What is the arithmetic mean of numbers. What is the arithmetic mean? How to find the arithmetic mean

Answer: everyone got one 4 pears.

Example 2. To courses in English on Monday 15 people came, on Tuesday - 10, on Wednesday - 12, on Thursday - 11, on Friday - 7, on Saturday - 14, on Sunday - 8. Find the average attendance of the courses for the week.
Solution: Let's find the arithmetic mean:

15 + 10 + 12 + 11 + 7 + 14 + 8	=	77	= 11
7		7

Answer: On average, people attended English language courses 11 person per day.

Example 3. A racer rode for two hours at 120 km/h and an hour at 90 km/h. Find the average speed of the car during the race.
Solution: Let's find the arithmetic average of the car speeds for each hour of travel:

120 + 120 + 90	=	330	= 110
3		3

Answer: the average speed of the car during the race was 110 km/h

Example 4. The arithmetic mean of 3 numbers is 6, and the arithmetic mean of 7 other numbers is 3. What is the arithmetic mean of these ten numbers?
Solution: Since the arithmetic mean of 3 numbers is 6, their sum is 6 3 = 18, similarly, the sum of the remaining 7 numbers is 7 3 = 21.
This means the sum of all 10 numbers will be 18 + 21 = 39, and the arithmetic mean is equal to

39	= 3.9
10

Answer: the arithmetic mean of 10 numbers is 3.9 .

It gets lost in calculating the average.

Average meaning set of numbers is equal to the sum of numbers S divided by the number of these numbers. That is, it turns out that average meaning equals: 19/4 = 4.75.

note

If you need to find the geometric mean for just two numbers, then you don’t need an engineering calculator: take the second root ( Square root) from any number can be done using the most ordinary calculator.

Helpful advice

Unlike the arithmetic mean, the geometric mean is not as strongly affected by large deviations and fluctuations between individual values ​​in the set of indicators under study.

Sources:

• Online calculator that calculates the geometric mean
• average geometric formula

Average value is one of the characteristics of a set of numbers. Represents a number that cannot be outside the range determined by the largest and lowest values in this set of numbers. Average arithmetic value is the most commonly used type of average.

Instructions

Add up all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on the specific calculation conditions, it is sometimes easier to divide each of the numbers by the number of values ​​in the set and sum the result.

Use, for example, included in the Windows OS if it is not possible to calculate the arithmetic average in your head. You can open it using the program launch dialog. To do this, press the hot keys WIN + R or click the Start button and select the Run command from the main menu. Then type calc in the input field and press Enter or click the OK button. The same can be done through the main menu - open it, go to the “All programs” section and in the “Standard” section and select the “Calculator” line.

Enter all the numbers in the set sequentially by pressing the Plus key after each of them (except the last one) or clicking the corresponding button in the calculator interface. You can also enter numbers either from the keyboard or by clicking the corresponding interface buttons.

Press the slash key or click this in the calculator interface after entering the last set value and type the number of numbers in the sequence. Then press the equal sign and the calculator will calculate and display the arithmetic mean.

You can use a table editor for the same purpose. Microsoft Excel. In this case, launch the editor and enter all the values ​​of the sequence of numbers into the adjacent cells. If, after entering each number, you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.

Click the cell next to the last number entered if you don't want to just see the average. Expand the Greek sigma (Σ) drop-down menu for the Edit commands on the Home tab. Select the line " Average" and the editor will insert the desired formula for calculating the arithmetic mean into the selected cell. Press the Enter key and the value will be calculated.

The arithmetic mean is one of the measures of central tendency, widely used in mathematics and statistical calculations. Find the average arithmetic number for several values ​​it is very simple, but each task has its own nuances, which are simply necessary to know in order to perform correct calculations.

What is an arithmetic mean

The arithmetic mean determines the average value for the entire original array of numbers. In other words, from a certain set of numbers, a value common to all elements is selected, mathematical comparison which with all elements is approximately equal in nature. The arithmetic average is used primarily in the preparation of financial and statistical reports or for calculating the results of similar experiments.

How to find the arithmetic mean

Finding the arithmetic mean for an array of numbers should begin by determining the algebraic sum of these values. For example, if the array contains the numbers 23, 43, 10, 74 and 34, then their algebraic sum will be equal to 184. When writing, the arithmetic mean is denoted by the letter μ (mu) or x (x with a bar). Next, the algebraic sum should be divided by the number of numbers in the array. In the example under consideration there were five numbers, so the arithmetic mean will be equal to 184/5 and will be 36.8.

Features of working with negative numbers

If the array contains negative numbers, then the arithmetic mean is found using a similar algorithm. The difference only exists when calculating in the programming environment, or if the problem has additional conditions. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps:

1. Finding the general arithmetic average using the standard method;
2. Finding the arithmetic mean of negative numbers.
3. Calculation of the arithmetic mean of positive numbers.

The responses for each action are written separated by commas.

Natural and decimal fractions

If an array of numbers is presented decimals, the solution is carried out using the method of calculating the arithmetic mean of integers, but the result is reduced according to the requirements of the problem for the accuracy of the answer.

