Average values are divided into two large classes: power means and structural means
Power averages:
Arithmetic
Harmonic
Geometric
Quadratic
A simple arithmetic mean is the average term, in determining which the total volume of a given characteristic in a set of data is equally distributed among all units included in this set. Thus, the average annual output per employee is the amount of output that would fall on each employee if the entire volume of output were equally distributed among all employees of the organization. The arithmetic mean simple value is calculated using the formula:
Simple arithmetic average- Equal to the ratio of the sum of individual values of a characteristic to the number of characteristics in the aggregate
Arithmetic average weighted
If the volume of the data set is large and represents a distribution series, then the weighted arithmetic mean is calculated. This is how the weighted average price per unit of production is determined: the total cost of production (the sum of the products of its quantity by the price of a unit of production) is divided by the total quantity of production.
Let's imagine this in the form of the following formula:
Weighted arithmetic average- is equal to the ratio of (the sum of the products of the value of a feature to the frequency of repetition of this feature) to (the sum of the frequencies of all features). It is used when variants of the population under study occur an unequal number of times.
Arithmetic mean for interval series
When calculating the arithmetic mean for an interval variation series, first determine the mean for each interval as the half-sum of the upper and lower limits, and then the mean of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the size of the intervals adjacent to them.
Averages calculated from interval series are approximate.
Averages calculated from interval series are approximate. The degree of their approximation depends on the extent to which the actual distribution of population units within the interval approaches uniform distribution.
When calculating averages, not only absolute but also relative values (frequency) can be used as weights:
Harmonic mean- used in cases where the individual values of the attribute and the product are known, but the frequencies are unknown.
In the example below - the yield is known, - the area is unknown (although it can be calculated by dividing the gross grain harvest by the yield), - the gross grain harvest is known.
The harmonic mean value can be determined using the following formula:
Harmonic mean formula:
Harmonic simple
In cases where the product is the same or equal to 1 (z = 1), the harmonic simple average is used for calculation, calculated using the formula:
The harmonic simple mean is an indicator that is the inverse of the arithmetic simple mean, calculated from the reciprocal values of the characteristic.
The geometric mean value makes it possible to preserve unchanged not the sum, but the product of the individual values of a given value. It can be determined by the following formula:
Geometric mean values are most often used when analyzing growth rates of economic indicators.
The harmonic mean is the arithmetic mean, calculated from the reciprocal values of the averaged characteristics. Depending on the nature of the available material, it is used when the weights have to be divided into options rather than multiplied, or, what is the same thing, multiplied by their inverse value. Thus, the harmonic mean is calculated when the volume characteristics are known (W=xf) and individual attribute values (x) and unknown weights (φ). Since feature volumes are the product of feature values (X) to the frequency f, then the frequency f is determined by removable = W: x.
The simple and weighted harmonic mean formulas are:
As you can see, the harmonic mean is a transformed form of the arithmetic mean. Instead of the harmonic mean, you can always calculate the arithmetic mean by first determining the weights of individual attribute values. When calculating the harmonic mean, the weights are the volumes of features.
The harmonic mean simple is used in cases where the volumes of phenomena for each attribute level.
For example, three combine operators are working to harvest grain crops. The first combine harvester spent 35 minutes harvesting 1 hectare during a 7-hour shift, the second - 31 minutes, the third - 33 minutes. It is necessary to determine the average labor cost for harvesting 1 hectare of grain crops.
Calculating the average time spent harvesting 1 hectare of grain crops using the simple arithmetic average formula would be correct
then, when all combine operators harvested 1 hectare or the same number of hectares of grain crops during a shift. However, during the shift, different areas of grain crops were harvested by individual combine operators.
The inappropriateness of using the arithmetic average formula is also explained by the fact that the indicator of labor costs per unit of work (harvesting 1 hectare of grain crops) is the inverse of the indicator of labor productivity (harvesting grain crops per unit of time).
