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Mode and median- a special kind of averages that are used to study the structure of the variation series. They are sometimes called structural averages, in contrast to the previously discussed power-law averages.

Fashion- this is the value of the attribute (variant), which is most often found in this population, i.e. has the highest frequency.

Fashion has a great practical application, and in some cases only fashion can characterize social phenomena.

Median is the variant that is in the middle of the ordered variation series.

The median shows the quantitative limit of the value of the variable characteristic, which is reached by half of the population units. The use of the median along with the average or instead of it is advisable if there are open intervals in the variation series, because the calculation of the median does not require the conditional establishment of the boundaries of open intervals, and therefore the absence of information about them does not affect the accuracy of the calculation of the median.

The median is also used when the indicators to be used as weights are unknown. The median is used instead of the arithmetic mean in statistical methods of product quality control. The sum of absolute deviations of options from the median is less than from any other number.

Consider the calculation of the mode and median in a discrete variational series :

Determine the mode and median.

Fashion Mo = 4 years, since this value corresponds to the highest frequency f = 5.

Those. Most of the workers have 4 years of experience.

In order to calculate the median, we first find half the sum of the frequencies. If the sum of the frequencies is an odd number, then we first add one to this sum, and then divide it in half:

The median will be the eighth option.

In order to find which option will be the eighth in number, we will accumulate frequencies until we get the sum of frequencies equal to or greater than half the sum of all frequencies. The corresponding option will be the median.

Me = 4 years.

Those. half of the workers have less than four years of experience, half more.

If the sum of the accumulated frequencies against one option is equal to half the sum of the frequencies, then the median is defined as the arithmetic average of this option and the next one.

Calculation of the mode and median in an interval variation series

The mode in the interval variation series is calculated by the formula

where X М0- initial border of the modal interval,

hm 0 is the value of the modal interval,

fm 0 , fm 0-1 , fm 0+1 - the frequency of the modal interval, respectively, preceding the modal and subsequent.

Modal The interval with the highest frequency is called.

Example 1

Groups by experience	Number of workers, people	Accumulated Frequencies

Determine the mode and median.

Modal interval, because it corresponds to the highest frequency f = 35. Then:

Hm 0 =6, fm 0 =35

Structural (positional) averages- these are average values that occupy a certain place (position) in a ranked variational series.

Fashion(Mo) is the value of the feature most frequently found in the study population.

For discrete variation series the mode will be the value of the options with the highest frequency

Example. Determine the mode from the available data (Table 7.5).

Table 7.5 - Distribution of women's shoes sold in a shoe store N, February 2013

According to Table. 5 shows that the highest frequency fmax= 28, it corresponds to the value of the feature x= 37 size. Consequently, Mo= 37 shoe size, i.e. it was this shoe size that was in the greatest demand, most often bought shoes of the 37th size.

AT first determined modal spacing, i.e. containing the mode - the interval with the highest frequency (in the case of an interval distribution with equal intervals, in the case of unequal intervals - by the highest density).

Mode is approximately considered the middle of the modal interval. The specific mode value for the interval series is determined by the formula:

where x Mo is the lower limit of the modal interval;

i Mo is the value of the modal interval;

f Mo is the frequency of the modal interval;

f Mo-1 is the frequency of the interval preceding the modal;

f Mo +1 is the frequency of the interval following the modal.

Example. Determine the mode from the available data (Table 7.6).

Table 7.6 - Distribution of employees by length of service

According to Table. 6 shows that the highest frequency fmax= 35, it corresponds to the interval: 6-8 years (modal interval). We define fashion by the formula:

years.

Consequently, Mo= 6.8 years, i.e. Most employees have 6.8 years of experience.

The name of the median is taken from geometry, where it refers to a segment connecting one of the vertices of the triangle with the midpoint of the opposite side and thus dividing the side of the triangle into two equal parts.

Median(Me) – is the value of the feature that falls in the middle of the ranged population. Otherwise, the median is a value that divides the number of an ordered variational series into two equal parts - one part has the values of the varying attribute less than the average variant, and the other has large values.

