Sample median formula. Structural characteristics of the variation distribution series

Median Me they call the value of the attribute that falls in the middle of the ranked series and divides it into two parts equal in number of units. Thus, in the ranked row of the distribution, one half of the row has attribute values exceeding the median, the other half is less than the median.

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

IN discrete variation series containing odd number units, the median is equal to the variant of the characteristic having the number:
,
where N is the number of population units.
In a discrete series consisting of an even number of population units, the median is defined as the average of the options having numbers and:
.
In the distribution of workers by length of service, the median is equal to the average of the options having numbers 10 in the ranked series: 2 = 5 and 10: 2 + 1 = 6. The options for the fifth and sixth characteristics are equal to 4 years, thus
of the year
When calculating the median in interval the row is first found median interval, (i.e. containing the median), for which accumulated frequencies or frequencies are used. The median is an interval whose accumulated frequency is equal to or greater than half the total volume of the population. The median value is then calculated using the formula:
,
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.
Let's calculate the median of the distribution of workers by salary (see lecture “Summary and grouping of statistical data”).
The median is the interval wages 800-900 UAH, since its cumulative frequency is 17, which exceeds half the sum of all frequencies (). Then
Me=800+100 UAH.
The obtained value indicates that half of the workers have wages below 875 UAH, but this is above the average.
To determine the median, you can use cumulative frequencies instead of cumulative frequencies.
The median, like the mode, does not depend on the extreme values of the variant, therefore it is also used to characterize the center in distribution series with uncertain boundaries.
Median property : the sum of absolute values of deviations from the median is less than from any other value (including from the arithmetic mean):

This property of the median is used in transport when designing the location of tram and trolleybus stops, gas stations, assembly points, etc.
Example. There are 10 garages along the 100 km long highway. To design the construction of a gas station, data was collected on the number of expected trips to the gas station for each garage.
Table 2 - Data on the number of trips to gas station for each garage.

It is necessary to install a gas station so that the total mileage of vehicles for refueling is minimal.
Option 1. If a gas station is placed in the middle of the highway, i.e. at the 50th kilometer (the center of the range of changes in the attribute), then the mileage, taking into account the number of trips, will be:
a) in one direction:
;
b) in the opposite:
;
c) total mileage in both directions: .

Option 2. If a gas station is placed on the middle section of the highway, determined by the arithmetic average formula taking into account the number of trips:

The median can be determined graphically, using the cumulate (see lecture “Summary and grouping of statistical data”). To do this, the last ordinate, equal to the sum of all frequencies or frequencies, is divided in half. From the resulting point, a perpendicular is restored until it intersects with the cumulate. The abscissa of the intersection point gives the median value.

Let us assume that we need to determine average level in the distribution of student scores or in the quality control data sample. To do this, you will need to calculate the median of a set of numbers using the MEDIAN function.

This function is one way to measure central tendency, that is, the location of the center of a set of numbers in a statistical distribution. There are three most common ways to determine central tendency.

Average value- this is a value that is an arithmetic mean, that is, it is calculated by adding a set of numbers and then dividing the resulting sum by their number. For example, the average of the numbers 2, 3, 3, 5, 7 and 10 is 5 (the result of dividing the sum of these numbers, which is 30, by their number, which is 6).

Median- a number that is the middle of a set of numbers: half the numbers have values greater than the median, and half the numbers have values less. For example, the median for the numbers 2, 3, 3, 5, 7 and 10 would be 4.

Fashion- the number most often found in a given set of numbers. For example, the mode for the numbers 2, 3, 3, 5, 7 and 10 would be 3.

With a symmetrical distribution of a set of numbers, all three values of central tendency will coincide. When the distribution of many numbers is biased, the values may be different.

The screenshots in this article are from Excel 2016. If you're using a different version, the interface may be slightly different, but the features will be the same.

Example

To make this example easier to understand, copy it onto a blank sheet of paper.

