How to determine the arithmetic mean. What is the arithmetic mean of numbers. The arithmetic mean has a number of properties that more fully reveal its essence and simplify the calculation.
The most common type of average is the arithmetic average.
simple arithmetic mean
The simple arithmetic mean is the average term, in determining which the total volume of a given attribute in the data is equally distributed among all units included in this population. Thus, the average annual output per worker is such a value of the volume of output that would fall on each employee if the entire volume of output was equally distributed among all employees of the organization. The arithmetic mean simple value is calculated by the formula:
simple arithmetic mean— Equal to the ratio of the sum of individual values of a feature to the number of features in the aggregate
Example 1. A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles per month.
Find the average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.
Arithmetic weighted average
If the volume of the data set is large and represents a distribution series, then a 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 and the price of a unit of production) is divided by the total quantity of production.
We represent this in the form of the following formula:
Weighted arithmetic mean- is equal to the ratio (the sum of the products of the attribute value to the frequency of repetition of this attribute) to (the sum of the frequencies of all attributes). It is used when the variants of the studied population occur an unequal number of times.
Example 2. Find the average wages of shop workers per month
The average wage can be obtained by dividing the total wage by the total number of workers:
Answer: 3.35 thousand rubles.
Arithmetic mean for an interval series
When calculating the arithmetic mean for an interval variation series, the average for each interval is first determined as the half-sum of the upper and lower boundaries, and then the average of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the value of the intervals adjacent to them.
Averages calculated from interval series are approximate.
Example 3. Determine the average age of students in the evening department.
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 uniform.
When calculating averages, not only absolute, but also relative values (frequency) can be used as weights:
The arithmetic mean has a number of properties that more fully reveal its essence and simplify the calculation:
1. The product of the average and the sum of the frequencies is always equal to the sum of the products of the variant and the frequencies, i.e.
2. The arithmetic mean of the sum of the varying quantities is equal to the sum of the arithmetic means of these quantities:
3. The algebraic sum of the deviations of the individual values of the attribute from the average is zero.
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.
Not only in various mathematical sciences, but also in everyday life, there are cases when you need to calculate the average of something. For example, the average cost of cucumbers on the market, the average height of a child, the average cost of staying in a hotel, etc.
All of this has long been thought of. scientific name- "average". This indicator is actively used in statistics to summarize the results. For example, the average age for having children, the average age of death among men and women, the average salary by region and in Russia as a whole.
For example, when adopting a law on raising the retirement age, the authorities just proceeded from the average age of death in our country.
Let's figure out what this indicator is.
The arithmetic mean is average of all available values. To calculate it, it is necessary to sum up all the numbers involved in the operation, and then divide by their total number.
For example, in 2017, children of different ages received a complete secondary education: 16, 17 and 18 years old. The arithmetic mean will be calculated as the sum of all ages divided by three. In total, the average age of a child who graduated from grade 11 was 17 years.
This example shows a primitive calculation using the example of three children. In fact, you need to summarize all the data available. That is, if we are talking about five children, then we sum up their age, for example, 17 + 17 + 18 + 16 + 17 and divide the result by five.
Similarly, any arithmetic mean is calculated for any operation. That is, if, for example, you need to calculate the average age of mothers who gave birth to their first child in 2017, then you will first need to sum up all the age indicators, and then divide by the total number of parents.
That is, in general the formula can be represented as follows:
Arithmetic mean = ( the sum of all available values)/total number of values involved in the operation.
Thus, the calculation is quite simple, even for schoolchildren. Difficulties may arise only because of the large number of respondents participating in the operation.
It is important to understand that the average is not just a number. It has a special physical meaning, which has been used in practice in the real world for many years.
It would be wrong to use the arithmetic mean only on paper, in a notebook or in computer programs. Otherwise, you can get a lot of meaningless and simply unrealistic values.
In fact, there are several middle ones. However, in each case, only one of them is correct. In each of the operations, you need to use only the kind of average that is needed, otherwise a huge mistake will be made.
What types of averages are used in practice? The most common averages are:
- Average;
- Geometric mean;
- Average harmonic.
These values most commonly used both in everyday life and in the sciences. Most often, of course, the first indicator is calculated.
Often this indicator is applied and calculated incorrectly in real conditions. Why it happens? In fact, the basis of the arithmetic mean is the application of the law of large numbers. In addition, the assumption is also applied, according to which the initial value is normally defined.
This means that around the presented in a number of values, there is most common deviation to any side. I.e. Big or small. For example, in a series of numbers 8,8,9,8,9,8,8, the deviation will be downwards, since there are more eights. And in the series: 17.17, 20,20,20,20,20, the deviation, on the contrary, will be upwards, since in this case there are still more “twenties”.
However, in most cases, such deviations are small and usually equal in probability. The essence of the problem is that in business, as in real life, the normality of distribution in practice can be found extremely rarely.
That is, for example, the time of servicing one client, the time that a client is expected to receive this service, the amount for which they will then conclude a contract, market share, income growth, etc., are those indicators that are not distributed evenly and normally. In some cases, it is undesirable to average them with the help of the arithmetic mean. Because that would be wrong.
In practice, the normality of the distribution can often be found in the presence of a large number of values ranging from hundreds to thousands. For example, the number of calls to the technical support of a large company can be distributed normally, both on paper and in fact.