For ranked series(i.e. ordered - built in ascending or descending order of individual attribute values) with an odd number of members ( n= odd) the median is the variant located in the center of the row. Ordinal number of the median ( N Me) is defined as follows:

N Me =(n+1)/ 2.

Example. In a series of 51 members, the median number is (51+1)/2 = 26, i.e. the median is the 26th option in the series.

For a ranked series with an even number of members ( n= even) - the median will be the arithmetic mean of the two values of the attribute located in the middle of the series. The serial numbers of the two central variants are determined as follows:

N Me 1 =n/ 2; N Me 2 =(n/ 2)+ 1.

Example. When n=50; N Me1 = 50/2 = 25; N Me2= (50/2)+1 = 26, i.e. the median is the average of the options in the 25th and 26th row in order.

AT discrete variation series the median is found by the accumulated frequency corresponding to the ordinal number of the median or exceeding it for the first time. Otherwise, according to the accumulated frequency equal to or for the first time exceeding half the sum of all frequencies of the series.

Example. Determine the median from the available data (Table 7.7).

Table 7.7 - Distribution of women's shoes sold in a shoe store N, February 2013

According to Table. 7 define the ordinal number of the median: N Me =( 67+1)/2=34.

Fashion. Median. How to calculate them (p. 1 of 2)

The cumulative frequency exceeding this value for the first time S= 41, it corresponds to the value of the feature x= 37 size. Consequently, Me= 37 shoe size, i.e. half of the pairs are bought smaller than size 37, and the other half are bought larger.

In this example, the mode and median are the same, but they may or may not be the same.

AT interval variation series cumulative frequencies are determined, according to the cumulative frequencies data are found median interval– the interval in which the accumulated frequency is half or for the first time exceeds half of the total sum of frequencies. The formula for determining the median in the interval series of the distribution is as follows:

where x Me is the lower limit of the median interval;

i Me is the value of the median interval;

∑fi is the sum of the frequencies of the series;

S Me-1 is the sum of the accumulated frequencies of the interval preceding the median;

f Me is the frequency of the median interval.

Example. Determine the median from the available data (Table 7.8).

Table 7.8 - Distribution of employees by length of service

According to Table. 8 define the ordinal number of the median: NMe=100/2=50. The cumulative frequency exceeding this value for the first time S= 82, it corresponds to an interval of 6-8 years (median interval). In this example, the modal and median intervals are the same, but they may or may not be the same. Let's determine the median by the formula:

years

Consequently, Me= 6.2 years, i.e. half of the employees have less than 6.2 years of experience and the other half have more.

Mode and median are widely used in various areas of the economy. Thus, the calculation of modal labor productivity, modal cost, etc. enables the economist to judge the currently prevailing level of them. This characteristic should be used to reveal the reserves of our economy. Fashion matters for solving practical problems. So, when planning the mass production of clothing and footwear, the size of the product is set, which is in greatest demand (modal size). The mode can be used as an approximate characteristic of the level of the studied trait instead of the arithmetic mean if the frequency distributions are close to symmetrical and have one non-flat top.

The median should be used as an average in cases where there is insufficient confidence in the homogeneity of the population under study. The median is affected not so much by the values themselves as by the number of cases at one level or another. It should also be noted that the median is always specific (for a large number of observations or in the case of an odd number of members of the population), because under Me some real real element of the population is implied, while the arithmetic average often takes on a value that none of the units of the population can take.

Main property Me in that the sum of absolute deviations of the trait values from the median is less than from any other value: . This property Me can be used, for example, when determining the construction site of public buildings, because Me determines the point that gives the shortest distance, say, kindergartens from the place of residence of parents, residents of the settlement from the cinema, when designing tram, trolleybus stops, etc.

In the system of structural indicators, the options that occupy a certain place in the ranked variation series (every fourth, fifth, tenth, twenty-fifth, etc.) act as indicators of the features of the distribution form. Similarly, with finding the median in the variational series, you can find the value of the feature for any unit of the ranked series in order.

