Statistical test

The rule by which the hypothesis R 0 is rejected or accepted is called statistical criterion. The name of the criterion, as a rule, contains a letter that denotes a specially compiled characteristic from paragraph 2 of the verification algorithm statistical hypothesis(see clause 4.1) calculated in the criterion. Under the conditions of this algorithm, the criterion would be called "V-criterion".

When testing statistical hypotheses, two types of errors are possible:

• - error of the first kind(you can reject the hypothesis I 0 when it is actually true);
• - type II error(you can accept the hypothesis I 0 when it is actually not true).

Probability A make a type one error is called the significance level of the criterion.

If for R denote the probability of making a Type II error, then (l - R) - the probability of not making a type II error, which is called the power of the criterion.

Goodness-of-fit x 2 Pearson

There are several types of statistical hypotheses:

• - about the law of distribution;
• - homogeneity of samples;
• - numerical values ​​of distribution parameters, etc.

We will consider the hypothesis about the law of distribution on the example of Pearson's x 2 goodness-of-fit test.

Concordance criterion called a statistical test for testing the null hypothesis about the alleged law of the unknown distribution.

Pearson's goodness-of-fit test is based on a comparison of empirical (observed) and theoretical frequencies of observations calculated under the assumption of a certain distribution law. Hypothesis # 0 here is formulated as follows: the general population is normally distributed according to the criterion under study.

Statistical Hypothesis Testing Algorithm #0 for Criteria x 1 Pearson:

• 1) we put forward the hypothesis R 0 - according to the criterion under study, the general population is distributed normally;
• 2) calculate the sample mean and sample standard deviation O V;

3) according to the available sample volume P we calculate a specially compiled characteristic ,

where: i, - empirical frequencies, - theoretical frequencies,

P - sample size,

h- the value of the interval (the difference between two adjacent options),

Normalized values ​​of the observed feature,

- table function. Also theoretical frequencies

can be calculated using the standard MS Excel function NORMDIST according to the formula ;

4) according to the sampling distribution, we determine the critical value of a specially compiled characteristic XL P

5) when hypothesis # 0 is rejected, when hypothesis # 0 is accepted.

Example. Consider the sign X- the value of testing indicators for convicts in one of the correctional colonies according to some psychological characteristic, presented as a variation series:

At a significance level of 0.05, test the hypothesis of a normal distribution population.

1. Based on the empirical distribution, you can put forward a hypothesis H 0: according to the criterion under study "the value of the test indicator for a given psychological characteristic", the general population

the number of children is normally distributed. Alternative Hypothesis 1: according to the studied feature “the value of the test indicator for this psychological characteristic”, the general population of convicts is not normally distributed.

2. Calculate numerical sample characteristics:

Intervals			x y y		X) sch

3. Calculate a specially composed characteristic j 2 . To do this, in the penultimate column of the previous table, we find the theoretical frequencies using the formula, and in the last column

let's calculate the characteristic % 2 . We get x 2 = 0,185.

For clarity, we construct a polygon of the empirical distribution and a normal curve for theoretical frequencies (Fig. 6).

Rice. 6.

4. Determine the number of degrees of freedom s: k = 5, t = 2, s = 5-2-1 = 2.

According to the table or using the standard MS Excel function "XI20BR" for the number of degrees of freedom 5 = 2 and the significance level a = 0.05 find the critical value of the criterion xl P .=5,99. For significance level A= 0.01 critical value of the criterion X%. = 9,2.

5. Observed value of the criterion X=0.185 less than all found values Hc R.-> therefore, the hypothesis R 0 is accepted at both significance levels. The discrepancy between the empirical and theoretical frequencies is insignificant. Therefore, the observational data are consistent with the hypothesis of a normal population distribution. Thus, according to the studied feature “the value of the test indicator for this psychological characteristic”, the general population of convicts is distributed normally.

• 1. Koryachko A.V., Kulichenko A.G. higher mathematics and mathematical methods in psychology: a guide to practical exercises for students of the psychological faculty. Ryazan, 1994.
• 2. Nasledov A.D. Mathematical Methods psychological research. Analysis and interpretation of data: Textbook, manual. SPb., 2008.
• 3. Sidorenko E.V. Methods of mathematical processing in psychology. SPb., 2010.
• 4. Soshnikova L.A. and others. Multivariate statistical analysis in the economy: Textbook, manual for universities. M., 1999.
• 5. Sukhodolsky E.V. Mathematical methods in psychology. Kharkov, 2004.
• 6. Shmoylova R.A., Minashkin V.E., Sadovnikova N.A. Workshop on the theory of statistics: Textbook, manual. M., 2009.

