Limit error of the average value formula. Mean and marginal sampling errors

To characterize the reliability of sample indicators, a distinction is made between mean and marginal sampling errors, which are characteristic only of sample observations. These indicators reflect the difference between the sample and the corresponding general indicators.

Average sample error is determined primarily by the sample size and depends on the structure and degree of variation of the trait under study.

The meaning of the mean sampling error is as follows. The calculated values ​​of the sample fraction (w) and the sample mean () are by their nature random variables. They can take on different values ​​depending on which specific units of the general population fall into the sample. For example, if, when determining the average age of employees of an enterprise, more young people are included in one sample, and older workers are included in another, then the sample means and sampling errors will be different. Average sampling error is determined by the formula:

(27) or - resampling. (28)

Where: μ is the average sampling error;

σ is the standard deviation of a trait in the general population;

n is the sample size.

The error value μ shows how the mean value of the feature, established by the sample, differs from the true value of the feature in the general population.

It follows from the formula that the sampling error is directly proportional to the standard deviation and inversely proportional to the square root of the number of units in the sample. This means, for example, that the greater the spread of the values ​​of a feature in the general population, that is, the greater the dispersion, the larger the sample size should be if we want to trust the results of a sample survey. Conversely, with a small variance, one can limit oneself to a small number of sample populations. The sampling error will then be within acceptable limits.

Since the size of the general population N during the sampling decreases during non-repeated selection, an additional factor is included in the formula for calculating the average sampling error

(one- ). The formula for the mean sampling error takes the following form:

The average error is smaller for non-repetitive sampling, which makes it more widely used.

Practical conclusions require a characterization of the general population based on sample results. Sample means and proportions are applied to the general population, taking into account the limit of their possible error, and with a level of probability that guarantees it. Given a specific level of probability, the value of the normalized deviation is chosen and the marginal sampling error is determined.

Reliability (confidence probability) of estimation X by X* called probability γ , with which the inequality

׀Х-Х*׀< δ, (30)

where δ is the marginal sampling error characterizing the width of the interval in which the value of the studied parameter of the general population is found with probability γ.

Trusted name the interval (X* - δ; X* + δ) that covers the investigated parameter X (that is, the value of the parameter X is inside this interval) with a given reliability γ.

Usually, the reliability of the estimate is set in advance, and a number close to one is taken as γ: 0.95; 0.99 or 0.999.

The limiting error δ is related to the average error μ as follows: , (31)

where: t is the confidence factor, depending on the probability P, with which it can be argued that the marginal error δ will not exceed the t-fold average error μ (it is also called the critical points or quantiles of Student's distribution).

As follows from the ratio, the marginal error is directly proportional to the average sampling error and the confidence coefficient, which depends on the given level of estimation reliability.

From the formula for the average sampling error and the ratio of the marginal and average errors, we obtain:

Taking into account the confidence probability, this formula will take the form.

As is known, in statistics there are two ways of observing mass phenomena, depending on the completeness of the coverage of the object: continuous and non-continuous. A variation of discontinuous observation is selective observation.

Under selective observation is understood as a non-continuous observation, in which units of the studied population, selected randomly, are subjected to statistical examination (observation).

Selective observation sets itself the task of characterizing the entire population of units for the examined part, subject to all the rules and principles of statistical observation and scientifically organized work on the selection of units.

The set of units selected for the survey in statistics is usually called sample population , and the set of units from which the selection is made is called general population . The main characteristics of the general and sample population are presented in Table 1.

When conducting selective observation, systematic and random errors occur. Systematic errors arise due to violation of the rules for selecting units in the sample. By changing the selection rules, such errors can be eliminated.

Random errors arise due to the discontinuous nature of the survey. Otherwise, they are called representativeness (representativeness) errors. Random errors are divided into average and marginal sampling errors, which are determined both when calculating the feature and when calculating the share.

The average and limit errors are related by the following relation :Δ = tμ, where Δ is the marginal sampling error, μ is the average sampling error, t is the confidence factor determined depending on the level of probability. Table 2 shows some values ​​of t taken from probability theory.

The value of the average sampling error is calculated differentially depending on the selection method and sampling procedure. The main formulas for calculating sampling errors are presented in Table 3.