Quartiles– attribute values dividing the ranged population into four equal parts. Distinguish the lower quartile ( Q1), average ( Q2) and upper ( Q 3). The lower quartile separates 1/4 of the population with the lowest values of the feature, the upper quartile separates 1/4 of the population with the highest values of the feature. This means that 25% of the population units will be smaller in value Q1; 25% units will be concluded between Q1 and Q2; 25% - between Q2 and Q 3; the remaining 25% outperform Q 3. The middle quartile ( Q2) is the median .

To calculate the quartiles for the interval series, the following formulas are used:

;

where x Q1– the lower limit of the interval containing the lower quartile (the interval is determined by the accumulated frequency, the first exceeding 25%);

x Q3– the lower limit of the interval containing the upper quartile (the interval is determined by the accumulated frequency, the first exceeding 75%);

S Q 1-1 is the cumulative frequency of the interval preceding the interval containing the lower quartile;

S Q 3-1 is the cumulative frequency of the interval preceding the interval containing the upper quartile;

fQ1 is the frequency of the interval containing the lower quartile;

fQ3 is the frequency of the interval containing the upper quartile.

Deciles are variant values that divide the ranked series into ten equal parts: 1st decile ( d1) divides the population 1/10 to 9/10, 2nd decile ( d2) - in the ratio of 2/10 to 8/10, etc. Deciles are calculated in the same way as the median and quartiles:

;

The use of the above characteristics in the analysis of variational distribution series allows one to deeply and in detail characterize the population under study.

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Structural averages

Along with power-law averages, structural averages are widely used.

The structure of statistical aggregates is different. At the same time, the more symmetrical the distribution of units of the population, the more qualitatively its composition according to the trait under study, the better, more reliably the average value of the trait characterizes the phenomenon under study. But for cases of sharp skewness (asymmetry) of the distribution series, the arithmetic mean is no longer so typical. For example, the average size of a deposit in savings banks is not of particular interest, since the bulk of deposits are below this level, and the average is significantly influenced by large deposits, which are few and which are not typical for the mass of deposits.

Fashion (statistics)

In such cases, statistics uses another system - the system of auxiliary structural averages. These include mode, median, as well as quartels, quintels, decels, percentels.

Fashion (Mo)- the most common value of the trait, and in a discrete variational series - this is the variant with the highest frequency.

In statistical practice, fashion is used in the study of incomes of the population, consumer demand, price registration and in the analysis of some technical and economic indicators of enterprises.

In some cases, it is the mode that is of interest, and not the arithmetic mean. Sometimes it is used instead of the arithmetic mean, for example, to characterize the structure of distribution series.

The order in which the mode is determined depends on the type of the distribution series. If the variable attribute is presented as a discrete series, then no calculations are required to determine the mode. In such a series, the mode will be the value of the feature that has the highest frequency.

If the value of the attribute is presented as an interval variation series with equal intervals, then the mode is determined by calculation using the formula:

where X Mo is the lower limit of the modal interval,

i Mo is the value of the modal interval,

f Mo , f Mo-1 , f Mo+1 are the frequencies of the modal, premodal (previous), and postmodal (following the modal) intervals, respectively.

Median (Me)- this is the value of the attribute, which is in the middle of the ranged variation series, where the individual values of the attribute (options) are arranged in ascending or descending order (by rank).

The median should be used as an average in cases where there is insufficient confidence in the homogeneity of the population under study. The median finds application in marketing activities. For example, the placement of elevators, primary wineries, canneries, the sum of the distances to which from raw material suppliers should be the smallest.

The median, like the mode, is defined in different ways. It depends on the structure of the distribution series.
To determine the median in discrete variational series:

1) find its serial number by the formula

N Me =
2) build a series of accumulated frequencies

3) find the accumulated frequency, which is equal to or exceeds the serial number of the median

4) of the variant corresponding to the given accumulated frequency is the median.