Advice: To switch between viewing the results and viewing the formulas that return those results, press CTRL+` (apostrophe) or on the tab Formulas in Group Formula dependencies click the button Show formulas.

To calculate the median in MS EXCEL, there is a special function MEDIAN(). In this article we will define the median and learn how to calculate it for a sample and for a given distribution law random variable.

Let's start with medians For samples(i.e. for a fixed set of values).

Sample median

Median(median) is a number that is the middle of a set of numbers: half of the numbers in the set are greater than median, and half the numbers are less than median.

To calculate medians necessary first (values in sample). For example, median for sample (2; 3; 3; 4 ; 5; 7; 10) will be 4. Because just in sample 7 values, three of them are less than 4 (i.e. 2; 3; 3), and three of them are greater (i.e. 5; 7; 10).

If the set contains an even number of numbers, then it is calculated for the two numbers in the middle of the set. For example, median for sample (2; 3; 3 ; 6 ; 7; 10) will be 4.5, because (3+6)/2=4.5.

For determining medians in MS EXCEL there is a function of the same name MEDIAN(), English version MEDIAN().

Median does not necessarily coincide with . A match occurs only if the values in the sample are distributed symmetrically with respect to average. For example, for samples (1; 2; 3 ; 4 ; 5; 6) median And average equal to 3.5.

If known Distribution function F(x) or probability density function p(X), That median can be found from the equation:

For example, having solved this equation analytically for the Lognormal distribution lnN(μ; σ 2), we obtain that median calculated using the formula =EXP(μ). When μ=0, the median is 1.

Pay attention to the point Distribution functions, for which F(x)=0.5(see picture above) . The abscissa of this point is equal to 1. This is the value of the median, which naturally coincides with the previously calculated value using the em formula.

In MS EXCEL median For lognormal distribution LnN(0;1) can be calculated using the formula =LOGNORM.REV(0.5,0,1).

Note: Recall that the integral of over the entire domain of specifying the random variable is equal to one.

Therefore, the median line (x=Median) divides the area under the graph probability density function into two equal parts.

TEST

On the topic: "Mode. Median. Methods for their calculation"

Introduction

Average values and associated 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 ones in the course.

The average is a very common summary measure in statistics. This is explained by the fact that only with the help of the average can a population be characterized by a quantitatively varying characteristic. In statistics, the average value is a generalizing characteristic of a set of similar phenomena based on some quantitatively varying characteristic. The average shows the level of this characteristic per unit of the population.

When studying social phenomena and trying to identify their characteristic, typical features in specific conditions of place and time, statisticians widely use average values. Using averages, you can compare different populations 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 average rates of dynamics, and the mean square is used only when calculating variation indices.

The arithmetic mean is the quotient of dividing the sum of the variants by their number. It is used in cases where the volume of a varying characteristic for the entire population is formed as the sum of the characteristic values of 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 characteristics in the aggregate is most often formed precisely as the sum of the characteristic values of 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 inverse values of the variant. It is used when, depending on the material, the weights have to be not multiplied, but divided into options or, what is the same thing, multiplied by their reciprocal 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 resorted to in cases where not the units of the population - the carriers of the characteristic - are used as weights, but the products of these units by the value of the characteristic.

1. Definition of mode and median in statistics

Arithmetic and harmonic means are generalizing characteristics of the population according to one or another varying characteristic. Auxiliary descriptive characteristics of the distribution of a varying characteristic are mode and median.

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

In statistics, the median is the option that is in the middle of the variation series. The median divides the series in half; on both sides of it (up and down) there are the same number of population units.

Mode and median, in contrast to power means, are specific characteristics; their meaning is assigned to any specific option in the variation series.

Mode is used in cases where it is necessary to characterize the most frequently occurring value of a characteristic. If you need, for example, to find out the most common wage rate at an enterprise, the price on the market at which it was sold greatest number goods, the size of shoes that is most in demand among consumers, etc., in these cases they resort to fashion.