The average time required to harvest 1 hectare of grain crops for all combiners will be determined as the ratio of the time spent by all combiners to the total number of hectares harvested. In our example, there is no information about the number of hectares actually harvested by each combine operator. However, these values can be calculated using the following relationship:
where the total time spent for each combine operator will be 420 minutes (7 years o 60 minutes).
Then the average time spent harvesting 1 hectare of grain crops can be determined by the formula:
Calculations can be greatly simplified if you use the harmonic mean prime formula:
So, for this set of combine operators, it takes an average of 32.9 minutes to harvest 1 hectare of grain crops.
We will consider the procedure for calculating the weighted harmonic average using the following example (Table 4.3).
Table 4.3. Data for calculating the weighted harmonic mean
Since the average yield is the ratio of the gross harvest to the area sown, we first determine the area sown with potatoes for each farm, and then the average yield:
According to one of the properties, the harmonic mean will not change if the volumes of phenomena, which are the weights of individual options, are multiplied or divided by any arbitrary number. This makes it possible to use not absolute indicators, but their specific weights when calculating it. Let’s say you need to determine the average selling price of potatoes using the following data (Table 4.4).
Table 4.4. Data for calculating the average selling price of potatoes
In the example given, there is no data on revenue from the sale of individual varieties of potatoes, which is the product of the sales price of 1 centner by the number of potatoes sold. Therefore, instead of the volumes of events, you can use their ratio, that is, the share of individual potato varieties in total revenue. Using the table data, we determine the average selling price of potatoes:
The harmonic mean is also used to determine the average yield for a group of homogeneous crops, if the gross harvest and yield of individual crops are known, to calculate the average percentage of implementation of the production plan and sales of products for a homogeneous population, if data on actually produced or sold products and the percentage of implementation of the plan are known. individual objects, etc.
Harmonic mean - is used when statistical information does not contain data on weights for individual variants of the population, but the products of the values of a varying characteristic by the corresponding weights are known.
The general formula for the weighted harmonic mean is as follows:
x – the value of the varying characteristic,
w – product of the value of a varying characteristic and its weight (xf)
In the event that the total volumes of phenomena, i.e. the products of feature values and their weights are equal, then the harmonic simple mean is applied:
x – individual values of the characteristic (options),
n – total number of options.
The harmonic mean is used for calculations when not the units of the population - the carriers of the characteristic - are used as weights, but the product of these units by the values of the characteristic (i.e. m = Xf). The average harmonic simple should be resorted to in cases of determining, for example, the average cost of labor, time, materials per unit of production, per one part for two (three, four, etc.) enterprises, workers engaged in the manufacture of the same type of product , the same part, product.
Geometric mean and chronological mean.
Geometric mean
If there are n growth coefficients, then the formula for the average coefficient is:
This is the geometric mean formula.
The geometric mean is equal to the root of degree n from the product of growth coefficients characterizing the ratio of the value of each subsequent period to the value of the previous one.
Chronological average is an average calculated from values that change over time. Used to calculate the average level of the moment series. In the event that the available data relate to fixed points in time at equal intervals, then the following formula is used:
X is the value of the series levels,
n – number of available indicators.
The average level of moment series of dynamics with unequally spaced dates is determined by the average chronological weighted formula:
= 
Where are the levels of the dynamics series
— duration of the time interval between levels
Mean square. Relationship between power averages.
If values expressed in the form of quadratic functions are subject to averaging, the average is applied quadratic. For example, using the root mean square, you can determine the diameters of pipes, wheels, etc.
The simple mean square is determined by taking the square root of the quotient of dividing the sum of squares of the individual values of the attribute by their number.
The weighted mean square is equal to:
Fashion concept. Calculation of mode for discrete and interval distribution series.
To characterize the structure of a statistical population, indicators called structural averages are used. These include mode and median.
Fashion (Mo) is the most common option. The mode is the value of the attribute that corresponds to the maximum point of the theoretical distribution curve.
Fashion represents the most frequently occurring or typical meaning.
Fashion is used in commercial practice to study consumer demand and record prices.
