Estimating variance from experimental data
For manual calculations, the variance can be estimated using formulas (1.3) and (1.5)  see paragraph 1.2. We need to subtract from each available value of the variable being investigated its arithmetic mean, build each difference into a square and add the resulting values.
In Table. 1.3. The necessary calculations for a group of women are reflected. Similar actions need to be done for a group of men. The amount received during the calculation must be divided by the number of subjects n or n  1, depending on whether we want whether we calculate the sample variance or estimate the variance of the population.
Table 1.3
Calculation of variance on the femininity scale  masculinity for a group of women
X 


21 
8.3 
68.89 
22 
7.3 
53.29 
22 
7.3 
53.29 
23 
6.3 
39.69 
23 
6.3 
39.69 
23 
4.3 
18.49 
27 
2.3 
5.29 
28 
1.3 
1.69 
30 
0.7 
0.49 
31 
1.7 
2.89 
31 
1.7 
2.89 
32 
2.7 
7.29 
32 
2.7 
7.29 
33 
3.7 
13.69 
33 
3.7 
13.69 
33 
3.7 
13.69 
34 
4.7 
22.09 
35 
5.7 
32.49 
35 
5.7 
32.49 
36 
6.7 
44.89 
Amount 
474.2 
The results of the calculations are presented in Table. 1.4. To obtain standard deviation values, it is necessary to extract the square root of each obtained value.
Table 1.4
Assessment of variance by sample and population for women and men groups
Dispersion 
Men 
Women 
For the general population 
27.75 
24.96 
For sample 
26.36 
23.71 
It should be noted that in practice the use of formulas (1.3) and (1.5) in calculations by hand or with the help of a simple pocket or desktop calculator is not very convenient. Therefore, we convert these formulas to the following form:
(1.8)
(19)
Now, to calculate the variance, it is sufficient to calculate the sums of the available values of the investigated variables, as well as the sum of the squares of these values (Table 1.5). Substituting the calculated sums into formulas (1.8) and (1.9), we obtain the variance values for the sample and the general population, indicated in Table. 1.4.
Table 1.5
Calculating variance by formulas (1.6) and (1.7)
Women ( X ) 
X 2 
Men ( Y ) 
2 
21 
441 
29 
841 
22 
484 
30 
900 
22 
484 
30 
900 
23 
529 
30 
900 
23 
529 
31 
961 
25 
625 
31 
961 
27 
729 
31 
961 
28 
784 
33 
1089 
30 
900 
35 
1225 
31 
961 
36 
1296 
31 
961 
36 
1296 
32 
1024 
39 
1521 
32 
1024 
39 
1521 
33 
1089 
40 
1600 
33 
1089 
41 
1681 
33 
1089 
41 
1681 
34 
1156 
41 
1681 
35 
1225 
43 
1849 
35 
1225 
44 
1936 
36 
1296 
44 
1936 
Σ X = 586 
Σ X 2 = 7644 
ΣΥ = 724 
Σ Υ 2 = 26,736 
Finding the distribution quartiles and, correspondingly, the semiinterquartile interval in manual calculations turns out to be similar to calculating the median. Only now it is necessary to find those values of our sample of data that cut off the first and last quarter of the distribution. So, if in the calculation of the median in our case we took the result of the subject with a conditional grade of 10.5, i.e. half of the whole sample + 0.5, then now to calculate the first quartile it is necessary to take half of the value + 0.5, i.e. the result of the subject with a conditional grade of 5.75. It can be seen that in the group of men the subjects who were in fifth and sixth places have the same result  31 points. This is the first quartile of the distribution. In the group of women, the subject who was in fifth place has a score of 23 points, and the subject, who was in sixth place, has 25 points. Consequently, the desired quartile is 24.5 points. Similarly, we calculate the third quartile. Its value is equal to 41 and 33 points for samples of men and women, respectively. Thus, the halfquartile interval in the group of men turns out to be equal to 5 points, and in the group of women  to 4,25 points.
However, it is more convenient and practical to use a computer to calculate variance, standard deviation and quartiles. The advantages of such processing become especially obvious if the task is to process a large amount of data on a set of variables.
In spreadsheets, for calculating variance, standard deviation, and quartiles, there are builtin functions that return the required values for both the sample and the population. For example, in MS Excel spreadsheets, the VAR and VARP functions serve to calculate the variance, and the STDEV and STDEV functions are used to calculate the standard deviation. To calculate the quartiles, use the QUARTILE function, which requires the arguments of the array of data to be examined and the number of the quartile to be calculated.
Computing variance and quartiles using the IBM statistical package SPSS Statistics differs little from the above estimate of expectation. Thus, the prepared data file is analyzed using the same descriptive statistics module.
To calculate the variance and standard deviation, you must select the same menu item in the descriptive statistics module, Frequencies and after selecting the variables for analysis, click the "Statistics" button. In the newly appeared window, this time, select Standard deviation & quot ;, Dispersion and, if necessary, "Quartiles" (see Figure 1.6).
Estimation of the anormal distribution
The abnormal distribution parameters, such as asymmetry and kurtosis, are generally tedious to calculate manually. Therefore, to calculate such parameters it is recommended to use computer technology.
To calculate the asymmetry of the distribution of a random variable in a sample and the population in the spreadsheet Excel , the functions of the SKOS and SKSO.G, and to calculate excesses  the function EXCESS.
The calculation of asymmetry and kurtosis in the statistical package IBM SPSS Statistics does not differ from the calculation of other distribution parameters. In the already familiar window "Frequencies: Statistics mark the parameters you need for us Excess and Asymmetry (see Figure 1.6).
In Table. 1.6 reflects the results of calculations of asymmetry and kurtosis of the data in the men and women groups. Obviously, although the results differ from zero values (except for the asymmetry of the distribution of male data), they are generally within the standard error of measurement. Given the small number of data, it can be assumed that the parameters under investigation are in fact described by the law of normal distribution, which is not surprising given the fact that the scales of questionnaires are usually standardized according to this law of distribution of a random variable.
Table 1.6
Evaluation of asymmetry and excess of femininity distribution  masculinity in groups of men and women
Options abnormal distribution 
Men 
Women 
Asymmetry 
0.06 
0.45 
Excess 
1.58 
1.31 
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