However, only the quantity will not be enough, because in each specific situation you need to monitor and correct distribution. Only in this way it will be possible to correctly calculate the value of the arithmetic mean in the end.
The question of how to find the arithmetic mean arises among people of different ages, and not only among students. Sometimes we urgently need to find the arithmetic mean, but we can't remember how to do it. Then we frantically flip through school textbooks in mathematics, trying to find the information we need. But it's very simple!
To find the arithmetic mean of several numbers, add them together. After that, the resulting amount should be divided by the number of terms.
To make it more clear, let's figure out together how to find the arithmetic mean of numbers, using the example: 78, 115, 121 and 224. First we need to add these numbers: 78+115+121+224=538. Now the amount received, i.e. 538 should be divided by the number of terms: 538:4=134.5. So, the arithmetic mean of these numbers is 134.5.
Arithmetic mean of several numbers: find with Excel
Finding the arithmetic mean is very easy using Excel. This program allows you to avoid lengthy calculations and, accordingly, errors. To find the arithmetic mean of several numbers, write them in one column. Then select this column and select the sum (?) icon and the average tab from the Quick Access Toolbar. The arithmetic mean of these numbers will appear at the bottom of the highlighted column.
Arithmetic mean - a statistical indicator that shows the average value of a given data array. Such an indicator is calculated as a fraction, the numerator of which is the sum of all array values, and the denominator is their number. The arithmetic mean is an important coefficient that is used in household calculations.
The meaning of the coefficient
The arithmetic mean is an elementary indicator for comparing data and calculating an acceptable value. For example, a can of beer from a particular manufacturer is sold in different stores. But in one store it costs 67 rubles, in another - 70 rubles, in the third - 65 rubles, and in the last - 62 rubles. There is a rather large range of prices, so the buyer will be interested in the average cost of a can, so that when buying a product he can compare his costs. On average, a can of beer in the city has a price:
Average price = (67 + 70 + 65 + 62) / 4 = 66 rubles.
Knowing the average price, it is easy to determine where it is profitable to buy goods, and where you will have to overpay.
The arithmetic mean is constantly used in statistical calculations in cases where a homogeneous data set is analyzed. In the example above, this is the price of a can of beer of the same brand. However, we cannot compare the price of beer from different manufacturers or the prices of beer and lemonade, since in this case the spread of values will be greater, the average price will be blurred and unreliable, and the very meaning of the calculations will be distorted to the caricature "average temperature in the hospital." To calculate heterogeneous data arrays, the arithmetic weighted average is used, when each value receives its own weighting factor.
Calculating the arithmetic mean
The formula for calculations is extremely simple:
P = (a1 + a2 + … an) / n,
where an is the value of the quantity, n is the total number of values.
What can this indicator be used for? The first and obvious use of it is in statistics. Almost every statistical study uses the arithmetic mean. This can be the average age of marriage in Russia, the average mark in a subject for a student, or the average spending on groceries per day. As mentioned above, without taking into account the weights, the calculation of averages can give strange or absurd values.
For example, the President of the Russian Federation made a statement that, according to statistics, the average salary of a Russian is 27,000 rubles. For most people in Russia, this level of salary seemed absurd. It is not surprising if the calculation takes into account the income of oligarchs, heads of industrial enterprises, large bankers on the one hand and the salaries of teachers, cleaners and sellers on the other. Even average salaries in one specialty, for example, an accountant, will have serious differences in Moscow, Kostroma and Yekaterinburg.
How to calculate averages for heterogeneous data
In payroll situations, it is important to consider the weight of each value. This means that the salaries of oligarchs and bankers would be given a weight of, for example, 0.00001, and the salaries of salespeople would be 0.12. These are numbers from the ceiling, but they roughly illustrate the prevalence of oligarchs and salesmen in Russian society.
Thus, to calculate the average of averages or the average value in a heterogeneous data array, it is required to use the arithmetic weighted average. Otherwise, you will receive an average salary in Russia at the level of 27,000 rubles. If you want to know your average mark in mathematics or the average number of goals scored by a selected hockey player, then the arithmetic mean calculator will suit you.
Our program is a simple and convenient calculator for calculating the arithmetic mean. You only need to enter parameter values to perform calculations.
Let's look at a couple of examples
Average Grade Calculation
Many teachers use the arithmetic mean method to determine an annual grade in a subject. Let's imagine that a child gets the following quarter grades in math: 3, 3, 5, 4. What annual grade will the teacher give him? Let's use a calculator and calculate the arithmetic mean. First, select the appropriate number of fields and enter the grade values in the cells that appear:
(3 + 3 + 5 + 4) / 4 = 3,75
The teacher will round the value in favor of the student, and the student will receive a solid four for the year.
Calculation of eaten sweets
Let's illustrate some absurdity of the arithmetic mean. Imagine that Masha and Vova had 10 sweets. Masha ate 8 candies, and Vova only 2. How many candies did each child eat on average? Using a calculator, it is easy to calculate that on average, children ate 5 sweets each, which is completely untrue and common sense. This example shows that the arithmetic mean is important for meaningful datasets.
Conclusion
The calculation of the arithmetic mean is widely used in many scientific fields. This indicator is popular not only in statistical calculations, but also in physics, mechanics, economics, medicine or finance. Use our calculators as an assistant for solving arithmetic mean problems.