In a discrete series, mode is the variant with the highest frequency. In an interval variation series, the mode is considered to be the central variant of the interval, which has the highest frequency (particularity).
Within the interval, you need to find the value of the attribute that is the mode.
where xo is the lower limit of the modal interval;
h – the value of the modal interval;
fm – modal interval frequency;
ft-1 – frequency of the interval preceding the modal one;
fm+1 – frequency of the interval following the modal one.
The mode depends on the size of the groups and on the exact position of the group boundaries.
Mode is a number that actually occurs most often (is the value of
nnaya), in practice has the widest application (the most common type of buyer).
Harmonic mean— ϶ᴛᴏ the reciprocal of the arithmetic mean, ᴛ.ᴇ. consists of the inverse values of the characteristic.
Example 5. Calculation of the average percentage of plan completion. The following data is available:
In the example, indicators of the degree of implementation of the plan (options) act as a varying characteristic, and the plan takes weights (frequencies). In this case, the average is obtained as a weighted arithmetic average:
If, when determining the average degree of plan fulfillment, we take not the task as weight, but its actual implementation, then the arithmetic average in this case will give the wrong result:
The correct result when weighing according to the actual completion of the task will be given by the harmonic weighted average:
Where w— weights of the harmonic weighted average.
Conditions for using harmonic mean
The harmonic mean is used when not the units of the population (carriers of the characteristic) are used as weights, but the products of these units by the values of the characteristic, ᴛ.ᴇ. 
.
From this rule it follows that the harmonic mean in statistics is essentially a transformed arithmetic mean, which is used when the size of the population is unknown and it is necessary to weigh options by the volume of the characteristic.
2. If absolute values are used as weights, any intermediate action when calculating the average should give economically significant results.
For example, when calculating the average percentage of plan completion, we multiply the plan completion indicator by the plan target and obtain the actual plan completion. If the indicator of plan implementation is multiplied by its actual implementation, then from an economic point of view the result will be absurd. This means that the middle form was applied incorrectly).
Read also
When statistical information does not contain frequencies for individual variants of the population, but is presented as their product, i.e. the frequency must be separately calculated based on the known variant X and the product X f, the harmonic mean is used. Average… [read more].
The harmonic mean is a primitive form of the arithmetic mean. It is calculated in cases where the weights fi are not specified directly, but are included as a factor in one of the available indicators. Just like the arithmetic mean, the harmonic mean can be... [read more].
Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values of the attribute. Like the arithmetic mean, it can be simple and weighted. Characteristics of variation series, along with... [read more].
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Average values and indicators of variation
Along with the arithmetic mean, statistics uses the harmonic mean, the inverse of the arithmetic mean of the inverse values of the attribute. Like the arithmetic mean, it can be simple and weighted. Thus, the formula for calculating the average... [read more].
The essence and meaning of average values, their types The most common form of statistical indicator is the average value. An indicator in the form of an average value expresses the typical level of a characteristic in the aggregate. Widespread use of medium... [read more].
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When solving problems, calculating the average value begins with drawing up the initial relation - the logical verbal formula of the average. It is compiled on the basis of theoretical and logical analysis. Sometimes the arithmetic mean cannot be used. In this case, in... [read more].
If, according to the conditions of the problem, it is necessary that the sum of values reciprocal to the individual values of a characteristic remain unchanged during averaging, then the average value is a harmonic average. The formula for harmonic mean is: For example, a car with... [read more].
70. Harmonic mean
The harmonic mean of positive numbers o, b is a number whose inverse is the arithmetic mean between , i.e. number
Problem 358. Prove that the harmonic mean does not exceed the geometric mean.
Average values in statistics: essence, properties, types. Examples of problem solving
The inverse to the harmonic mean is the arithmetic mean of numbers; the inverse to the geometric mean is the geometric mean of numbers, so it remains to refer to the inequality about the arithmetic and geometric mean.