If the number of members of a discrete series is odd, then the median is in the middle of the series and divides this series into two equal parts according to the number of members of the series. The ordinal number of the median in this case is calculated by the formula:

NMe =(f + 1)2,

where  f – the number of members of the series.

In interval series, the median interval is first determined. For this, just as in discrete series, the ordinal number of the median is calculated. The accumulated frequency, which is equal to the number of the median or the first one exceeds it, corresponds to the median interval in the interval variation series. Let's denote this accumulated frequency as S Me . The median is directly calculated using the formula:

,
where is the lower limit of the median interval

- the value of the median interval

is the cumulative frequency of the interval preceding the median

— frequency of the median interval

Graphical definition of mode and median
Mode and median in an interval series can be determined graphically.

The mode is determined from the histogram of the distribution. To do this, the tallest rectangle is selected, which in this case is modal. Then we connect the right vertex of the modal rectangle with the upper right corner of the previous rectangle. And the left vertex of the modal rectangle is with the upper left corner of the subsequent rectangle. Further, from the point of their intersection, a perpendicular is lowered to the abscissa axis. The abscissa of the point of intersection of these lines will be the distribution mode (Fig. 1). The median is calculated from the cumulate (Fig. 2). To determine it, from a point on the scale of accumulated frequencies (frequencies) corresponding to 50%, a straight line is drawn parallel to the abscissa axis until it intersects with the cumulate. Then, from the point of intersection of the specified straight line with the cumulate, a perpendicular is lowered to the abscissa axis. The abscissa of the intersection point is the median.

Indicators of variation in statistics.

In the process of statistical analysis, a situation may arise when the values of the average values coincide, and the populations on the basis of which they are calculated consist of units whose characteristic values differ quite sharply from each other. In this case, the indicators of variation are calculated.

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On the topic: "Mode. Median. Methods for calculating them"

Introduction

Mean values and related indicators of variation play a very important role in statistics, which is due to the subject of its study. Therefore, this topic is one of the central in the course.

The average is a very common generalizing indicator in statistics. This is explained by the fact that only with the help of the average it is possible to characterize the population according to a quantitatively varying attribute. An average value in statistics is a generalizing characteristic of a set of phenomena of the same type according to some quantitatively varying attribute. The average shows the level of this attribute, related to the unit of the population.

Studying social phenomena and seeking to identify their characteristic, typical features in specific conditions of place and time, statisticians make extensive use of average values. With the help of averages, different populations can be compared with each other according to varying characteristics.

Averages used in statistics belong to the class of power averages. Of the power averages, the arithmetic mean is most often used, less often the harmonic mean; the harmonic mean is used only when calculating the average rates of dynamics, and the mean square - only when calculating the variation indicators.

The arithmetic mean is the quotient of dividing the sum of the options by their number. It is used in cases where the volume of a variable attribute for the entire population is formed as the sum of the attribute values for its individual units. The arithmetic mean is the most common type of average, since it corresponds to the nature of social phenomena, where the volume of varying signs in the aggregate is most often formed precisely as the sum of the values of the attribute in individual units of the population.

According to its defining property, the harmonic mean should be used when the total volume of the attribute is formed as the sum of the reciprocal values of the variant. It is used when, depending on the material available, the weights do not have to be multiplied, but divided into options or, what is the same, multiplied by their inverse value. The harmonic mean in these cases is the reciprocal of the arithmetic mean of the reciprocal values of the attribute.

The harmonic mean should be used in those cases when the weights are not the units of the population - the carriers of the feature, but the products of these units and the value of the feature.

1. Definition of mode and median in statistics

The arithmetic and harmonic means are the generalizing characteristics of the population according to one or another varying attribute. Auxiliary descriptive characteristics of the distribution of a variable attribute are the mode and the median.

In statistics, fashion is the value of a feature (variant) that is most often found in a given population. In the variation series, this will be the variant with the highest frequency.

The median in statistics is called the variant, which is in the middle of the variation series. The median divides the series in half, on both sides of it (up and down) there is the same number of population units.

Mode and median, in contrast to the exponential averages, are specific characteristics, their value is any particular variant in the variation series.

