How to get the arithmetic mean. How to find the arithmetic mean, and where it can be useful in everyday life

What is the arithmetic mean

The arithmetic mean of several quantities is the ratio of the sum of these quantities to their number.

The arithmetic mean of a certain series of numbers is the sum of all these numbers divided by the number of terms. Thus, the arithmetic mean is the average value of a 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 complicated in calculating or finding the arithmetic mean of several numbers; it is enough to add 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 look at this process in more detail. What do we need to do to calculate the arithmetic mean and obtain 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 and their sum is obtained. Naturally, if the numbers are simple and there are a small number of them, then the calculations can be made by writing them by hand. But 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 will get a result, which will be the arithmetic mean of this series.

Why do you need the arithmetic mean?

The arithmetic mean can be useful not only for solving examples and problems in mathematics lessons, but for other purposes necessary in everyday life person. Such goals can be calculating the arithmetic average to calculate the average financial expense per month, or to calculate the time you spend on the road, also in order to find out attendance, productivity, speed of movement, yield and much more.

So, for example, let's try to calculate how much time you spend traveling to school. Every time you go to school or return home, you spend on travel different times, because when you are in a hurry, you walk faster, and therefore the journey takes less time. But when returning home, you can walk slowly, communicating with classmates, admiring nature, and therefore the journey will take more time.

Therefore, you will not be able to accurately determine the time spent on the road, but thanks to the arithmetic average, you can approximately find out the time you spend on the road.

Let's assume 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, and your journey took the same amount of time on Thursday, and on Friday you were in no hurry and returned for a whole half an hour.

Let's find the arithmetic mean, adding time, for all five days. So,

15 + 20 + 25 + 25 + 30 = 115

Now divide this amount by the number of days

Thanks to this method, you learned that the journey from home to school takes approximately twenty-three minutes of your time.

Homework

1.Using simple calculations, find the average arithmetic number weekly attendance of students in your class.

2. Find the arithmetic mean:

3. Solve the problem:

The arithmetic mean is the sum of numbers divided by the number of these same numbers. And finding the arithmetic mean is very simple.

As follows from the definition, we must take the numbers, add them and divide by their number.

Let's give an example: we are given the numbers 1, 3, 5, 7 and we need to find the arithmetic mean of these numbers.

first add these numbers (1+3+5+7) and get 16
We need to divide the resulting result by 4 (quantity): 16/4 and get the result 4.

So, the arithmetic mean of the numbers 1, 3, 5 and 7 is 4.

Arithmetic mean - the average value among the given indicators.

It is found by dividing the sum of all indicators by their number.

For example, I have 5 apples weighing 200, 250, 180, 220 and 230 grams.

We find the average weight of 1 apple as follows:

we are looking for the total weight of all apples (the sum of all indicators) - it is equal to 1080 grams,
divide the total weight by the number of apples 1080:5 = 216 grams. This is the arithmetic mean.

This is the most commonly used indicator in statistics.

An arithmetic mean is a number added together and divided by their number, the resulting answer is the arithmetic mean.

For example: Katya put 50 rubles in the piggy bank, Maxim 100 rubles, and Sasha put 150 rubles in the piggy bank. 50 + 100 + 150 = 300 rubles in the piggy bank, now we divide this amount by three (three people put money in). So 300: 3 = 100 rubles. These 100 rubles will be the arithmetically average, each of them put in the piggy bank.

There is such a simple example: one person eats meat, another person eats cabbage, and the arithmetically average they both eat cabbage rolls.

The average salary is calculated in the same way...

The arithmetic mean is the sum of all values and divided by their number.

For example the numbers 2, 3, 5, 6. You need to add them up 2+ 3+ 5 + 6 = 16

We divide 16 by 4 and get the answer 4.

4 is the arithmetic mean of these numbers.

The arithmetic mean of several numbers is the sum of these numbers divided by their number.

x avg arithmetic mean

S sum of numbers

n number of numbers.

For example, we need to find the arithmetic mean of the numbers 3, 4, 5 and 6.

To do this, we need to add them up and divide the resulting amount by 4:

(3 + 4 + 5 + 6) : 4 = 18: 4 = 4,5.