Indicator	Designation and formula
	Population	Sample population
Mean feature error for random resampling
Mean share error for random resampling
Limit error of a feature in case of random re-selection
Marginal Share Error in Random Reselection
Average error of a feature for random non-repetitive selection
Mean share error in random non-repetitive selection
Limit error of a feature with random non-repetitive selection
Marginal share error for random non-repetitive selection

The calculation of the average and marginal sampling errors allows you to determine the possible limits in which the characteristics of the general population will be .

For example, for a sample mean, such limits are set based on the following relationships:

Limits of the share of the trait in the general population p.

Examples of solving problems on the topic "Sampling observation in statistics"

Task 1 . There is information on the output of products (works, services) obtained on the basis of 10% sample observation of enterprises in the region:

Determine: 1) for the enterprises included in the sample: a) the average size of output per enterprise; b) dispersion of the volume of production; c) the share of enterprises with a production volume of more than 400 thousand rubles; 2) for the region as a whole, with a probability of 0.954, the limits within which one can expect: a) the average volume of production per enterprise; b) the share of enterprises with a production volume of more than 400 thousand rubles; 3) the total volume of output in the region.

Decision

To solve the problem, we expand the proposed table.

1) For enterprises included in the sample, the average size of output per enterprise

110800/400 = 277 thousand rubles

We calculate the dispersion of the volume of production in a simplified way σ 2 = 35640000/400 - 277 2 = 89100 - 76229 = 12371.

The number of enterprises whose production volume exceeds 400 thousand rubles. equals 36+12 = 48, and their share is equal to ω = 48:400 = 0.12 = 12%.

2) From the theory of probability it is known that with a probability P=0.954 the confidence factor t=2. Marginal sampling error

2√12371:400 = 11.12 thousand rubles

Let's set the boundaries of the general average: 277-11.12 ≤Xav ≤ 277+11.12; 265.88 ≤Xav ≤ 288.12

Marginal sampling error of the share of enterprises

2√0,12*0,88/400 = 0,03

Let's define the boundaries of the general share: 0.12-0.03≤ p ≤0.12+0.03; 0.09≤ p≤0.15

3) Since the considered group of enterprises is 10% of the total number of enterprises in the region, there are 4,000 enterprises in the region as a whole. Then the total volume of output in the region lies within 265.88×4000≤Q≤288.12×4000; 1063520 ≤ Q ≤ 1152480

Task 2 . According to the results of a control audit by the tax authorities of 400 business structures, 140 of them do not fully indicate the income subject to taxation in their tax returns. Determine in the general population (for the entire region) the share of business structures that hid part of their tax revenues with a probability of 0.954.

Decision

According to the condition of the problem, the number of units in the sample population is n=400, the number of units with the considered feature is m=140, the probability is P=0.954.

From the theory of probability it is known that with the probability P=0.954 the confidence factor t=2.

The proportion of units that have the indicated attribute is determined by the formula: p=w+∆p, where w = m/n=140/400=0.35=35%,
and the limit error of the feature ∆p is obtained from the formula: ∆p= t √w(1-w)/n = 2√0.35×0.65/400 ≈ 0.5 = 5%

Then p = 35±5%.

Answer : The share of business structures that hid part of their tax income with a probability of 0.954 is 35±5%.

The concept of selective observation.

Selective such an observation is called in which the characteristic of the entire set of units is given according to some of their parts, selected in a random order.

Reasons for using selective observation:

1. Saving material, labor, financial resources and time.

2. The selected observation often leads to an increase in the accuracy of the data, since a decrease in the number of observation units sharply reduces the errors in registering the values ​​of a sign (misprints, undercounting, double counting ...).

3. Selective observation is the only possible one if the observation is accompanied by complete or partial damage to the observed objects (quality of batches of eggs, tissue strength, etc.).

That part of the units that are selected for observation is usually called sample population or simply sampling, and the entire set of units from which the selection is made - general population.

The following system of designation of indicators for the selected and general population has been adopted.

Depending on the application of the selection technique, the sample is divided into serial (nested) and typological.