Mode is used in cases where it is necessary to characterize the most frequently occurring value of a feature.

5.5 Mode and median. Their calculation in discrete and interval variational series

If it is necessary, for example, to find out the most common wage in the enterprise, the market price at which the largest number of goods were sold, the size of shoes that are most in demand among consumers, etc., in these cases resort to fashion.

The median is interesting in that it shows the quantitative limit of the value of the variable characteristic, which was reached by half of the members of the population. Let the average salary of bank employees amount to 650,000 rubles. per month. This characteristic can be supplemented if we say that half of the workers received a salary of 700,000 rubles. and higher, i.e. let's take the median. The mode and median are typical characteristics in cases where the populations are homogeneous and large in number.

Finding the Mode and Median in a Discrete Variation Series

Finding the mode and median in a variational series, where the attribute values are given by certain numbers, is not very difficult. Consider table 1. with the distribution of families by the number of children.

Table 1. Distribution of families by number of children

Obviously, in this example, the fashion will be a family with two children, since this value of options corresponds to the largest number of families. There may be distributions where all variants are equally frequent, in which case there is no fashion, or, in other words, all variants can be said to be equally modal. In other cases, not one, but two options may be the highest frequency. Then there will be two modes, the distribution will be bimodal. Bimodal distributions may indicate the qualitative heterogeneity of the population according to the trait under study.

To find the median in a discrete variation series, you need to divide the sum of frequencies in half and add ½ to the result. So, in the distribution of 185 families by the number of children, the median will be: 185/2 + ½ = 93, i.e. The 93rd option, which divides the ordered row in half. What is the meaning of the 93rd option? In order to find out, you need to accumulate frequencies, starting from the smallest options. The sum of the frequencies of the 1st and 2nd option is 40. It is clear that there are no 93 options here. If we add the frequency of the 3rd option to 40, then we get the sum equal to 40 + 75 = 115. Therefore, the 93rd option corresponds to the third value of the variable attribute, and the median will be a family with two children.

Mode and median in this example coincided. If we had an even sum of frequencies (for example, 184), then applying the above formula, we get the number of the median options, 184/2 + ½ = 92.5. Since there are no fractional options, the result indicates that the median is in the middle between 92 and 93 options.

3. Calculation of the mode and median in the interval variation series

The descriptive nature of the mode and median is due to the fact that they do not compensate for individual deviations. They always correspond to a certain variant. Therefore, the mode and median do not require calculations to find them if all the values of the attribute are known. However, in the interval variation series, calculations are used to find the approximate value of the mode and median within a certain interval.

To calculate a certain value of the modal value of a sign enclosed in an interval, the following formula is used:

M o \u003d X Mo + i Mo * (f Mo - f Mo-1) / ((f Mo - f Mo-1) + (f Mo - f Mo + 1)),

Where X Mo is the minimum limit of the modal interval;

i Mo is the value of the modal interval;

fMo is the frequency of the modal interval;

f Mo-1 - the frequency of the interval preceding the modal;

f Mo+1 is the frequency of the interval following the modal.

We will show the calculation of the mode using the example given in Table 2.

Table 2. Distribution of workers of the enterprise according to the implementation of production standards

To find the mode, we first determine the modal interval of the given series. It can be seen from the example that the highest frequency corresponds to the interval where the variant lies in the range from 100 to 105. This is the modal interval. The value of the modal interval is 5.

Substituting the numerical values from table 2. into the above formula, we get:

M o \u003d 100 + 5 * (104 -12) / ((104 - 12) + (104 - 98)) \u003d 108.8

The meaning of this formula is as follows: the value of that part of the modal interval, which must be added to its minimum boundary, is determined depending on the magnitude of the frequencies of the previous and subsequent intervals. In this case, we add 8.8 to 100, i.e. more than half of the interval, because the frequency of the previous interval is less than the frequency of the subsequent interval.