The median is interesting in that it shows the quantitative limit of the value of a varying characteristic, which half of the members of the population have reached. Let the average salary of bank employees be 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 give the median. Mode and median are typical characteristics in cases where populations are homogeneous and large in number.

2. Finding the mode and median in a discrete variation series

Finding the mode and median in a variation series, where the values of a characteristic are given by certain numbers, is not very difficult. Let's look at Table 1 with the distribution of families by 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 option value corresponds to the largest number of families. There may be distributions where all options occur equally often, in which case there is no mode, or, in other words, we can say that all options are equally modal. In other cases, not one, but two options may be of the highest frequency. Then there will be two modes, the distribution will be bimodal. Bimodal distributions may indicate qualitative heterogeneity of the population according to the characteristic being studied.

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 smallest options. The sum of the frequencies of the 1st and 2nd options is 40. It is clear that there are no 93 options here. If we add the frequency of the 3rd option to 40, we get a sum equal to 40 + 75 = 115. Therefore, the 93rd option corresponds to the third value of the varying characteristic, and the median will be a family with two children.

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

3. Calculation of mode and median in 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 specific option. Therefore, the mode and median do not require calculations to find if all the values of the attribute are known. However, in an 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 characteristic contained in an interval, use the formula:

M o = X Mo + i Mo *(f Mo – f Mo-1)/((f Mo – f Mo-1) + (f Mo – f Mo+1)),

Where XMo is the minimum boundary of the modal interval;

i Mo – the value of the modal interval;

f Mo – frequency of the modal interval;

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

f Mo+1 – frequency of the interval following the modal one.

Let us show the calculation of the mode using the example given in Table 2.

Table 2. Distribution of enterprise workers by fulfillment of production standards

To find the mode, we first determine the modal interval this series. The example shows that the highest frequency corresponds to the interval where the variants lie in the range from 100 to 105. This is the modal interval. The modal interval value is 5.

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

M o = 100 + 5 * (104 -12)/((104 – 12) + (104 – 98)) = 108.8

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

Let's now calculate the median. To find the median in an interval variation series, we first determine the interval in which it is located (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 gradually summing frequencies, starting from the interval with the lowest value of the attribute. Half of the sum of frequencies is 250 (500:2). Therefore, according to Table 3, the median interval will be the interval with a salary value of 350,000 rubles. up to 400,000 rub.

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

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

Due to the fact that the researcher does not have data on sales volume at each exchange office, calculating the arithmetic average in order to determine average price for a dollar is not practical.

Median of a series of numbers

However, it is possible to determine the value of the attribute, which is called the median (Me). Median

in our example

Median number: NoMe = ;

Fashion

Table 3.6.

f— sum of frequencies of the series;

S cumulative frequencies

12_

S—accumulated frequencies.

In Fig. 3.2. Shown is a histogram of the distribution of banks by profit margin (according to Table 3.6.).

x - profit amount, million rubles,

f is the number of banks.

"MEDIAN OF ORDERED SERIES"

Text HTML version of the publication

Algebra lesson notes in 7th grade

Lesson topic: “MEDIAN OF AN ORDERED SERIES.”

teacher of the Ozyornaya school, branch of the MCOU Burkovskaya secondary school Eremenko Tatyana Alekseevna
Goals:
the concept of median as a statistical characteristic of an ordered series; develop the ability to find the median for ordered series with an even and odd number of terms; to develop the ability to interpret the values of the median depending on the practical situation, to consolidate the concept of the arithmetic mean of a set of numbers. Develop skills independent work. Develop an interest in mathematics.
During the classes

Oral work.
The rows are given: 1) 4; 1; 8; 5; 1; 2) ; 9; 3; 0.5; ; 3) 6; 0.2; ; 4; 6; 7.3; 6. Find: a) the largest and smallest values of each series; b) the scope of each row; c) the mode of each row.
II. Explanation of new material.
Work according to the textbook. 1. Let's consider the problem from paragraph 10 of the textbook. What does ordered series mean? I would like to emphasize that before finding the median, you must always order the data series. 2.On the board we get acquainted with the rules for finding the median for series with an even and odd number of terms:
Median

orderly

row
numbers
With

odd

number

members
is the number written in the middle, and
median

ordered series
numbers
with an even number of members
is called the arithmetic mean of two numbers written in the middle.
Median

arbitrary

row
is called the median 1 3 1 7 5 4 of the corresponding ordered series.
I note that indicators - average arithmetic, mode and median by

differently

characterize

data,

received

result

observations.