I remember taking the final test in mathematics

So there it was necessary to find the arithmetic mean.

It's good that good people They told me what to do, otherwise there would be trouble.

For example, we have 4 numbers.

Add up the numbers and divide by their number (in this case 4)

For example the numbers 2,6,1,1. Add 2+6+1+1 and divide by 4 = 2.5

As you can see, nothing complicated. So the arithmetic mean is the average of all numbers.

We know this from school. Who had good teacher in mathematics, it was possible to remember this simple action the first time.

When finding the arithmetic mean, you need to add up all the available numbers and divide by their number.

For example, I bought 1 kg of apples, 2 kg of bananas, 3 kg of oranges and 1 kg of kiwi at the store. How many kilograms of fruit did I buy on average?

7/4= 1.8 kilograms. This will be the arithmetic mean.

The arithmetic mean is the average number between several numbers.

For example, between the numbers 2 and 4, the average number is 3.

The formula for finding the arithmetic mean is:

You need to add up all the numbers and divide by the number of these numbers:

For example, we have 3 numbers: 2, 5 and 8.

Finding the arithmetic mean:

X=(2+5+8)/3=15/3=5

The scope of application of the arithmetic mean is quite wide.

For example, knowing the coordinates of two points on a segment, you can find the coordinates of the middle of this segment.

For example, the coordinates of the segment: (X1,Y1,Z1)-(X2,Y2,Z2).

Let us denote the middle of this segment by coordinates X3,Y3,Z3.

We separately find the midpoint for each coordinate:

The arithmetic mean is the average of the given...

Those. simply we have the number of sticks different lengths and we want to know their average value..

It is logical that for this we bring them together, getting a long stick, and then divide it into the required number of parts..

Here comes the arithmetic mean...

This is how the formula is derived: Sa=(S(1)+..S(n))/n..

Arithmetic is considered the most elementary branch of mathematics and studies simple steps with numbers. Therefore, the arithmetic mean is also very easy to find. Let's start with a definition. The arithmetic mean is a value that shows which number is closest to the truth after several successive operations of the same type. For example, when running a hundred meters, a person shows a different time each time, but the average value will be within, for example, 12 seconds. Finding the arithmetic mean in this way comes down to sequentially summing all the numbers in a certain series (race results) and dividing this sum by the number of these races (attempts, numbers). In formula form it looks like this:

Sarif = (Х1+Х2+..+Хn)/n

As a mathematician, I am interested in questions on this subject.

I'll start with the history of the issue. Average values have been thought about since ancient times. Arithmetic mean, geometric mean, harmonic mean. These concepts are proposed in ancient Greece Pythagoreans.

And now the question that interests us. What is meant by arithmetic mean of several numbers:

So, to find the arithmetic mean of numbers, you need to add all the numbers and divide the resulting sum by the number of terms.

The formula is:

Example. Find the arithmetic mean of the numbers: 100, 175, 325.

Let's use the formula for finding the arithmetic mean of three numbers (that is, instead of n there will be 3; you need to add up all 3 numbers and divide the resulting sum by their number, i.e. by 3). We have: x=(100+175+325)/3=600/3=200.

The concept of arithmetic mean means the result of a simple sequence of calculations average size for a series of numbers determined in advance. It should be noted that this value in given time widely used by specialists in a number of industries. For example, formulas are known when carrying out calculations by economists or workers in the statistical industry, where a value of this type is required. In addition, this indicator is actively used in a number of other industries that are related to the above.

One of the features of the calculations given value is the simplicity of the procedure. Carry out calculations Anyone can do it. To do this you don't need to have special education. Often there is no need to use computer technology.

To answer the question of how to find the arithmetic mean, consider a number of situations.

The simplest option for calculating this value is to calculate it for two numbers. The calculation procedure in this case is very simple:

Initially, you need to carry out the operation of adding the selected numbers. This can often be done, as they say, manually, without using electronic equipment.
After addition is performed and its result is obtained, division must be performed. This operation involves dividing the sum of two added numbers by two - the number of added numbers. It is this action that will allow you to obtain the required value.