Let's calculate the median now. To find the median in the interval variation series, we first determine the interval in which it is located (the median interval). Such an interval will be one whose cumulative frequency is equal to or greater than half the sum of the frequencies. Cumulative frequencies are formed by gradual summation of frequencies, starting from the interval with the smallest feature value. Half the sum of the frequencies we have is 250 (500:2). Therefore, according to table 3. the median interval will be the interval with the value of wages from 350,000 rubles. up to 400,000 rubles.

Table 3. Calculation of the median in the interval variation series

Before this interval, the sum of the accumulated frequencies was 160. Therefore, in order to obtain the value of the median, it is necessary to add another 90 units (250 - 160).

When determining the value of the median, it is assumed that the value of units within the boundaries of the interval is distributed evenly. Therefore, if 115 units in this interval are distributed evenly in an interval equal to 50, then 90 units will correspond to the following value:

Fashion in statistics

Median (statistic)

Median (statistic), in mathematical statistics, a number that characterizes a sample (for example, a set of numbers). If all the elements in the sample are different, then the median is the number of the sample such that exactly half of the elements in the sample are greater than it and the other half are less than it.

In a more general case, the median can be found by ordering the elements of the sample in ascending or descending order and taking the middle element. For example, the sample (11, 9, 3, 5, 5) after ordering turns into (3, 5, 5, 9, 11) and its median is the number 5. If the sample has an even number of elements, the median may not be uniquely determined: for numerical data, the half-sum of two adjacent values is most often used (that is, the median of the set (1, 3, 5, 7) is taken equal to 4).

In other words, the median in statistics is the value that divides the series in half in such a way that on both sides of it (up or down) the same number of units of the given population is located. Because of this property, this indicator has several other names: the 50th percentile or the 0.5 quantile.

The median is used instead of the arithmetic mean when the extreme variants of the ranked series (smallest and largest) in comparison with the rest turn out to be excessively large or excessively small.

The MEDIAN function measures the central trend, which is the center of a set of numbers in a statistical distribution. There are three most common ways to determine the central trend:

Mean- the arithmetic mean, which is calculated by adding a set of numbers, followed by dividing the resulting sum by their number.
For example, the average for the numbers 2, 3, 3, 5, 7, and 10 is 5, which is the result of dividing their sum, which is 30, by their number, which is 6.
Median- a number that is the middle of a set of numbers: half of the numbers have values greater than the median, and half of the numbers are smaller.
For example, the median for numbers 2, 3, 3, 5, 7 and 10 is 4.
Fashion is the number that occurs most frequently in the given set of numbers.
For example, the mode for the numbers 2, 3, 3, 5, 7 and 10 is 3.

Median (statistic), in mathematical statistics - a number that characterizes a sample (for example, a set of numbers). If all the elements in the sample are different, then the median is the number of the sample such that exactly half of the elements in the sample are greater than it and the other half are less than it. In a more general case, the median can be found by ordering the elements of the sample in ascending or descending order and taking the middle element. For example, the sample (11, 9, 3, 5, 5) after ordering turns into (3, 5, 5, 9, 11) and its median is the number 5. If the sample has an even number of elements, the median may not be uniquely determined: for numerical data, the half-sum of two adjacent values is most often used (that is, the median of the set (1, 3, 5, 7) is taken equal to 4).

The MEDIAN function measures the central trend, which is the center of a set of numbers in a statistical distribution. There are three most common ways to determine the central trend:

Mean- the arithmetic mean, which is calculated by adding a set of numbers, followed by dividing the resulting sum by their number.
For example, the average for the numbers 2, 3, 3, 5, 7, and 10 is 5, which is the result of dividing their sum, which is 30, by their number, which is 6.
Median- a number that is the middle of a set of numbers: half of the numbers have values greater than the median, and half of the numbers have smaller values.
For example, the median for numbers 2, 3, 3, 5, 7 and 10 is 4.
Fashion- the number that occurs most frequently in a given set of numbers.
For example, the mode for the numbers 2, 3, 3, 5, 7 and 10 is 3.

Fashion- this is the value of the attribute (variant), which is most often found in this population, i.e. has the highest frequency.