· When typological sampling, the general population is divided into types (groups, districts), and then a random selection of units from each type is made.

· At serial the sample is chosen not by units, but by certain series, groups, areas within which continuous observation is carried out.

There are two ways to select units in a sample:

- reselection

each unit in the sample is returned to the general population and has a chance to be re-sampled.

- non-repetitive selection

the selected unit is not returned to the population, and the remaining units are more likely to be included in the sample. Non-repetitive sampling gives more accurate results, but sometimes it cannot be done (consumer demand study).

The quality of the results of sampling depends on the extent to which the composition of the sample represents the general population, in other words, on how much the sample representative(representative). To ensure the representativeness of the sample, it is necessary to observe the principle of random selection of units.

Sampling error

The concept and types of sampling errors

Since the statistical population under study consists of units with varying characteristics, the composition of the sample population may differ to some extent from the composition of the general population.

The discrepancy between the characteristics of the sample and the general population is sampling error.

Types of sampling errors

The main task of the sampling method is to study random errors of representativeness.

Average sampling error

The random error of representativeness depends on the following facts (it is assumed that there are no registration errors):

1. The larger the sample size, ceteris paribus, the smaller the sampling error, i.e. sampling error is inversely proportional to its size.

2. The smaller the variation of the attribute, the smaller the sampling error. If the sign does not vary at all, and, consequently, the variance is zero, then there will be no sampling error, because any unit of the population will accurately characterize the entire population on this basis. Thus, the sampling error is directly proportional to the magnitude of the variance.

In mathematical statistics, it is proved that the value of the average error of a random resampling can be determined by the formula

However, it should be borne in mind that the magnitude of the dispersion in the general population s2 we do not know, because selective observation. We can only calculate the variance in the sample population S2. The ratio between the variances of the general and sample population is expressed by the formula:

(6.2)

If a n large, therefore

s2 = S2

And the formula for the average resampling error (6.1.) will take the form:

But here we have considered only the sampling error for the mean value of the feature of interest. There is also an indicator of the proportion of units with a feature of interest. The calculation of the error of this indicator has its own characteristics.

The variance for the characteristic share indicator is determined by the formula:

S 2 \u003d w (1-w) (6.4)

Then the average sampling error for the measure of the share of the feature will be equal to:

(6.5)

The proof of formulas (6.3) and (6.5) starts from the resampling scheme. Usually, the sample is organized in a non-repetitive way. Because with non-repetitive selection, the size of the general population N is abbreviated in the sampling code, then an additional factor is included in the sampling error formulas , and the formulas take the form:

(6.6)

(6.7)

Example 1. Let's determine how much the sample and general indicators differ according to the data of a 10% non-repeated sample of student performance.

Calculation of the non-re-sampling error for the mean:

n= 100 N= 1000

Find the sample variance using the formula:

Here, the value is not known, which can be found as an ordinary weighted average:

Thus,

Those. we can say that the average score of all students () is 3.65 ± 0.07

Now let's calculate the proportion of students in the general population studying for "4" and "5".

Based on the sample, we will find the proportion of students who received grades “4” and “5”.

(or 64%)

The calculation of the non-re-sampling error for the share is made according to the formula:

(or 4.5%)

Thus, the share of students enrolled in "4" and "5" in the general population ( P) is 0.64±0.045 (or 64%±4.5%).

Marginal sampling error

The fact that the general average and the general share will not go beyond certain limits can be asserted not with absolute certainty, but only with a certain degree of probability.

In mathematical statistics, it is proved that general characteristics deviate from the sample ones by the amount of sampling error (± m), only with a probability of 0.683. With regard to sample studies, this is understood to mean that the values ​​of the limits can be guaranteed only in 683 cases out of 1000. In the remaining 317 cases, the values ​​of these limits will be different.

The probability of judgment can be increased by expanding the limits of deviation by taking as a measure the average sampling error, increased by t once.

Those. with a certain degree of probability, we can assert that the deviations of the sample characteristics from the general ones will not exceed a certain value, which is called the marginal sampling error D (delta):

where t– confidence factor (error multiplicity factor), determined depending on the confidence level with which it is necessary to guarantee the results of a sample study.

