Estimation of variance according to the experimental data, Estimation...

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 semi-interquartile 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 half-quartile 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 built-in 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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