III. Formation of skills and abilities.
1st group. Exercises on applying formulas for finding the median of an ordered and unordered series. 1.
№ 186.
Solution: a) Number of members of the series P= 9; median Meh= 41; b) P= 7, the row is ordered, Meh= 207; V) P= 6, the row is ordered, Meh= = 21; G) P= 8, the row is ordered, Meh= = 2.9. Answer: a) 41; b) 207; at 21; d) 2.9. Students comment on how to find the median. 2. Find the arithmetic mean and median of a series of numbers: a) 27, 29, 23, 31, 21, 34; V) ; 1. b) 56, 58, 64, 66, 62, 74. Solution: To find the median, it is necessary to order each row: a) 21, 23, 27, 29, 31, 34. P = 6; X = = 27,5; Meh= = 28; 20 22 2 + 2, 6 3, 2 2 + 1125 ; ; ; 3636 21 23 27 29 31 34 165 66 +++++ = 27 29 2 + b) 56, 58, 62, 64, 66, 74.

How to find the median in statistics

P = 6; X = 63,3; Meh= = 63; V) ; 1. P = 5; X = : 5 = 3: 5 = 0,6; Meh = . 3.
№ 188
(orally). Answer: yes; b) no; c) no; d) yes. 4. Knowing that an ordered series contains T numbers, where T– an odd number, indicate the number of the member that is the median if T equals: a) 5; b) 17; c) 47; d) 201. Answer: a) 3; b) 9; c) 24; d) 101. 2nd group. Practical tasks on finding the median of the corresponding series and interpreting the result obtained. 1.
№ 189.
Solution: Number of series members P= 12. To find the median, the series must be ordered: 136, 149, 156, 158, 168, 174, 178, 179, 185, 185, 185, 194. Median of the series Meh= = 176. Monthly output was greater than the median for the following members of the artel: 56 58 62 64 66 74 380 66 +++++ =≈ 62 64 2 + 1125 ; ; ; 3636 1125 12456 18 1:5:5 6336 6 6 ++++ ⎛⎞ ++++ = = ⎜⎟ ⎝⎠ 2 3 67 174 178 22 xx+ + = 1) Kvitko; 4) Bobkov; 2) Baranov; 5) Rilov; 3) Antonov; 6) Astafiev. Answer: 176. 2.
№ 192.
Solution: Let's sort the data series: 30, 31, 32, 32, 32, 32, 32, 32, 33, 35, 35, 36, 36, 36, 38, 38, 38, 40, 40, 42; number of series members P= 20. Swing A = x max – x min = 42 – 30 = 12. Fashion Mo= 32 (this value occurs 6 times - more often than others). Median Meh= = 35. In this case, the range shows the greatest variation in the time for processing the part; the mode shows the most typical processing time value; median – processing time, which was not exceeded by half of the turners. Answer: 12; 32; 35.
IV. Lesson summary.
– What is the median of a series of numbers called? – Can the median of a series of numbers not coincide with any of the numbers in the series? – What number is the median of an ordered series containing 2 P numbers? 2 P– 1 numbers? – How to find the median of an unordered series?
Homework:
№ 187, № 190, № 191, № 254. 10 11 35 35 22 xx + + =

To the section basic general education

Mode and median

Average values also include mode and median.

The median and mode are often used as an average characteristic in those populations where the calculation of the average (arithmetic, harmonic, etc.) is impossible or impractical.