Problem 359. The numbers are positive. Prove that
Solution. The required inequality can be rewritten in the form
that is, it is necessary to prove that the arithmetic mean of the numbers is greater than or equal to their harmonic mean. This becomes clear if we insert the geometric mean between them:
the last inequality reduces to an inequality about the arithmetic and geometric mean of numbers.
Another solution uses the following trick. We will prove a more general inequality (called the Cauchy-Bunyakovsky inequality)
(if we substitute it into it we get what we need).
To prove the Cauchy-Bunyakovsky inequality, consider the quadratic trinomial
Opening the brackets in it and grouping the terms by powers of x, we get the trinomial
For any x, this trinomial is non-negative - after all, it is a sum of squares. This means that its discriminant is not greater than zero, i.e.
How did you like this trick?
Example : It is required to determine the average age of a part-time student using the data specified in the following table:
|
Age of students, years ( X) |
Number of students, people ( f) |
average value of the interval (x’,xcentral) |
xi*fi |
|
26 and older |
|||
|
Total: |
To calculate the average in interval series, first determine the average value of the interval as the half-sum of the upper and lower limits, and then calculate the average using the arithmetic weighted average formula.
Above is an example with equal intervals, with the 1st and last being open.
.
Answer: The average student age is 22.6 years, or approximately 23 years.
Harmonic mean has a more complex structure than the arithmetic mean. Used in cases where statistical information does not contain frequencies for individual values of the attribute, and is represented by the product of the attribute value by frequency . The harmonic mean as a type of power mean looks like this:
Depending on the form of presentation of the source data, the harmonic mean can be calculated as simple or weighted. If the source data is not grouped, then average harmonic simple :
It is used in cases of determining, for example, the average cost of labor, materials, etc.
Harmonic mean simple and weighted
per unit of production across several enterprises.
When working with grouped data, use weighted harmonic mean:
Geometric meanapplies in cases where when the total volume of the averaged feature is a multiplicative quantity,those. is determined not by summing, but by multiplying the individual values of the characteristic.
Shape of geometric weighted mean in practical calculations not applicable .
Mean square used in cases where, when replacing individual values of a characteristic with an average value, it is necessary to keep the sum of squares of the original values unchanged .
home scope of its use – measurement of the degree of fluctuation of individual values of a characteristic relative to the arithmetic mean(standard deviation). In addition, the mean square is used in cases where it is necessary to calculate the average value of a characteristic expressed in square or cubic units of measurement (when calculating the average value of square sections, average diameters of pipes, trunks, etc.).
The root mean square is calculated in two forms:
All power means differ from each other in the values of the exponent. Wherein, the higher the exponent, the morequantitative value of the average:
This property of power averages is called property of majorance of averages.
Harmonic mean value
Provided that the values k = –1 are substituted into the general formula (6.1), we can obtain harmonic mean value, which has a simple and weighted form.
For the ranked series, the harmonic mean is used simple a value that can be written as follows.
where n is the total number of options; – reverse meaning options.
Let’s say there is evidence that when transporting potatoes, the speed of a car with a load is 30 km/h, without a load – 60 km/h. You need to find the average speed of the car. At first glance, it seems like a completely simple solution to the problem: apply the method of the arithmetic average of a simple value, i.e. 
However, if we keep in mind that the speed of movement is equal to the distance traveled divided by the time spent, then it is quite obvious that the result (45 km/h) turns out to be inaccurate, since it takes a car with and without a load to travel the same path ( round trip) the time required will vary significantly. Consequently, a more accurate average speed of a vehicle with and without load can be calculated using the harmonic mean simple value:
Thus, the average speed of a car with and without cargo is not 45, but 40 km/h.
In discrete or interval series, the harmonic mean is used weighted size:
where W is the product of options and frequency (weighted option, xf).