When working with natural fractions they should be reduced to a common denominator, which is multiplied by the number of numbers in the array. The numerator of the answer will be the sum of the given numerators of the original fractional elements.

Engineering calculator.

Instructions

Keep in mind that in general, the geometric mean of numbers is found by multiplying these numbers and taking the root of the power from them, which corresponds to the number of numbers. For example, if you need to find the geometric mean of five numbers, then you will need to extract the root of the power from the product.

To find the geometric mean of two numbers, use the basic rule. Find their product, then take the square root of it, since the number is two, which corresponds to the power of the root. For example, in order to find the geometric mean of the numbers 16 and 4, find their product 16 4=64. From the resulting number, extract the square root √64=8. This will be the desired value. Please note that the arithmetic mean of these two numbers is greater than and equal to 10. If the entire root is not extracted, round the result to the desired order.

To find the geometric mean of more than two numbers, also use the basic rule. To do this, find the product of all numbers for which you need to find the geometric mean. From the resulting product, extract the root of the power equal to the number of numbers. For example, to find the geometric mean of the numbers 2, 4, and 64, find their product. 2 4 64=512. Since you need to find the result of the geometric mean of three numbers, take the third root from the product. It is difficult to do this verbally, so use an engineering calculator. For this purpose it has a button "x^y". Dial the number 512, press the "x^y" button, then dial the number 3 and press the "1/x" button, to find the value of 1/3, press the "=" button. We get the result of raising 512 to the 1/3 power, which corresponds to the third root. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.

Using an engineering calculator, you can find the geometric mean in another way. Find the log button on your keyboard. After that, take the logarithm for each of the numbers, find their sum and divide it by the number of numbers. Take the antilogarithm from the resulting number. This will be the geometric mean of the numbers. For example, in order to find the geometric mean of the same numbers 2, 4 and 64, perform a set of operations on the calculator. Dial the number 2, then press the log button, press the "+" button, dial the number 4 and press log and "+" again, dial 64, press log and "=". The result will be a number equal to the sum decimal logarithms numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers for which the geometric mean is sought. From the result, take the antilogarithm by switching the case button and use the same log key. The result will be the number 8, this is the desired geometric mean.

Most of all in eq. In practice, we have to use the arithmetic mean, which can be calculated as the simple and weighted arithmetic mean.

Arithmetic average (SA)-n The most common type of average. It is used in cases where the volume of a varying characteristic for the entire population is the sum of the values ​​of the characteristics of its individual units. Social phenomena are characterized by the additivity (totality) of the volumes of a varying characteristic; this determines the scope of application of SA and explains its prevalence as a general indicator, for example: the general salary fund is the sum of the salaries of all employees.

To calculate SA, you need to divide the sum of all feature values ​​by their number. SA is used in 2 forms.

Let's first consider a simple arithmetic average.

1-CA simple (initial, defining form) is equal to the simple sum of the individual values ​​of the characteristic being averaged, divided by the total number of these values ​​(used when there are ungrouped index values ​​of the characteristic):

The calculations made can be generalized into the following formula:

(1)

Where - the average value of the varying characteristic, i.e., the simple arithmetic average;

means summation, i.e. the addition of individual characteristics;

x- individual values ​​of a varying characteristic, which are called variants;

n - number of units of the population

Example 1, it is required to find the average output of one worker (mechanic), if it is known how many parts each of 15 workers produced, i.e. given a series of ind. attribute values, pcs.: 21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.

Simple SA is calculated using formula (1), pcs.:

Example2. Let's calculate SA based on conditional data for 20 stores included in the trading company (Table 1). Table 1

Distribution of stores of the trading company "Vesna" by sales area, sq. M

Store no.		Store no.

To calculate the average store area ( ) it is necessary to add up the areas of all stores and divide the resulting result by the number of stores:

Thus, the average store area for this group of retail enterprises is 71 sq.m.

Therefore, to determine a simple SA, you need to divide the sum of all values ​​of a given attribute by the number of units possessing this attribute.

2

Where f 1 , f 2 , … ,f n – weight (frequency of repetition of identical signs);

– the sum of the products of the magnitude of features and their frequencies;

– the total number of population units.

- SA weighted - With The middle of options that are repeated a different number of times, or, as they say, have different weights. The weights are the number of units in different groups aggregates (identical options are combined into a group). SA weighted – average of grouped values x 1 , x 2 , .., x n, – calculated: (2)

Where X- options;

f- frequency (weight).

Weighted SA is the quotient of dividing the sum of the products of options and their corresponding frequencies by the sum of all frequencies. Frequencies ( f) appearing in the SA formula are usually called scales, as a result of which the SA calculated taking into account the weights is called weighted.

We will illustrate the technique of calculating weighted SA using example 1 discussed above. To do this, we will group the initial data and place them in the table.

The average of the grouped data is determined as follows: first, the options are multiplied by the frequencies, then the products are added and the resulting sum is divided by the sum of the frequencies.

According to formula (2), the weighted SA is equal, pcs.:

Distribution of workers for parts production

P

The data presented in the previous example 2 can be combined into homogeneous groups, which are presented in table. Table

Distribution of Vesna stores by sales area, sq. m

Thus, the result was the same. However, this will already be a weighted arithmetic mean value.

In the previous example, we calculated the arithmetic average provided that the absolute frequencies (number of stores) are known. However, in a number of cases, absolute frequencies are absent, but relative frequencies are known, or, as they are commonly called, frequencies that show the proportion or the proportion of frequencies in the entire set.

When calculating SA weighted use frequencies allows you to simplify calculations when the frequency is expressed in large, multi-digit numbers. The calculation is made in the same way, however, since the average value turns out to be increased by 100 times, the result should be divided by 100.

Then the formula for the arithmetic weighted average will look like:

Where d– frequency, i.e. the share of each frequency in the total sum of all frequencies.

(3)

In our example 2, we first determine the share of stores by group in the total number of stores of the Vesna company. So, for the first group the specific gravity corresponds to 10%
. We get the following data Table3

As the number of elements of a set of numbers tends stationary random process the arithmetic mean tends to infinity mathematical expectation random variable.

Introduction

Let's denote the set of numbers X = (x 1 , x 2 , …, x n), then the sample mean is usually indicated by a horizontal bar over the variable (pronounced " x with a line").

To denote the arithmetic mean of the entire set of numbers, it is usually used greek letter μ. For random variable, for which the average value is determined, μ is probability average or expected value random variable. If the set X is a collection random numbers with a probabilistic mean μ, then for any sample x i from this set μ = E( x i) There is expected value this sample.

In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) is that μ is a typical variable, because you can see a sample rather than the whole general population. Therefore, if the sample is random (in terms of probability theory), then x ¯ (\displaystyle (\bar (x)))(but not μ) can be interpreted as random variable, having probability distribution on the sample (probability distribution of the mean).

Both of these quantities are calculated in the same way:

x ¯ = 1 n ∑ i = 1 n x i = 1 n (x 1 + ⋯ + x n) .

(\displaystyle (\bar (x))=(\frac (1)(n))\sum _(i=1)^(n)x_(i)=(\frac (1)(n))(x_ (1)+\cdots +x_(n)).)

• Examples

For three numbers, you need to add them and divide by 3:

• x 1 + x 2 + x 3 3 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3))(3)).) For

four numbers

you need to add them up and divide them by 4:

x 1 + x 2 + x 3 + x 4 4 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3)+x_(4))(4)).) Continuous random variable If there is an integral of some function f (x) (\displaystyle f(x)) [a; :

b ] (\displaystyle )

determined through Definite integral

definite integral

f (x) ¯ [ a ;

b ] = 1 b − a ∫ a b f (x) d x . What is meant here is that b > a . Some problems of using the average Lack of robustness

A classic example is calculating average income. The arithmetic mean can be misinterpreted as medians, which may lead to the conclusion that there are more people with high incomes than there actually are. “Average” income is interpreted to mean that most people have incomes around this number. This “average” (in the sense of the arithmetic mean) income is higher than the incomes of most people, since a high income with a large deviation from the average makes the arithmetic mean highly skewed (in contrast, the average income at the median “resists” such skew). However, this "average" income says nothing about the number of people near the median income (and says nothing about the number of people near the modal income). However, if you take the concepts of “average” and “most people” lightly, you can draw the incorrect conclusion that most people have incomes higher than they actually are. For example, a report on the "average" net income in Medina, State Washington, calculated as the arithmetic average of all annual net incomes of the residents, will give a surprisingly large number due to Bill Gates. Consider the sample (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six values ​​are below this mean.

Compound interest

If the numbers multiply, but not fold, need to use geometric mean, not the arithmetic mean. Most often this incident occurs when calculating return on investment in finance.

For example, if a stock fell 10% in the first year and rose 30% in the second, then it is incorrect to calculate the “average” increase over those two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, which gives an annual growth rate of only about 8.16653826392% ≈ 8.2%.

The reason for this is that percentages have a new starting point each time: 30% is 30% from a number less than the price at the beginning of the first year: if a stock started out at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock rose 30%, it would be worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the shares rose by only $5.1 in 2 years, average height gives 8.2% final result $35.1:

[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic average of 10% in the same way, we will not get the actual value: [$30 (1 + 0.1) (1 + 0.1) = $36.3].

Compound interest at the end of 2 years: 90% * 130% = 117%, that is, the total increase is 17%, and the average annual compound interest 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%), that is, an average annual increase of 8.2%.

Directions

Main article: Destination statistics

When calculating the average arithmetic values some variable that changes cyclically (for example, phase or corner), special care should be taken. For example, the average of 1 and 359 would be 1 ∘ + 359 ∘ 2 = (\displaystyle (\frac (1^(\circ )+359^(\circ ))(2))=) 180. This number is incorrect for two reasons.

The average value for a cyclic variable calculated using the above formula will be artificially shifted relative to the real average towards the middle of the numerical range. Because of this, the average is calculated in a different way, namely, the number with the smallest variance ( center point). Also, instead of subtraction, the modular distance (that is, the circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on the circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).