Fashion has a great practical application, and in some cases only fashion can characterize social phenomena.

Median is the variant that is in the middle of the ordered variation series.

Consider the calculation of the mode and median in a discrete variational series :

Determine the mode and median.

Fashion Mo = 4 years, since this value corresponds to the highest frequency f = 5.

Those. Most of the workers have 4 years of experience.

In order to calculate the median, we first find half the sum of the frequencies. If the sum of the frequencies is an odd number, then we first add one to this sum, and then divide it in half:

The median will be the eighth option.

Me = 4 years.

Those. half of the workers have less than four years of experience, half more.

If the sum of the accumulated frequencies against one option is equal to half the sum of the frequencies, then the median is defined as the arithmetic average of this option and the next one.

Calculation of the mode and median in an interval variation series

The mode in the interval variation series is calculated by the formula

where X М0- initial border of the modal interval,

hm 0 is the value of the modal interval,

fm 0 , fm 0-1 , fm 0+1 - the frequency of the modal interval, respectively, preceding the modal and subsequent.

Modal The interval with the highest frequency is called.

Example 1

Groups by experience	Number of workers, people	Accumulated Frequencies

Determine the mode and median.

Modal interval, because it corresponds to the highest frequency f = 35. Then:

Hm 0 =6, fm 0 =35

hm 0 =2, fm 0-1 =20

fm 0+1 =11

Conclusion: The largest number of workers has an experience of approximately 6.7 years.

For an interval series, Me is calculated using the following formula:

where Hm e- the lower border of the medial interval,

hm e- the size of the medial interval,

- half the sum of frequencies,

fm e is the frequency of the median interval,

Sm e-1 is the sum of the accumulated frequencies of the interval preceding the median.

The median interval is such an interval to which the cumulative frequency corresponds, equal to or greater than half the sum of the frequencies.

Let's define the median for our example.

since 82>50, then the median interval .

Hm e =6, fm e =35,

hm e =2, Sm e-1 =47,

Conclusion: Half of the workers have less than 6.16 years of experience, and half have more than 6.16 years of experience.

Note. In this lesson, we set out problems in geometry about the median of a triangle. If you need to solve a problem in geometry, which is not here - write about it in the forum. Almost certainly the course will be supplemented.

A task. Find the length of the median of a triangle in terms of its sides

The sides of the triangle are 8, 9 and 13 centimeters. The median is drawn to the longest side of the triangle. Determine the median of a triangle based on the dimensions of its sides.

Solution.

The problem has two ways of solving. The first one, which is not liked by high school teachers, but is the most versatile.

Method 1.

Let's apply Stewart's Theorem, according to which the square of the median is equal to one fourth of the sum of twice the squares of the sides, from which the square of the side to which the median is drawn is subtracted.

M c 2 = (2a 2 + 2b 2 - c 2) / 4

Respectively

M c 2 \u003d (2 * 8 2 + 2 * 9 2 - 13 2) / 4
m c 2 = 30.25
m c = 5.5 cm

Method 2.

The second solution that teachers at school love is the additional construction of a triangle to a parallelogram and the solution through the parallelogram diagonal theorem.

We extend the sides of the triangle and the median by completing them to a parallelogram. In this case, the median BO of the triangle ABC will be equal to half the diagonal of the resulting parallelogram, and the two sides of the triangle AB, BC will be equal to its sides. The third side of triangle AC, to which the median was drawn, is the second diagonal of the resulting parallelogram.

According to the theorem, the sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of its sides.

2(a 2 +b 2)=d 1 2 +d 2 2

Let's denote the diagonal of the parallelogram, which is formed by the continuation of the median of the original triangle as x, we get:

2(8 2 + 9 2) = 13 2 + x 2
290 = 169 + x2
x2 = 290 - 169
x2 = 121
x = 11

Since the desired median is equal to half the diagonal of the parallelogram, then the value of the median of the triangle will be 11/2 = 5.5 cm

Answer: 5.5 cm

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