The topic of arithmetic and geometric mean is included in the mathematics program for grades 6-7. Since the paragraph is quite simple to understand, it is quickly passed, and by the end of the school year, students forget it. But knowledge in basic statistics is needed to pass the exam, as well as for international SAT exams. And for everyday life, developed analytical thinking never hurts.
How to calculate the arithmetic and geometric mean of numbers
Suppose there is a series of numbers: 11, 4, and 3. The arithmetic mean is the sum of all numbers divided by the number of given numbers. That is, in the case of numbers 11, 4, 3, the answer will be 6. How is 6 obtained?
Solution: (11 + 4 + 3) / 3 = 6
The denominator must contain a number equal to the number of numbers whose average is to be found. The sum is divisible by 3, since there are three terms.
Now we need to deal with the geometric mean. Let's say there is a series of numbers: 4, 2 and 8.
The geometric mean is the product of all given numbers, which is under a root with a degree equal to the number of given numbers. That is, in the case of numbers 4, 2 and 8, the answer is 4. Here's how it happened:
Solution: ∛(4 × 2 × 8) = 4
In both options, whole answers were obtained, since special numbers were taken as an example. This is not always the case. In most cases, the answer has to be rounded or left at the root. For example, for the numbers 11, 7, and 20, the arithmetic mean is ≈ 12.67, and the geometric mean is ∛1540. And for the numbers 6 and 5, the answers, respectively, will be 5.5 and √30.
Can it happen that the arithmetic mean becomes equal to the geometric mean?
Of course it can. But only in two cases. If there is a series of numbers consisting only of either ones or zeros. It is also noteworthy that the answer does not depend on their number.
Proof with units: (1 + 1 + 1) / 3 = 3 / 3 = 1 (arithmetic mean).
∛(1 × 1 × 1) = ∛1 = 1 (geometric mean).
Proof with zeros: (0 + 0) / 2=0 (arithmetic mean).
√(0 × 0) = 0 (geometric mean).
There is no other option and there cannot be.
Method of averages
3.1 Essence and meaning of averages in statistics. Types of averages
Average value in statistics, a generalized characteristic of qualitatively homogeneous phenomena and processes according to some varying attribute is called, which shows the level of the attribute, related to the unit of the population. average value abstract, because characterizes the value of the attribute for some impersonal unit of the population.Essence of average magnitude lies in the fact that the general and necessary, i.e., the tendency and regularity in the development of mass phenomena, are revealed through the individual and the accidental. Features that generalize in average values are inherent in all units of the population. Due to this, the average value is of great importance for identifying patterns inherent in mass phenomena and not noticeable in individual units of the population.
General principles for the use of averages:
a reasonable choice of the population unit for which the average value is calculated is necessary;
when determining the average value, it is necessary to proceed from the qualitative content of the averaged trait, take into account the relationship of the studied traits, as well as the data available for calculation;
average values should be calculated according to qualitatively homogeneous aggregates, which are obtained by the grouping method, which involves the calculation of a system of generalizing indicators;
overall averages should be supported by group averages.
Depending on the nature of the primary data, the scope and method of calculation in statistics, the following are distinguished: main types of averages:
1) power averages(arithmetic mean, harmonic, geometric, root mean square and cubic);
2) structural (non-parametric) averages(mode and median).
In statistics, the correct characterization of the population under study on the basis of varying characteristics in each individual case is given only by a well-defined type of average. The question of what type of average should be applied in a particular case is resolved by a specific analysis of the population under study, as well as based on the principle of meaningfulness of the results when summing up or when weighing. These and other principles are expressed in statistics the theory of averages.
For example, the arithmetic mean and the harmonic mean are used to characterize the mean value of a variable trait in the population being studied. The geometric mean is used only when calculating the average rate of dynamics, and the mean square only when calculating the variation indicators.
Formulas for calculating average values are presented in Table 3.1.
Table 3.1 - Formulas for calculating average values
|
Types of averages |
Calculation formulas |
|
|
simple |
weighted |
|
|
1. Arithmetic mean |
|
|
|
2. Average harmonic | ||
|
3. Geometric mean | ||
|
4. Root Mean Square |
|
|
Designations:- quantities for which the average is calculated; - average, where the line above indicates that the averaging of individual values takes place; - frequency (repeatability of individual trait values).
Obviously, different averages are derived from the general formula for the power mean (3.1) :
,
(3.1)
for k = + 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = +2 - root mean square.