Let's consider example. The labor intensity of producing 1 ton of potatoes in the first division of the agricultural organization is 10 man-hours, in the second - 30 man-hours. In both divisions, 30 thousand man-hours were spent on potato production. It is necessary to calculate the average arithmetic labor intensity of potatoes in an agricultural organization. It seems that the average labor intensity is easy to find as half the sum of the labor intensity of potatoes in two divisions, that is, using the method of arithmetic simple average:
However, this solution makes two mistakes. The first, fundamental mistake is that when calculating the average labor intensity using the arithmetic simple average method, the essence of the labor intensity itself is not taken into account, which is found as the ratio of direct labor costs to the volume of production. The second mistake is that the solution did not take into account the specific volume of labor costs for potato production given according to the conditions of the problem (30 thousand each).
Harmonic mean
person-hour in both departments). This allows one to calculate the frequencies (weights) for the potato labor intensity and thus find the arithmetic average weighted labor intensity, which will be successfully replaced by applying the harmonic weighted average:
Thus, the average labor intensity of potatoes in an agricultural organization is not 20, as was calculated above, but 15 people. h/t.
The harmonic mean value is used mainly in cases where the variants of the series are represented by inverse values, and the frequencies (weights) are hidden in the total volume of the characteristic being studied.
Structural averages
In some cases, to obtain a general characteristic of a statistical population for any criterion, it is necessary to use the so-called structural average. These include fashion And median.
Fashion represents the variant most often found in a given statistical population. In a ranked series, the mode, as a rule, is not determined, since each option corresponds to a frequency equal to unity.
The mode in a discrete series corresponds to the variant with the highest frequency, while a random variable can have several modes. If one of them is present, the distribution of the statistical population is usually called unimodal, in the presence of two modes - bimodal, three or more modes - multimodal. The presence of several modes often means the combination of statistical units of different quality in one set.
The mode for an interval series with equal intervals is calculated by the formula
(6.12)
where xmo sub> is the lower limit of the modal interval; i mo – interval value;
f mo – frequency of the modal interval; f dmo – frequency of the premodal interval; f zmo – frequency of the submodal interval.
Let’s say that market prices for apples in the regional centers of the region are as follows (Table 6.8). Using these data, it is necessary to calculate the trend in market prices for potatoes.
Table 6.8. Market prices for apples
From the data in table. 6.8 shows that the maximum number of markets is concentrated in the third interval, and the distribution of the statistical population is unimodal. To calculate the fashion of market prices for apples, we use formula (6.12):
Thus, the modal market price for apples in the regional centers of the region is 1690 rubles/kg.
The modal option when characterizing a statistical population can be used in cases where calculating the average value is difficult or impossible, for example, in market conditions when studying supply and demand, price levels, etc.
Median– options located in the middle of the variation series. The median in the ranked series is found as follows. First, calculate the number of median options:
where n me is the number of median options; n is the total number of options in the series.
Secondly, in the ranked series the value of the median of the options is determined: if the total number of options is odd, then the median corresponds to the number calculated by formula (6.13).
Let's say the ranked series consists of 99 units distributed by sugar beet yield. The median number of options is found using formula (6.13): 
.
This means that number 50 is the desired median yield, which is equal to, for example, 500 c/ha.
If the total number of variants is even, then the median is equal to half the sum of two adjacent median variants. For example, in the ranked series there are 100 statistical units, again distributed by sugar beet yield. Consequently, in such a series there are two median numbers, as can be seen from the following calculation using formula (6.13):
This means that in this case, numbers 50 and 51 are considered medians, and the median yield of sugar beet, for example, can be calculated as the following half-sum of two adjacent yields, i.e.
For a discrete distribution series, the median is calculated from the accumulated frequencies: first, the half-sum of the accumulated frequencies is found; secondly, they determine whether this half-sum corresponds to a specific option, which will be the median.
For example, the annual milk yield of cows is distributed in the form of a discrete series in which the sum of the accumulated frequencies is 200 units and, accordingly, the half-sum is 100 units.
This median number is in the group of statistical units of a discrete series and corresponds to the annual milk yield of cows of 5000 kg of milk, which is the median of the discrete series.
In an interval variation series, the median is calculated using the formula
, (6.14)
where M e is the median of the interval series; x me – lower limit of the median interval; i me – the value of the median interval; Σf – the sum of accumulated frequencies in the interval series; f n – accumulated frequency of the pre-median interval; f me – frequency of the median interval.
To calculate the median in an interval series, we will use the following data (Table 6.9).
Table 6.9.
Potato yield in personal plots
Households
From the data in table. 6.9, first of all, it is clear that the fourth interval is the median. In addition, a simple calculation shows that the sum of the accumulated frequencies (total number of farms) is 200 units, and the accumulated frequency of the pre-median interval is 90 units.
Let's use formula (6.14) and calculate the median potato yield:
Thus, the median potato yield in private household plots is 256 c/ha.
The use of the median has a specific character. Thus, if the variation series is relatively small, then the value of the arithmetic mean may be influenced by random fluctuations of the extreme variants, which will not affect the size of the median.
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The most common form of statistical indicator is averagemagnitude. An indicator in the form of an average value expresses the typical level of a characteristic in the aggregate. The widespread use of average values is explained by the fact that they allow one to compare the values of a characteristic among units belonging to different populations. For example, you can compare the average length of a working day, the average wage category of workers, the average wage level for different enterprises.
The essence of average values is that they cancel out deviations in the values of a characteristic in individual units of the population, caused by the action of random factors. Therefore, average values must be calculated for sufficiently large populations (in accordance with the law of large numbers). The reliability of average values also depends on the variability of the attribute values in the aggregate. In general, the smaller the variation of a characteristic and the larger the population from which the average value is determined, the more reliable it is.
The typicality of the average value is also directly related to homogeneity of the statistical population. The average value will only reflect the typical level of the attribute when it is calculated from a qualitatively homogeneous population. Otherwise, the average method is used in combination with the grouping method. If the population is heterogeneous, then the general averages are replaced or supplemented by group averages calculated for qualitatively homogeneous groups.
Selecting the type of averages is determined by the economic content of the indicator under study and the source data. The following types of averages are most often used in statistics: power averages (arithmetic, harmonic, geometric, quadratic, cubic, etc.), chronological average, and structural averages (mode and median).
Arithmetic mean most often found in socio-economic research. The arithmetic average is used in the form of a simple average and a weighted average.
Calculated from ungrouped data based on formula (4.1):
Where x- individual values of the characteristic (options);
n- the number of units in the population.
Example. It is required to find the average output of a worker in a brigade consisting of 15 people, if the number of products produced by one worker (pieces) is known: 21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.
Simple arithmetic mean calculated from ungrouped data based on formula (4.2):
where f is the frequency of repetition of the corresponding value of the attribute (variant);
∑f is the total number of population units (∑f = n).
Example. Based on the available data on the distribution of workers in a team according to the number of products they produce, it is necessary to find the average output of a worker in the team.
Note 1. The average value of a characteristic in the aggregate can be calculated both on the basis of individual values of the characteristic and on the basis of group (private) averages calculated for individual parts of the population. In this case, the arithmetic weighted average formula is used, and group (partial) averages ( x j).
Example. There is data on the average length of service of workers in the plant's workshops. It is required to determine the average length of service of workers for the plant as a whole.
Note 2. In the case when the values of the characteristic being averaged are specified in the form of intervals, when calculating the arithmetic mean value, the average values of these intervals are taken as the values of the characteristic in groups ( X’). Thus, the interval series is converted into a discrete series. In this case, the value of open intervals, if any (as a rule, these are the first and last), is conditionally equated to the value of the intervals adjacent to them.
Example. There is data on the distribution of enterprise workers by wage level.
Harmonic mean value is a modification of the arithmetic mean. It is used in cases where individual values of a characteristic are known, i.e. variants ( x), and the product of the variant and the frequency (xf = M), but the frequencies themselves are unknown ( f).
The weighted harmonic mean is calculated using formula (4.3):
Example. It is required to determine the average wages of employees of an association consisting of three enterprises, if the wage fund and the average wages of employees for each enterprise are known.