• Gmurman V.E. Theory of Probability and Mathematical Statistics. S. 465.

The width of the interval will be:

Xmax - the maximum value of the grouping feature in the aggregate.
Xmin - the minimum value of the grouping feature.
Let's define the boundaries of the group.

Group number	Bottom line	Upper bound
1	43	45.83
2	45.83	48.66
3	48.66	51.49
4	51.49	54.32
5	54.32	57.15
6	57.15	60

The same feature value serves as the upper and lower boundaries of two adjacent (previous and subsequent) groups.
For each value of the series, we calculate how many times it falls into a particular interval. To do this, sort the series in ascending order.

43	43 - 45.83	1
48.5	45.83 - 48.66	1
49	48.66 - 51.49	1
49	48.66 - 51.49	2
49.5	48.66 - 51.49	3
50	48.66 - 51.49	4
50	48.66 - 51.49	5
50.5	48.66 - 51.49	6
51.5	51.49 - 54.32	1
51.5	51.49 - 54.32	2
52	51.49 - 54.32	3
52	51.49 - 54.32	4
52	51.49 - 54.32	5
52	51.49 - 54.32	6
52	51.49 - 54.32	7
52	51.49 - 54.32	8
52	51.49 - 54.32	9
52.5	51.49 - 54.32	10
52.5	51.49 - 54.32	11
53	51.49 - 54.32	12
53	51.49 - 54.32	13
53	51.49 - 54.32	14
53.5	51.49 - 54.32	15
54	51.49 - 54.32	16
54	51.49 - 54.32	17
54	51.49 - 54.32	18
54.5	54.32 - 57.15	1
54.5	54.32 - 57.15	2
55.5	54.32 - 57.15	3
57	54.32 - 57.15	4
57.5	57.15 - 59.98	1
57.5	57.15 - 59.98	2
58	57.15 - 59.98	3
58	57.15 - 59.98	4
58.5	57.15 - 59.98	5
60	57.15 - 59.98	6

The results of the grouping will be presented in the form of a table:

Groups	population number	Frequency f i
43 - 45.83	1	1
45.83 - 48.66	2	1
48.66 - 51.49	3,4,5,6,7,8	6
51.49 - 54.32	9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26	18
54.32 - 57.15	27,28,29,30	4
57.15 - 59.98	31,32,33,34,35,36	6

Table for calculating indicators.

Groups	x i	Quantity, fi	x i * f i	Cumulative frequency, S	|x - x cf |*f	(x - x sr) 2 *f	Frequency, f i /n
43 - 45.83	44.42	1	44.42	1	8.88	78.91	0.0278
45.83 - 48.66	47.25	1	47.25	2	6.05	36.64	0.0278
48.66 - 51.49	50.08	6	300.45	8	19.34	62.33	0.17
51.49 - 54.32	52.91	18	952.29	26	7.07	2.78	0.5
54.32 - 57.15	55.74	4	222.94	30	9.75	23.75	0.11
57.15 - 59.98	58.57	6	351.39	36	31.6	166.44	0.17
		36	1918.73		82.7	370.86	1

To evaluate the distribution series, we find the following indicators:
Distribution Center Metrics.
weighted average


Fashion
Mode is the most common value of a feature in units of a given population.

where x 0 is the beginning of the modal interval; h is the value of the interval; f 2 -frequency corresponding to the modal interval; f 1 - premodal frequency; f 3 - postmodal frequency.
We choose 51.49 as the beginning of the interval, since this interval accounts for the largest number.

The most common value of the series is 52.8
Median
The median divides the sample into two parts: half the option is less than the median, half is more.
IN interval series distribution, you can immediately specify only the interval in which the mode or median will be located. The median corresponds to the option in the middle of the range. The median is the interval 51.49 - 54.32, because in this interval, the accumulated frequency S is greater than the median number (the first interval is called the median, the accumulated frequency S of which exceeds half of the total sum of frequencies).


Thus, 50% of the population units will be less than 53.06
Variation indicators.
Absolute Variation Rates.
The range of variation is the difference between the maximum and minimum values ​​of the attribute of the primary series.
R = X max - X min
R = 60 - 43 = 17
Average linear deviation- calculated in order to take into account the differences of all units of the studied population.