Averages are either simple or weighted. weighted averages values are called that take into account that some variants of the attribute values may have different numbers; in this regard, each option has to be multiplied by this number. In this case, the “weights” are the numbers of population units in different groups, i.e. each option is "weighted" by its frequency. The frequency f is called statistical weight or weighing average.
Eventually correct choice of average assumes the following sequence:
a) the establishment of a generalizing indicator of the population;
b) determination of a mathematical ratio of values for a given generalizing indicator;
c) replacement of individual values by average values;
d) calculation of the average using the corresponding equation.
3.2 Arithmetic mean and its properties and calculation technique. Average harmonic
Arithmetic mean- the most common type of medium size; it is calculated in those cases when the volume of the averaged attribute is formed as the sum of its values for individual units of the studied statistical population.
The most important properties of the arithmetic mean:
1. The product of the average and the sum of frequencies is always equal to the sum of the products of the variant (individual values) and frequencies.
2. If any arbitrary number is subtracted (added) from each option, then the new average will decrease (increase) by the same number.
3. If each option is multiplied (divided) by some arbitrary number, then the new average will increase (decrease) by the same amount
4. If all frequencies (weights) are divided or multiplied by any number, then the arithmetic mean will not change from this.
5. The sum of deviations of individual options from the arithmetic mean is always zero.
It is possible to subtract an arbitrary constant value from all values of the attribute (better is the value of the middle option or options with the highest frequency), reduce the resulting differences by a common factor (preferably by the value of the interval), and express the frequencies in particulars (in percent) and multiply the calculated average by the common factor and add an arbitrary constant value. This method of calculating the arithmetic mean is called method of calculation from conditional zero .
Geometric mean finds its application in determining the average growth rate (average growth rates), when the individual values of the trait are presented as relative values. It is also used if it is necessary to find the average between the minimum and maximum values of a characteristic (for example, between 100 and 1000000).
root mean square used to measure the variation of a trait in the population (calculation of the standard deviation).
In statistics it works Majority rule for means:
X harm.< Х геом. < Х арифм. < Х квадр. < Х куб.
3.3 Structural means (mode and median)
To determine the structure of the population, special averages are used, which include the median and mode, or the so-called structural averages. If the arithmetic mean is calculated based on the use of all variants of the attribute values, then the median and mode characterize the value of the variant that occupies a certain average position in the ranked variation series
Fashion- the most typical, most often encountered value of the attribute. For discrete series the mode will be the one with the highest frequency. To define fashion interval series first determine the modal interval (interval having the highest frequency). Then, within this interval, the value of the feature is found, which can be a mode.
To find a specific value of the mode of the interval series, it is necessary to use the formula (3.2)
(3.2)
where X Mo is the lower limit of the modal interval; i Mo - the value of the modal interval; f Mo is the frequency of the modal interval; f Mo-1 - the frequency of the interval preceding the modal; f Mo+1 - the frequency of the interval following the modal.
Fashion is widely used in marketing activities in the study of consumer demand, especially in determining the sizes of clothes and shoes that are in greatest demand, while regulating pricing policy.
Median - the value of the variable attribute, falling in the middle of the ranged population. For ranked series with an odd number individual values (for example, 1, 2, 3, 6, 7, 9, 10) the median will be the value that is located in the center of the series, i.e. the fourth value is 6. For ranked series with an even number individual values (for example, 1, 5, 7, 10, 11, 14) the median will be the arithmetic mean value, which is calculated from two adjacent values. For our case, the median is (7+10)/2= 8.5.
Thus, to find the median, it is first necessary to determine its ordinal number (its position in the ranked series) using formulas (3.3):
(if there are no frequencies)
N Me= 
(if there are frequencies)
(3.3)
where n is the number of units in the population.
The numerical value of the median interval series determined by the accumulated frequencies in a discrete variational series. To do this, you must first specify the interval for finding the median in the interval series of the distribution. The median is the first interval where the sum of the accumulated frequencies exceeds half of the observations of the total number of all observations.
The numerical value of the median is usually determined by the formula (3.4)
(3.4)
where x Me - the lower limit of the median interval; iMe - the value of the interval; SMe -1 - the accumulated frequency of the interval that precedes the median; fMe is the frequency of the median interval.