The harmonic mean, which is simple in statistics practice, is used extremely rarely. In cases where xf = Mm = const, the weighted harmonic mean turns into a simple harmonic mean (4.4):
Example. Two cars traveled the same route. At the same time, one of them was moving at a speed of 60 km/h, the second - at a speed of 80 km/h. It is required to determine the average speed of the cars along the way.
Other types of power averages. Average chronological
The geometric mean is used to calculate the average dynamics. The geometric mean is used in the form of a simple average (for ungrouped data) and a weighted average (for grouped data).
Geometric mean simple (4.5):
where n is the number of attribute values;
P is the sign of the product.
Weighted geometric mean(4.6):
Root mean square value used when calculating variation indices. It is used in a simple and weighted form.
Simple mean square (4.7):
Weighted mean square (4.8):
The average cubic value is used when calculating indicators asymmetry And excess. It is used in simple weighed form.
Average cubic simple (4.9):
Average cubic weighted (4.10):
The average chronological value is used to calculate the average level of the time series (4.11):
Structural averages
In addition to the average values discussed above, statistics uses structural averages, which include mode and median.
Fashion(Mo) is the value of the characteristic being studied (variant), which is most often found in the aggregate. In a discrete series The mode is determined quite simply - by the maximum frequency indicator. In an interval variation series, the mode approximately corresponds to the center of the modal interval, that is, the interval that has a high frequency (frequency).
The specific mode value is calculated using formula (4.12):
where is the lower limit of the modal interval;
modal interval width;
frequency corresponding to the modal interval;
frequency of the interval preceding the modal;
frequency of the interval following the modal.
The median (Me) is the value of the attribute located in the middle of the ranked series. By ranked we mean a series ordered in ascending or descending order of attribute values. The median divides the ranked series into two parts, one of which has attribute values no greater than the median, and the other no less.
For a ranked series with an odd number of members, the median is the option located in the center of the series. The position of the median is determined by the serial number of the unit of the series in accordance with formula (4.13):
where n is the number of members of the ranked series.
For a ranked series with an even number of members, the median is the arithmetic mean of two adjacent values located in the center of the series.
In an interval variation series, the following formula (4.14) is used to find the median:
where is the lower limit of the median interval;
width of the median interval;
accumulated frequency of the interval preceding the median;
frequency of the median interval.
Example. Work team consisting of 9 people, have the following tariffs digits: 4; 3; 4; 5; 3; 3; 6; 2;6. It is required to determine the modal and median values of the tariff category.
Since this brigade has the most workers of the 3rd category, this category will be modal, i.e. Mo = 3.
To determine the median Let's rank the original series in ascending order of attribute values:
2; 3; 3; 3; 4; 4; 5; 6; 6.
The central value in this series is the fifth value of the attribute. Accordingly, Me = 4.
Example.It is required to determine the modal and median tariff category of factory workers based on the data from the following distribution row.
Since the original distribution series is discrete, the modal value is determined by the maximum frequency indicator. In this example, the plant has the most workers of the 3rd category (f max = 30), i.e. this discharge is modal (Mo = 3).
Let's determine the position of the median. The initial distribution series is constructed on the basis of a ranked series, ordered by increasing values of the attribute. The middle of the series is between the 50th and 51st serial numbers of the attribute values. Let's find out which group the workers with these serial numbers belong to. To do this, let's calculate the accumulated frequencies. The accumulated frequencies indicate that the median value of the tariff category is equal to three (Me = 3), since the values of the characteristic with serial numbers from 39 to 68, including 50 and 51, are equal 3.
Example. It is required to determine the modal and median wages of factory workers based on the data from the following distribution series.
Since the initial distribution series is interval, the modal value of wages is calculated using the formula. In this case, the modal interval is 360-420 with a maximum frequency of 30.
The median salary value is also calculated using the formula. In this case, the median is the interval 360-420, the accumulated frequency of which is 70, while the accumulated frequency of the previous interval was only 40 with a total number of units equal to 100.