For example, a sample survey of 12 commercial currency exchange offices in Omsk made it possible to record different prices per dollar when it is sold (data as of October 10, 1995 at the exchange rate of the dollar -4493 rubles).

Due to the fact that the researcher does not have data on the sales volume at each exchange office, calculating the arithmetic average to determine the average price per dollar is impractical. However, it is possible to determine the value of the attribute, which is called the median (Me). Median lies in the middle of the ranked row and divides it in half.

The calculation of the median for ungrouped data is as follows:

a) arrange the individual values of the characteristic in ascending order:

4500 4500 4535 4540 4550 4560 4560 4560 4560 4570 4570 4570

b) determine the ordinal number of the median using the formula:

in our example this means that the median in this case is located between the sixth and seventh values of the attribute in the ranked series, since the series has an even number of individual values. Thus, Me is equal to the arithmetic mean of the neighboring values: 4550, 4560.

c) consider the procedure for calculating the median in the case of an odd number of individual values.

Let's say we observe not 12, but 11 currency exchange points, then the ranked series will look like this (discard the 12th point):

4500 4500 4535 4540 4550 4560 4560 4560 4560 4570 4570

Median number: NoMe = ;

in sixth place is = 4560, which is the median: Me = 4560. On both sides of it there are the same number of points.

Fashion— this is the most common value of a characteristic among units of a given population. It corresponds to a specific attribute value.

In our case, the modal price per dollar can be called 4560 rubles: this value is repeated 4 times, more often than all others.

In practice, the mode and median are usually found using grouped data. As a result of the grouping, a series of distributions of banks was obtained according to the amount of profit received for the year (Table 3.6.).

Table 3.6.

Grouping of banks by the amount of profit received for the year

To determine the median, you need to calculate the sum of the cumulative frequencies. The total increase continues until cumulative amount frequencies exceeding half the sum of frequencies. In our example, the sum of accumulated frequencies (12) exceeds half of all values (20:2). This value corresponds to the median interval, which contains the median (5.5 - 6.4). Let's determine its value using the formula:

where is the initial value of the interval containing the median;

— the value of the median interval;

f— sum of frequencies of the series;

— the sum of cumulative frequencies preceding the median interval;

— frequency of the median interval.

Thus, 50% of banks have a profit of 6.1 million rubles, and 50% of banks have a profit of more than 6.1 million rubles.

The highest frequency also corresponds to the interval 5.5 - 6.4, i.e. the mode must be in this interval. We determine its value using the formula:

where is the initial value of the interval containing the mode;

— the value of the modal interval;

— frequency of the modal interval;

— frequency of the interval preceding the modal;

— frequency of the interval following the modal one.

The given mode formula can be used in variation series with equal intervals.

Thus, in this population, the most common profit size is 6.10 million rubles.

The median and mode can be determined graphically. The median is determined by the cumulate (Fig. 3.1.). To construct it, it is necessary to calculate the cumulative frequencies and frequencies. Cumulative frequencies show how many population units have attribute values no greater than the value under consideration, and are determined by sequential summation of interval frequencies. When constructing a cumulative interval distribution series, the lower limit of the first interval corresponds to a frequency equal to zero, and the upper limit corresponds to the entire frequency of a given interval. The upper limit of the second interval corresponds to a cumulative frequency equal to the sum of the frequencies of the first two intervals, etc.

Let's construct a cumulative curve according to the data in Table. 6 on the distribution of banks by profit margin.

S cumulative frequencies

12_

3.7-4.6 4.6-5.5 5.5-6.4 6.4-7.3 7.3-8.2 X profit

Rice. 3.1. Cumulates of the distribution series of banks by profit margin:

x - profit amount, million rubles,

S—accumulated frequencies.

To determine the median, the height of the largest ordinate, which corresponds to the total population size, is divided in half. A straight line is drawn through the resulting point, parallel to the abscissa axis, until it intersects with the cumulate. The abscissa of the intersection point is the median.