In practice, tables are used where the probabilities are calculated for various values t. Let's take a look at some of them.

For example, if in our example we want to increase the probability of judgment to 0.954, then we take t= 2 and thus change the limits of deviations of the average score of all students and the proportion of students enrolled in "4" and "5".

That is, (6.9)

That is, (6.10)

During selective observation, it should be ensured accident unit selection. Each unit must have an equal opportunity to be selected with the others. This is what random sampling is based on.

To proper random sample refers to the selection of units from the entire general population (without preliminary dividing it into any groups) by drawing lots (mainly) or some other similar method, for example, using a table of random numbers. Random selection This selection is not random. The principle of randomness suggests that the inclusion or exclusion of an object from the sample cannot be influenced by any factor other than chance. An example actually random selection can serve as circulations of winnings: from the total number of issued tickets, a certain part of the numbers that account for winnings is randomly selected. Moreover, all numbers are provided with an equal opportunity to get into the sample. In this case, the number of units selected in the sample set is usually determined based on the accepted proportion of the sample.

Sample share is the ratio of the number of units of the sample population to the number of units of the general population:

So, with a 5% sample from a batch of parts in 1000 units. sample size P is 50 units, and with a 10% sample - 100 units. etc. With the correct scientific organization of sampling, representativeness errors can be reduced to minimum values, as a result, selective observation becomes sufficiently accurate.

Proper random selection "in its pure form" is rarely used in the practice of selective observation, but it is the starting point among all other types of selection, it contains and implements the basic principles of selective observation.

Let us consider some questions of the theory of the sampling method and the error formula for a simple random sample.

When applying the sampling method in statistics, two main types of generalizing indicators are usually used: the average value of a quantitative trait and the relative value of the alternative feature(the proportion or proportion of units in the statistical population that differ from all other units of this population only by the presence of the trait being studied).

Sample share (w), or frequency, is determined by the ratio of the number of units that have the characteristic under study t, to the total number of sampling units P:

For example, if out of 100 sample details ( n=100), 95 parts turned out to be standard (t=95), then the sample fraction

w=95/100=0,95 .

To characterize the reliability of sample indicators, there are middle and marginal sampling error.

Sampling error ? or, in other words, the representativeness error is the difference between the corresponding sample and general characteristics:

*

*

Sampling error is characteristic only of selective observations. The greater the value of this error, the more the sample indicators differ from the corresponding general indicators.

The sample mean and the sample share are inherently random variables, which can take on different values ​​depending on which units of the population were included in the sample. Therefore, sampling errors are also random variables and can take on different values. Therefore, determine the average of the possible errors - the average sample error.

What does it depend on mean sampling error? Subject to the principle of random selection, the average sampling error is determined primarily sample size: the larger the population, ceteris paribus, the smaller the average sampling error. Covering a sample survey with an increasing number of units of the general population, we more and more accurately characterize the entire population.

The mean sampling error also depends on degree of variation studied trait. The degree of variation, as you know, is characterized by dispersion? 2 or w(1-w)-- for an alternative feature. The smaller the variation of the feature, and hence the variance, the smaller the average sampling error, and vice versa. With zero dispersion (the attribute does not vary), the average sampling error is zero, i.e., any unit of the general population will accurately characterize the entire population according to this attribute.

The dependence of the average sampling error on its volume and the degree of variation of the attribute is reflected in the formulas that can be used to calculate the average sampling error under conditions of sample observation, when the general characteristics ( x, p) are unknown, and therefore, it is not possible to find the real sampling error directly from the formulas (form. 1), (form. 2).

W With random selection average errors theoretically calculated by the following formulas:

* for the average quantitative trait

* for share (alternative characteristic)

Since practically the variance of the attribute in the general population? 2 is not exactly known, in practice they use the value of the variance S 2 calculated for the sample population on the basis of the law of large numbers, according to which the sample population with a sufficiently large sample size accurately reproduces the characteristics of the general population.