Each value of the series differs from the other by no more than 2.3
Dispersion- characterizes the measure of spread around its mean value (measure of dispersion, i.e. deviation from the mean).


Unbiased estimator of variance is a consistent estimate of the variance.


Standard deviation.

Each value of the series differs from the average value of 53.3 by no more than 3.21
Estimating the standard deviation.

Relative measures of variation.
The relative indicators of variation include: oscillation coefficient, linear coefficient variations, relative linear deviation.
The coefficient of variation- a measure of the relative spread of population values: shows what proportion of the average value of this quantity is its average spread.

Since v ≤ 30%, the population is homogeneous and the variation is weak. The results obtained can be trusted.
Linear coefficient of variation or Relative linear deviation- characterizes the proportion of the average value of the sign of absolute deviations from the average value.

Testing hypotheses about the type of distribution.
1. Let's test the hypothesis that X is distributed over normal law using Pearson's goodness-of-fit test.

where p i is the probability of hitting i-th interval random variable, distributed according to the hypothetical law
To calculate the probabilities p i, we apply the formula and the table of the Laplace function

Where
s = 3.21, xav = 53.3
The theoretical (expected) frequency is n i = np i , where n = 36

Group intervals	Observed frequency n i	x 1 \u003d (x i - x cf) / s	x 2 \u003d (x i + 1 - x cf) / s	Ф(x 1)	Ф(x 2)	Probability of hitting the i-th interval, p i \u003d Ф (x 2) - Ф (x 1)	Expected frequency, 36p i	Terms of the Pearson statistics, K i
43 - 45.83	1	-3.16	-2.29	-0.5	-0.49	0.01	0.36	1.14
45.83 - 48.66	1	-2.29	-1.42	-0.49	-0.42	0.0657	2.37	0.79
48.66 - 51.49	6	-1.42	-0.56	-0.42	-0.21	0.21	7.61	0.34
51.49 - 54.32	18	-0.56	0.31	-0.21	0.13	0.34	12.16	2.8
54.32 - 57.15	4	0.31	1.18	0.13	0.38	0.26	9.27	3
57.15 - 59.98	6	1.18	2.06	0.38	0.48	0.0973	3.5	1.78
	36							9.84

Let us define the boundary of the critical region. Since the Pearson statistic measures the difference between the empirical and theoretical distributions, the larger its observed value of K obs, the stronger the argument against the main hypothesis.
Therefore, the critical region for this statistic is always right-handed:

Empirical Frequencies

ni

Probabilities
pi

Theoretical frequencies
npi

(ni-npi)2

Pearson's criterion

Pearson's criterion, or criterion χ 2- the most commonly used criterion for testing the hypothesis about the law of distribution. In many practical problems, the exact distribution law is unknown, that is, it is a hypothesis that requires statistical verification.

Denote by X the random variable under study. Let it be necessary to test the hypothesis H 0 that this random variable obeys the distribution law F(x) . To test the hypothesis, we will make a sample consisting of n independent observations on a random variable X. Using the sample, we can build an empirical distribution F * (x) of the random variable under study. Empirical comparison F * (x) and theoretical distributions are made using a specially selected random variable - the goodness of fit criterion. One of these criteria is the Pearson criterion.

Criterion statistics

To check the criterion, statistics are introduced:

Where - estimated probability of hitting i-th interval, - corresponding empirical value, n i- number of sample elements from i-th interval.

This value, in turn, is random (due to the randomness of X) and must obey the distribution χ 2 .

Criteria Rule

Before formulating a rule for accepting or rejecting a hypothesis, it is necessary to take into account that Pearson's criterion has a right-sided critical region.

Rule. If the resulting statistics exceeds the quantile of the distribution law of a given significance level with or with degrees of freedom , where k is the number of observations or the number of intervals (for the case of an interval variation series), and p is the number of estimated parameters of the distribution law , then the hypothesis is rejected. Otherwise, the hypothesis is accepted at the given significance level.

Literature

• Kendall M, Stuart A. Statistical inferences and connections. - M.: Nauka, 1973.

see also

• Pearson's criterion on the site of Novosibirsk State University
• Chi-square type criteria on the site of the Novosibirsk State Technical University (Recommendations for standardization R 50.1.033–2001)
• On the choice of the number of intervals on the site of the Novosibirsk State Technical University
• About the Nikulin criterion on the website of the Novosibirsk State Technical University

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Books

• Criteria for checking the deviation of the distribution from the uniform law. Application guide: monograph, Lemeshko B.Yu.