The mode is determined by the distribution histogram. The histogram is built like this:

Equal segments are plotted on the abscissa axis, which on the accepted scale correspond to the size of the intervals of the variation series. Rectangles are constructed on the segments, the areas of which are proportional to the frequencies (or frequencies) of the interval.

Median in statistics

3.2. Shown is a histogram of the distribution of banks by profit margin (according to Table 3.6.).

3.7-4.6 4.6-5.5 5.5-6.4 6.4-7.3 7.3-8.2 X

Rice. 3.2. Distribution of commercial banks by profit margin:

x - profit amount, million rubles,

f is the number of banks.

To determine the mode, we connect the right vertex of the modal rectangle to the upper right corner of the previous rectangle, and the left vertex of the modal rectangle to the upper left corner of the subsequent rectangle. The abscissa of the intersection point of these lines will be the distribution mode.

Median (statistics)

Median (statistics), V mathematical statistics— a number characterizing a sample (for example, a set of numbers). If all the sample elements are different, then the median is the sample number such that exactly half of the sample elements are greater than it, and the other half are less than it. More generally, the median can be found by ordering the elements of a 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, a median in statistics is a value that divides a series in half in such a way that on both sides of it (down or up) there are the same number of units in a given population.

Task No. 1. Calculation of arithmetic mean, modal and median values

Because of this property, this indicator has several other names: 50th percentile or 0.5 quantile.

Average value
Median
Fashion

Median (statistics)

Median (statistics), in mathematical statistics, a number characterizing a sample (for example, a set of numbers). If all the sample elements are different, then the median is the sample number such that exactly half of the sample elements are greater than it, and the other half are less than it. More generally, the median can be found by ordering the elements of a 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.

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

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

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

Average value- arithmetic mean, which is calculated by adding a set of numbers and then dividing the resulting sum by their number.
For example, the average of the numbers 2, 3, 3, 5, 7 and 10 is 5, which is the result of dividing their sum of 30 by their sum of 6.
Median- a number that is the middle of a set of numbers: half the numbers have values greater than the median, and half the numbers have values less.
For example, the median for the numbers 2, 3, 3, 5, 7 and 10 would be 4.
Fashion- the number most often found in a given set of numbers.
For example, the mode for the numbers 2, 3, 3, 5, 7 and 10 would be 3.

Algebra lesson in 7th grade.

Topic: “Median as a statistical characteristic.”

Teacher Egorova N.I.

The purpose of the lesson: to form in students an idea of the median of a set of numbers and the ability to calculate it for simple numerical sets, to consolidate the concept of the arithmetic mean of a set of numbers.

Lesson type: explanation of new material.

During the classes

1. Organizational moment.

Inform the topic of the lesson and formulate its goals.

2. Updating previous knowledge.

Questions for students:

What is the arithmetic mean of a set of numbers?

Where is the arithmetic mean located within a set of numbers?

What characterizes the arithmetic mean of a set of numbers?

Where is the arithmetic mean of a set of numbers often used?

Oral tasks:

Find the arithmetic mean of a set of numbers:

Checking homework.

Textbook: No. 169, No. 172.

3. Studying new material.

In the previous lesson, we became acquainted with such a statistical characteristic as the arithmetic mean of a set of numbers. Today we will devote a lesson to another statistical characteristic - the median.

Not only the arithmetic mean shows where on the number line the numbers of any set are located and where their center is. Another indicator is the median.

The median of a set of numbers is the number that divides the set into two equal parts. Instead of “median,” you could say “middle.”

First, let's look at examples of how to find the median, and then give a strict definition.

Consider the following oral example using a projector

At the end school year 11 7th grade students passed the 100 meter running standard. The following results were recorded:

After the guys ran the distance, Petya approached the teacher and asked what his result was.

“Most average result: 16.9 seconds,” the teacher replied.