Thus, calculation formulas middle sampling errors random resampling will be as follows:

* for the average quantitative trait

* for share (alternative characteristic)

However, the variance of the sample population is not equal to the variance of the general population, and therefore, the average sampling errors calculated using the formulas (form. 5) and (form. 6) will be approximate. But in the theory of probability it is proved that the general variance is expressed through the elective by the following relation:

As P/(n-1) for sufficiently large P -- value close to unity, it can be assumed that, and therefore, in practical calculations of the average sampling errors, formulas (form. 5) and (form. 6) can be used. And only in cases of a small sample (when the sample size does not exceed 30) it is necessary to take into account the coefficient P/(n-1) and calculate small sample mean error according to the formula:

W X With random non-repetitive selection in the above formulas for calculating the average sampling errors, it is necessary to multiply the root expression by 1-(n / N), since the number of units in the general population is reduced in the process of non-repetitive sampling. Therefore, for a non-repetitive selection calculation formulas mean sampling error will take the following form:

* for the average quantitative trait

* for share (alternative characteristic)

. (form. 10)

As P always less N, then the additional factor 1-( n/n) will always be less than one. It follows from this that the average error with non-repetitive selection will always be less than with repeated selection. At the same time, with a relatively small percentage of the sample, this factor is close to one (for example, with a 5% sample it is 0.95; with a 2% sample it is 0.98, etc.). Therefore, sometimes in practice, formulas (forms 5) and (forms 6) are used to determine the average sampling error without the specified multiplier, although the sample is organized as a non-repeating one. This occurs when the number of units of the general population N is unknown or unlimited, or when P very little compared to N, and in essence, the introduction of an additional factor, close in value to one, will practically not affect the value of the average sampling error.

Mechanical sampling consists in the fact that the selection of units in the sample from the general, divided by a neutral criterion into equal intervals (groups), is carried out in such a way that only one unit is selected from each such group in the sample. To avoid systematic error, the unit that is in the middle of each group should be selected.

When organizing mechanical selection, the units of the population are pre-arranged (usually in a list) in a certain order (for example, alphabetically, by location, in ascending or descending order of the values ​​of any indicator that is not associated with the property under study, etc.). etc.), after which a given number of units is selected mechanically, at a certain interval. In this case, the size of the interval in the general population is equal to the reciprocal of the sample share. So, with a 2% sample, every 50th unit (1: 0.02) is selected and checked, with a 5% sample, every 20th unit (1: 0.05), for example, descending detail from the machine.

With a sufficiently large population, the mechanical selection in terms of the accuracy of the results is close to proper random. Therefore, to determine the average error of a mechanical sample, the formulas for self-random non-repetitive sampling are used (form. 9), (form. 10).

To select units from a heterogeneous population, the so-called typical sample , which is used in cases where all units of the general population can be divided into several qualitatively homogeneous, similar groups according to the characteristics that affect the indicators under study.

When surveying enterprises, such groups can be, for example, industry and sub-sector, forms of ownership. Then, from each typical group, an individual selection of units into the sample is made by a random or mechanical sample.

A typical sample is usually used in the study of complex statistical populations. For example, in a sample survey of the family budgets of workers and employees in certain sectors of the economy, labor productivity of workers in an enterprise, represented by separate groups by qualification.

A typical sample gives more accurate results compared to other methods of selecting units in a sample set. Typification of the general population ensures the representativeness of such a sample, the representation of each typological group in it, which makes it possible to exclude the influence of intergroup dispersion on the average sample error.

When determining average error of a typical sample as an indicator of variation is the average of the intragroup variances.

The mean sampling error are found by the formulas:

* for the average quantitative trait

(reselection); (form. 11)

(irreversible selection); (form. 12)

* for share (alternative characteristic)

(reselection); (form.13)

(non-repetitive selection), (form. 14)

where is the average of the intra-group variances for the sample population;

The average of the intra-group variances of the share (alternative trait) in the sample population.

serial sampling involves random selection from the general population not of individual units, but of their equal groups (nests, series) in order to subject all units without exception to observation in such groups.

The use of serial sampling is due to the fact that many goods for their transportation, storage and sale are packed in packs, boxes, etc. Therefore, when controlling the quality of packaged goods, it is more rational to check several packages (series) than to select the required amount of goods from all packages.

Since within groups (series) all units without exception are examined, the average sampling error (when selecting equal series) depends only on the intergroup (interseries) variance.