Within the found interval, the median is also calculated using the formula Me = xl e, where the second factor on the right side of the equation shows the location of the median within the median interval, and x is the length of this interval. The median divides the variation series in half by frequency. Define more quartiles , which divide the variation series into 4 parts of equal size in probability, and deciles dividing the series into 10 equal parts.
What is the arithmetic mean
The arithmetic mean of several values is the ratio of the sum of these values to their number.
The arithmetic mean of a certain series of numbers is called the sum of all these numbers, divided by the number of terms. Thus, the arithmetic mean is the average value of the number series.
What is the arithmetic mean of several numbers? And they are equal to the sum of these numbers, which is divided by the number of terms in this sum.
How to find the arithmetic mean
There is nothing difficult in calculating or finding the arithmetic mean of several numbers, it is enough to add up all the numbers presented, and divide the resulting sum by the number of terms. The result obtained will be the arithmetic mean of these numbers.
Let's consider this process in more detail. What do we need to do to calculate the arithmetic mean and get the final result of this number.
First, to calculate it, you need to determine a set of numbers or their number. This set can include large and small numbers, and their number can be anything.
Secondly, all these numbers need to be added up and get their sum. Naturally, if the numbers are simple and their number is small, then the calculations can be done by writing by hand. And if the set of numbers is impressive, then it is better to use a calculator or spreadsheet.
And, fourthly, the amount obtained from addition must be divided by the number of numbers. As a result, we get the result, which will be the arithmetic mean of this series.
What is the arithmetic mean for?
The arithmetic mean can be useful not only for solving examples and problems in mathematics lessons, but for other purposes necessary in a person’s daily life. Such goals can be the calculation of the arithmetic mean to calculate the average expense of finance per month, or to calculate the time you spend on the road, also in order to find out attendance, productivity, speed, productivity and much more.
So, for example, let's try to calculate how much time you spend commuting to school. Going to school or returning home, you spend different time on the road each time, because when you are in a hurry, you go faster, and therefore the road takes less time. But, returning home, you can go slowly, talking with classmates, admiring nature, and therefore it will take more time for the road.
Therefore, you will not be able to accurately determine the time spent on the road, but thanks to the arithmetic mean, you can approximately find out the time you spend on the road.
Suppose that on the first day after the weekend, you spent fifteen minutes on the way from home to school, on the second day your journey took twenty minutes, on Wednesday you covered the distance in twenty-five minutes, in the same time you made your way on Thursday, and on Friday you were in no hurry and returned for half an hour.
Let's find the arithmetic mean, adding the time, for all five days. So,
15 + 20 + 25 + 25 + 30 = 115
Now divide this amount by the number of days
Through this method, you have learned that the journey from home to school takes approximately twenty-three minutes of your time.
Homework
1. Using simple calculations, find the arithmetic average of the attendance of students in your class per week.
2. Find the arithmetic mean:
3. Solve the problem:
Topic 5. Averages as statistical indicators
The concept of average. Scope of average values in a statistical study
Average values are used at the stage of processing and summarizing the obtained primary statistical data. The need to determine the average values is due to the fact that for different units of the studied populations, the individual values of the same trait, as a rule, are not the same.
Average value call an indicator that characterizes the generalized value of a feature or a group of features in the study population.
If a population with qualitatively homogeneous characteristics is being studied, then the average value appears here as typical average. For example, for groups of workers in a certain industry with a fixed level of income, a typical average spending on basic necessities is determined, i.e. the typical average generalizes the qualitatively homogeneous values of the attribute in the given population, which is the share of expenditures of workers in this group on essential goods.
In the study of a population with qualitatively heterogeneous characteristics, the atypical average indicators may come to the fore. Such, for example, are the average indicators of the produced national income per capita (different age groups), the average yields of grain crops throughout Russia (areas of different climatic zones and different grain crops), the average birth rates of the population in all regions of the country, the average temperature for a certain period, etc. Here, average values generalize qualitatively heterogeneous values of features or systemic spatial aggregates (international community, continent, state, region, district, etc.) or dynamic aggregates extended in time (century, decade, year, season, etc.) . These averages are called system averages.
Thus, the meaning of average values consists in their generalizing function. The average value replaces a large number of individual values of a trait, revealing common properties inherent in all units of the population. This, in turn, makes it possible to avoid random causes and to identify common patterns due to common causes.