"Why?" – Petya was surprised. – After all, the arithmetic average of all the results is approximately 18.3 seconds, and I ran more than a second better. And in general, Katya’s result (18.4) is much closer to the average than mine.”

“Your result is average, since five people ran better than you, and five - worse. That is, you are right in the middle,” said the teacher.

Write down an algorithm for finding the median of a set of numbers:

Arrange a number set (make a ranked series).

Simultaneously cross out the “largest” and “smallest” numbers of a given set of numbers until one or two numbers remain.

If there is one number left, then it is the median.

If there are two numbers left, then the median will be the arithmetic mean of the two remaining numbers.

Invite students to independently formulate the definition of the median of a set of numbers, then read the definition of the median in the textbook (p. 40), then solve No. 186 (a, b), No. 187 (a) of the textbook (p. 41).

Comment:

Draw students' attention to an important fact: the median is practically insensitive to significant deviations of individual extreme values of sets of numbers. In statistics, this property is called stability. The stability of a statistical indicator is a very important property; it insures us against random errors and individual unreliable data.

4. Consolidation of the studied material.

Problem solving.

Let's denote x-arithmetic mean, Me-median.

Set of numbers: 1,3,5,7,9.

x=(1+3+5+7+9):5=25:5=5,

Set of numbers: 1,3,5,7,14.

x=(1+3+5+7+14):5=30:5=6.

a) Set of numbers: 3,4,11,17,21

b) Set of numbers: 17,18,19,25,28

c) Set of numbers: 25, 25, 27, 28, 29, 40, 50

Conclusion: the median of a set of numbers consisting of an odd number of members is equal to the number in the middle.

a) A set of numbers: 2, 4, 8, 9.

Me = (4+8):2=12:2=6

b) A set of numbers: 1,3,5,7,8,9.

Me = (5+7):2=12:2=6

The median of a set of numbers containing an even number of terms is equal to half the sum of the two numbers in the middle.

The student received the following grades in algebra during the quarter:

5, 4, 2, 5, 5, 4, 4, 5, 5, 5.

Find the mean and median of this set.

Let's find the average score, that is, the arithmetic mean:

x= (5+4+2+5+5+4+4+5+5+5): 10=44:10 = 4.4

Let's find the median of this set of numbers:

Let's order the set of numbers: 2,4,4,4,5,5,5,5,5,5

There are only 10 numbers, to find the median you need to take the two middle numbers and find their half-sum.

Me = (5+5):2 = 5

Question for students: If you were a teacher, what grade would you give this student for the quarter? Justify your answer.

The president of the company receives a salary of 300,000 rubles. three of his deputies receive 150,000 rubles each, forty employees - 50,000 rubles each. and the cleaning lady's salary is 10,000 rubles. Find the arithmetic mean and median of salaries in the company. Which of these characteristics is more beneficial for the president to use for advertising purposes?

x = (300000+3·150000+40·50000+10000):(1+3+40+1) = 2760000:45=61333.33 (rub.)

No. 6. Orally.

A) How many numbers are there in a set if its ninth term is its median?

B) How many numbers are there in a set if its median is the arithmetic mean of the 7th and 8th terms?

C) In a set of seven numbers, the largest number is increased by 14. Will this change the arithmetic mean and median?

D) Each of the numbers in the set is increased by 3. What happens to the arithmetic mean and median?

Sweets in the store are sold by weight. To find out how many candies are contained in one kilogram, Masha decided to find the weight of one candy. She weighed several candies and got the following results:

12, 13, 14, 12, 15, 16, 14, 13, 11.

Both characteristics are suitable for estimating the weight of one candy, because they are not very different from each other.

So, to characterize statistical information, the arithmetic mean and median are used. In many cases, one of the characteristics may not have any meaningful meaning (for example, having information about the time of road accidents, it hardly makes sense to talk about the arithmetic mean of these data).

Homework: paragraph 10, No. 186 (c, d), No. 190.

5. Lesson summary. Reflection.

"Statistical research: collection and grouping of statistical data"

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