W The mean sampling error for the mean score during serial selection, they are found by the formulas:

(reselection); (form.15)

(non-repetitive selection), (form. 16)

where r- number of selected series; R- total number of episodes.

The intergroup variance of the serial sample is calculated as follows:

where is the average i- th series; - the general average for the entire sample population.

W Average sampling error for share (alternative feature) in serial selection:

(reselection); (form. 17)

(non-repetitive selection). (form. 18)

Intergroup(inter-series) variance of the serial sample share determined by the formula:

, (form. 19)

where is the share of the feature in i th series; - the total share of the trait in the entire sample.

In the practice of statistical surveys, in addition to the previously considered selection methods, their combination is used (combined selection).

The concept of selective observation.

With the statistical method of observation, it is possible to use two methods of observation: continuous, covering all units of the population, and selective (non-continuous).

By sampling is meant a research method associated with the establishment of generalizing indicators of the population for some of its parts based on the method of random selection.

With selective observation, a relatively small part of the entire population (5-10%) is subjected to examination.

The totality to be examined is called general population.

The part of the units selected from the general population that is subject to the survey is called sample population or sample.

Indicators characterizing the general and sample population:

1) Share of an alternative sign;

AT population the proportion of units that have some alternative attribute is denoted by the letter "P".

AT sampling frame the proportion of units that have some alternative attribute is denoted by the letter "w".

2) The average size of the sign;

AT population the average size of a feature is denoted by a letter (general average).

AT sampling frame the average size of a feature is denoted by a letter (sample mean).

Definition of sampling error.

Selective observation is based on the principle of equal possibility of getting units of the general population into the sample. This avoids systematic observational errors. However, due to the fact that the population under study consists of units with varying characteristics, the composition of the sample may differ from the composition of the general population, causing discrepancies between the general and sample characteristics.

Such discrepancies are called representativeness errors or sampling errors.

Determining the sampling error is the main task to be solved during selective observation.

In mathematical statistics, it is proved that the average sampling error is determined by the formula:

Where m is the sampling error;

s 2 0 is the variance of the general population;

n is the number of sample units.

In practice, the sample population variance s 2 is used to determine the mean sampling error.

There is an equality between the general and sample variances:

(2).

It can be seen from formula (2) that the general variance is greater than the sample variance by the value (). However, for a sufficiently large sample size, this ratio is close to unity, so we can write that

However, this formula for determining the mean sampling error is only applicable to resampling.

In practice, it is usually used non-repetitive selection and the mean sampling error is calculated slightly differently, as the sample size shrinks over the course of the study:

(4)

where n is the size of the sample;

N is the size of the general population;

s 2 - sample variance.

For the proportion of an alternative feature, the average sampling error at no-reselection is determined by the formula:

(5), where

w (1-w) - the average error of the sample share of the alternative attribute;

w is the share of the alternative feature of the sample population.

At re-selection the average error of the share of an alternative attribute is determined by a simplified formula:

(6)

If a the sample size does not exceed 5%, the average error of the sample share and the sample mean is determined by simplified formulas (3) and (6).

Determination of the mean error of the sample mean and sample share is necessary to establish the possible values ​​of the general mean (x) and the general share (P) based on the sample mean (x) and sample share (w).

One of the possible values ​​within which the general average is located is determined by the formula:

For the general share, this interval can be written as :

(8)

The characteristics of the share and the mean in the general population obtained in this way differ from the value of the sample share and the sample mean by the value m. However, this can not be guaranteed with complete certainty, but only with a certain degree of probability.

In mathematical statistics, it is proved that the limits of the values ​​of the characteristics of the general and sample mean differ by m only with a probability of 0.683. Therefore, only in 683 cases out of 1000 the general average is within x= x m x, in other cases, it will go beyond these limits.

The probability of judgments can be increased by expanding the limits of deviations by taking as a measure the average sampling error, increased by t times.

The factor t is called the confidence factor. It is determined depending on the confidence level with which it is necessary to guarantee the results of the study.

Mathematician A.M. Lyapushev calculated various values ​​of t, which are usually given in ready-made tables.