraph 2).
2. Find deviations from the mean:
3. Squaring the deviation of each option from the mean:
4. Multiply squared deviations by weights (frequencies):
5. Summarize the received works:
6. The resulting amount is divided by the sum of the weights (frequencies):
Let's put the calculations in a table:
| Deposit amount, rub. | Number of contributors, f | The middle of the interval, x | |||
|---|---|---|---|---|---|
| 200-400 | 32 | 300 | -480 | 230400 | 7372800 |
| 400-600 | 56 | 500 | -280 | 78400 | 4390400 |
| 600-800 | 120 | 700 | -80 | 6400 | 768000 |
| 800-1000 | 104 | 900 | 120 | 14400 | 1497600 |
| 1000-1200 | 88 | 1100 | 320 | 102400 | 9011200 |
| Total | 400 | - | - | - | 23040000 |