Types of average values and methods for their calculation
At the stage of statistical processing, a variety of research tasks can be set, for the solution of which it is necessary to choose the appropriate average. In this case, it is necessary to be guided by the following rule: the values \u200b\u200bthat represent the numerator and denominator of the average must be logically related to each other.
power averages;
structural averages.
Let us introduce the following notation:
The values for which the average is calculated;
Average, where the line above indicates that the averaging of individual values takes place;
Frequency (repeatability of individual trait values).
Various means are derived from the general power mean formula:
(5.1)
for k = 1 - arithmetic mean; k = -1 - harmonic mean; k = 0 - geometric mean; k = -2 - root mean square.
Averages are either simple or weighted. weighted averages are called quantities that take into account that some variants of the values of the attribute may have different numbers, and therefore each variant has to be multiplied by this number. In other words, the "weights" are the numbers of population units in different groups, i.e. each option is "weighted" by its frequency. The frequency f is called statistical weight or weight average.
Arithmetic mean- the most common type of medium. It is used when the calculation is carried out on ungrouped statistical data, where you want to get the average summand. The arithmetic mean is such an average value of a feature, upon receipt of which the total volume of the feature in the population remains unchanged.
The arithmetic mean formula (simple) has the form
where n is the population size.
For example, the average salary of employees of an enterprise is calculated as the arithmetic average:
The determining indicators here are the wages of each employee and the number of employees of the enterprise. When calculating the average, the total amount of wages remained the same, but distributed, as it were, equally among all workers. For example, it is necessary to calculate the average salary of employees of a small company where 8 people are employed:
When calculating averages, individual values of the attribute that is averaged can be repeated, so the average is calculated using grouped data. In this case, we are talking about using arithmetic mean weighted, which looks like
(5.3)
So, we need to calculate the average share price of a joint-stock company at the stock exchange. It is known that transactions were carried out within 5 days (5 transactions), the number of shares sold at the sales rate was distributed as follows:
1 - 800 ac. - 1010 rubles
2 - 650 ac. - 990 rub.
3 - 700 ak. - 1015 rubles.
4 - 550 ac. - 900 rub.
5 - 850 ak. - 1150 rubles.
The initial ratio for determining the average share price is the ratio of the total amount of transactions (TCA) to the number of shares sold (KPA):
OSS = 1010 800+990 650+1015 700+900 550+1150 850= 3 634 500;
CPA = 800+650+700+550+850=3550.
In this case, the average share price was equal to
It is necessary to know the properties of the arithmetic mean, which is very important both for its use and for its calculation. There are three main properties that most of all led to the widespread use of the arithmetic mean in statistical and economic calculations.
Property one (zero): the sum of positive deviations of the individual values of the trait from its mean value is equal to the sum of negative deviations. This is a very important property, since it shows that any deviations (both with + and with -) due to random causes will be mutually canceled.
Proof:
The second property (minimum): the sum of the squared deviations of the individual values of the attribute from the arithmetic mean is less than from any other number (a), i.e. is the minimum number.
Proof.
Compose the sum of the squared deviations from the variable a:
(5.4)
To find the extremum of this function, it is necessary to equate its derivative with respect to a to zero:
From here we get:
(5.5)
Therefore, the extremum of the sum of squared deviations is reached at . This extremum is the minimum, since the function cannot have a maximum.
Third property: the arithmetic mean of a constant is equal to this constant: at a = const.
In addition to these three most important properties of the arithmetic mean, there are so-called design properties, which are gradually losing their significance due to the use of electronic computers:
if the individual value of the attribute of each unit is multiplied or divided by a constant number, then the arithmetic mean will increase or decrease by the same amount;
the arithmetic mean will not change if the weight (frequency) of each feature value is divided by a constant number;
if the individual values of the attribute of each unit are reduced or increased by the same amount, then the arithmetic mean will decrease or increase by the same amount.
Average harmonic. This average is called the inverse arithmetic average, since this value is used when k = -1.
Simple harmonic mean is used when the weights of the characteristic values are the same. Its formula can be derived from the base formula by substituting k = -1:
For example, we need to calculate the average speed of two cars that have traveled the same path, but at different speeds: the first at 100 km/h, the second at 90 km/h. Using the harmonic mean method, we calculate the average speed:
In statistical practice, harmonic weighted is more often used, the formula of which has the form
This formula is used in cases where the weights (or volumes of phenomena) for each attribute are not equal. In the original ratio, the numerator is known to calculate the average, but the denominator is unknown.