Formula

Thus, the formula for calculating the required value in the case of two will look like as follows:

(A+B)/2

This formula uses the following notation:

A and B are pre-selected numbers for which you need to find a value.

Finding the value for three

Calculating this value in a situation where three numbers are selected will not differ much from the previous option:

To do this, select the numbers needed in the calculation and add them to get total amount.
After this amount three will be found, you need to perform the division procedure again. In this case, the resulting amount must be divided by three, which corresponds to the number of selected numbers.

Formula

Thus, the formula necessary for calculating the arithmetic three will look like this:

(A+B+C)/3

In this formula The following notation is accepted:

A, B and C are the numbers for which you will need to find the arithmetic mean.

Calculating the arithmetic mean of four

As can already be seen by analogy with the previous options, the calculation of this value for a quantity equal to four will be in the following order:

Four numbers are selected for which the average must be calculated arithmetic value. Next, summation is performed and the final result of this procedure is found.
Now, to get the final result, you should take the resulting sum of four and divide it by four. The received data will be the required value.

Formula

From the sequence of actions described above for finding the arithmetic mean for four, you can obtain the following formula:

(A+B+C+E)/4

In this formula the variables have the following meaning:

A, B, C and E are those for which it is necessary to find the value of the arithmetic mean.

Using this formula, it will always be possible to calculate the required value for given quantity numbers.

Calculating the arithmetic mean of five

Performing this operation will require a certain algorithm of actions.

First of all, you need to select five numbers for which the arithmetic mean will be calculated. After this selection These numbers, as in the previous options, just need to be added and get the final amount.
The resulting amount will need to be divided by their number by five, which will allow you to get the required value.

Formula

Thus, similarly to the previously considered options, we obtain the following formula for calculating the arithmetic mean:

(A+B+C+E+P)/5

In this formula, the variables are designated as follows:

A, B, C, E and P are numbers for which it is necessary to obtain the arithmetic mean.

Universal calculation formula

Conducting a review various options formulas to calculate the arithmetic mean, you can pay attention to the fact that they have a common pattern.

Therefore, it will be more practical to use a general formula to find the arithmetic mean. After all, there are situations when the number and magnitude of calculations can be very large. Therefore, it would be wiser to use a universal formula and not output it every time individual technology to calculate this value.

The main thing when determining the formula is principle of calculating the arithmetic mean O.

This principle, as can be seen from the examples given, looks like this:

The number of numbers that are specified to obtain the required value is counted. This operation can be carried out either manually with a small number of numbers or using computer technology.
The selected numbers are summed. This operation in most situations is performed using computer technology, since numbers can consist of two, three or more digits.
The amount obtained by adding the selected numbers must be divided by their number. This value is determined at the initial stage of calculating the arithmetic mean.

Thus, general formula to calculate the arithmetic mean of a series of selected numbers will look like this:

(A+B+…+N)/N

This formula contains the following variables:

A and B are numbers that are selected in advance to calculate their arithmetic mean.

N is the number of numbers that were taken to calculate the required value.

By substituting the selected numbers into this formula each time, we can always obtain the required value of the arithmetic mean.

As you can see, finding the arithmetic mean is a simple procedure. However, you must be careful about the calculations performed and check the results obtained. This approach is explained by the fact that even in the simplest situations there is a possibility of receiving an error, which can then affect further calculations. In this regard, it is recommended to use computer technology that is capable of performing calculations of any complexity.

In order to find the average value in Excel (no matter whether it is a numeric, text, percentage or other value), there are many functions. And each of them has its own characteristics and advantages. Indeed, in this task certain conditions may be set.

For example, the average values of a series of numbers in Excel are calculated using statistical functions. You can also manually enter your own formula. Let's consider various options.

How to find the arithmetic mean of numbers?

To find the arithmetic mean, you need to add up all the numbers in the set and divide the sum by the quantity. For example, a student’s grades in computer science: 3, 4, 3, 5, 5. What is included in the quarter: 4. We found the arithmetic mean using the formula: =(3+4+3+5+5)/5.

How to quickly do this using Excel functions? Let's take for example a series of random numbers in a string:

Or: make the active cell and simply enter the formula manually: =AVERAGE(A1:A8).

Now let's see what else the AVERAGE function can do.

Let's find the arithmetic mean of the first two and last three numbers. Formula: =AVERAGE(A1:B1,F1:H1). Result:

Condition average

The condition for finding the arithmetic mean can be a numerical criterion or a text one. We will use the function: =AVERAGEIF().

Find the arithmetic mean of numbers that are greater than or equal to 10.

Function: =AVERAGEIF(A1:A8,">=10")

The third argument – “Averaging range” – is omitted. First of all, it is not required. Secondly, the range analyzed by the program contains ONLY numeric values. The cells specified in the first argument will be searched according to the condition specified in the second argument.

Attention! The search criterion can be specified in the cell. And make a link to it in the formula.

Let's find the average value of the numbers using the text criterion. For example, the average sales of the product “tables”.

The function will look like this: =AVERAGEIF($A$2:$A$12,A7,$B$2:$B$12). Range – a column with product names. The search criterion is a link to a cell with the word “tables” (you can insert the word “tables” instead of link A7). Averaging range – those cells from which data will be taken to calculate the average value.

As a result of calculating the function, we obtain the following value:

Attention! For a text criterion (condition), the averaging range must be specified.

How to calculate the weighted average price in Excel?

How did we find out the weighted average price?

Formula: =SUMPRODUCT(C2:C12,B2:B12)/SUM(C2:C12).

Using the SUMPRODUCT formula, we find out the total revenue after selling the entire quantity of goods. And the SUM function sums up the quantity of goods. Dividing the total revenue from the sale of goods by total quantity units of goods, we found the weighted average price. This indicator takes into account the “weight” of each price. Its share in the total mass of values.

Standard deviation: formula in Excel

There are standard deviations for the general population and for the sample. In the first case, this is the root of the general variance. In the second, from the sample variance.

To calculate this statistical indicator a dispersion formula is compiled. The root is extracted from it. But in Excel there is a ready-made function for finding the standard deviation.

The standard deviation is tied to the scale of the source data. This is not enough for a figurative representation of the variation of the analyzed range. To get relative level scatter of data, the coefficient of variation is calculated:

standard deviation / arithmetic mean

The formula in Excel looks like this:

STDEV (range of values) / AVERAGE (range of values).

The coefficient of variation is calculated as a percentage. Therefore, we set the percentage format in the cell.

The most common type of average is the arithmetic mean.

Simple arithmetic mean

A simple arithmetic mean is the average term, in determining which the total volume of this characteristic in the data is distributed equally among all units included in the given population. So, average annual output products per worker - this is the value of the volume of production that would fall on each worker if the entire volume of output were in to the same degree distributed among all employees of the organization. The arithmetic mean simple value is calculated using the formula:

Simple arithmetic average— Equal to the ratio of the sum of individual values of a characteristic to the number of characteristics in the aggregate

. A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles per month.

Find average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.

Arithmetic average weighted

If the volume of the data set is large and represents a distribution series, then the weighted arithmetic mean is calculated. This is how the weighted average price per unit of production is determined: the total cost of production (the sum of the products of its quantity by the price of a unit of production) is divided by the total quantity of production.

Let's imagine this in the form of the following formula:

Weighted arithmetic average— equal to the ratio of (the sum of the products of the value of a feature to the frequency of repetition of this feature) to (the sum of the frequencies of all features). It is used when variants of the population under study occur an unequal number of times.

. Find the average salary of workshop workers per month

The average salary can be obtained by dividing the total wages on total number workers:

Answer: 3.35 thousand rubles.

Arithmetic mean for interval series

When calculating the arithmetic mean for an interval variation series, first determine the mean for each interval as the half-sum of the upper and lower limits, and then the mean of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the size of the intervals adjacent to them.

Averages calculated from interval series are approximate.

Example 3. Define middle age evening students.

Averages calculated from interval series are approximate. The degree of their approximation depends on the extent to which the actual distribution of population units within the interval approaches uniformity.

When calculating averages, not only absolute but also relative values (frequency) can be used as weights: