In mathematics, the arithmetic mean of numbers (or simply the average) is the sum of all the numbers in a given set divided by their number. This is the most generalized and widespread concept of the average value. As you already understood, in order to find you need to sum up all the numbers given to you, and divide the result by the number of terms.
What is the arithmetic mean?
Let's look at an example.
Example 1. Numbers are given: 6, 7, 11. You need to find their average value.
Solution.
First, let's find the sum of all given numbers.
Now we divide the resulting sum by the number of terms. Since we have three terms, respectively, we will divide by three.
Therefore, the average of 6, 7, and 11 is 8. Why 8? Yes, because the sum of 6, 7 and 11 will be the same as three eights. This is clearly seen in the illustration.
The average value is somewhat reminiscent of the "alignment" of a series of numbers. As you can see, the piles of pencils have become one level.
Consider another example to consolidate the knowledge gained.
Example 2 Numbers are given: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.
Solution.
We find the sum.
3 + 7 + 5 + 13 + 20 + 23 + 39 + 23 + 40 + 23 + 14 + 12 + 56 + 23 + 29 = 330
Divide by the number of terms (in this case, 15).
Therefore, the average value of this series of numbers is 22.
Now consider negative numbers. Let's remember how to sum them up. For example, you have two numbers 1 and -4. Let's find their sum.
1 + (-4) = 1 - 4 = -3
Knowing this, consider another example.
Example 3 Find the average value of a series of numbers: 3, -7, 5, 13, -2.
Solution.
Finding the sum of numbers.
3 + (-7) + 5 + 13 + (-2) = 12
Since there are 5 terms, we divide the resulting sum by 5.
Therefore, the arithmetic mean of the numbers 3, -7, 5, 13, -2 is 2.4.
In our time of technological progress, it is much more convenient to use computer programs to find the average value. Microsoft Office Excel is one of them. Finding the average in Excel is quick and easy. Moreover, this program is included in the software package from Microsoft Office. Let's consider a brief instruction, value using this program.
In order to calculate the average value of a series of numbers, you must use the AVERAGE function. The syntax for this function is:
=Average(argument1, argument2, ... argument255)
where argument1, argument2, ... argument255 are either numbers or cell references (cells mean ranges and arrays).
To make it clearer, let's test the knowledge gained.
- Enter the numbers 11, 12, 13, 14, 15, 16 in cells C1 - C6.
- Select cell C7 by clicking on it. In this cell, we will display the average value.
- Click on the "Formulas" tab.
- Select More Functions > Statistical to open
- Select AVERAGE. After that, a dialog box should open.
- Select and drag cells C1-C6 there to set the range in the dialog box.
- Confirm your actions with the "OK" button.
- If you did everything correctly, in cell C7 you should have the answer - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will be displayed in the formula bar.
It is very useful to use this function for accounting, invoices, or when you just need to find the average of a very long range of numbers. Therefore, it is often used in offices and large companies. This allows you to keep the records in order and makes it possible to quickly calculate something (for example, the average income per month). You can also use Excel to find the mean of a function.
When the number of elements of the set of numbers of a stationary random process tends to infinity, the arithmetic mean tends to the mathematical expectation of a random variable.
Introduction
Denote the set of numbers X = (x 1 , x 2 , …, x n), then the sample mean is usually denoted by a horizontal bar over the variable (, pronounced " x with a dash").
The Greek letter μ is usually used to denote the arithmetic mean of the entire population of numbers. For a random variable , for which the mean value is defined, μ is probability mean or the mathematical expectation of a random variable. If the set X is a collection of random numbers with a probability mean μ, then for any sample x i from this collection μ = E( x i) is the expectation of this sample.
In practice, the difference between μ and x ¯ (\displaystyle (\bar (x))) in that μ is a typical variable, because you can see the sample rather than the entire population. Therefore, if the sample is presented randomly (in terms of probability theory), then x ¯ (\displaystyle (\bar (x)))(but not μ) can be treated as a random variable having a 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:
- For four numbers, you need to add them and divide by 4:
Continuous random variable
If there is an integral of some function f (x) (\displaystyle f(x)) one variable, then the arithmetic mean of this function on the segment [ a ; b] (\displaystyle) is defined through a definite integral:
f (x) ¯ [ a ; b ] = 1 b − a ∫ a b f (x) d x . (\displaystyle (\overline (f(x)))_()=(\frac (1)(b-a))\int _(a)^(b)f(x)dx.)Here it is implied that b > a . (\displaystyle b>a.)
Some problems of using the average
Lack of robustness
Although the arithmetic mean is often used as means or central trends, this concept does not apply to robust statistics, which means that the arithmetic mean is heavily influenced by "large deviations". It is noteworthy that for distributions with a large skewness, the arithmetic mean may not correspond to the concept of “average”, and the values of the mean from robust statistics (for example, the median) may better describe the central trend.
The classic example is the calculation of the average income. The arithmetic mean can be misinterpreted as the median, which can lead to the conclusion that there are more people with more income than there really are. "Mean" income is interpreted in such a way that most people's incomes are close to this number. This "average" (in the sense of the arithmetic mean) income is higher than the income of most people, since a high income with a large deviation from the average makes the arithmetic mean strongly skewed (in contrast, the median income "resists" such a 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 the concepts of "average" and "majority" are taken lightly, then one can incorrectly conclude that most people have incomes higher than they actually are. For example, a report on the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of 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 of the six values are below this mean.
Compound interest
If numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often, this incident happens when calculating the return on investment in finance.
For example, if stocks fell 10% in the first year and rose 30% in the second year, then it is incorrect to calculate the "average" increase over these two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, from which the annual growth is 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 the stock started at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock is up 30%, it is worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only grown by $5.1 in 2 years, an average increase of 8.2% gives a final result of $35.1:
[$30 (1 - 0.1) (1 + 0.3) = $30 (1 + 0.082) (1 + 0.082) = $35.1]. If we use the arithmetic mean 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 year 2: 90% * 130% \u003d 117%, that is, a total increase of 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 arithmetic mean of some variable that changes cyclically (for example, phase or angle), special care should be taken. For example, the average of the numbers 1 and 359 will be equal to 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 according to the above formula, will be artificially shifted relative to the real average to 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) is chosen as the average value. Also, instead of subtracting, modulo distance (i.e., circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on a circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).
In order to find the average value in Excel (whether it is a numerical, textual, percentage or other value), there are many functions. And each of them has its own characteristics and advantages. After all, certain conditions can be set in this task.
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 add all the numbers in the set and divide the sum by the number. For example, a student's grades in computer science: 3, 4, 3, 5, 5. What goes for a quarter: 4. We found the arithmetic mean using the formula: \u003d (3 + 4 + 3 + 5 + 5) / 5.
How to do it quickly using Excel functions? Take for example a series of random numbers in a string:
Or: make the cell active and simply manually enter the formula: =AVERAGE(A1:A8).
Now let's see what else the AVERAGE function can do.
Find the arithmetic mean of the first two and last three numbers. Formula: =AVERAGE(A1:B1;F1:H1). Result:
Average by condition
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 result of using the AVERAGEIF function on the condition ">=10": The third argument - "Averaging range" - is omitted. First, it is not required. Secondly, the range parsed by the program contains ONLY numeric values. In the cells specified in the first argument, the search will be performed according to the condition specified in the second argument.
Attention! The search criterion can be specified in a cell. And in the formula to make a reference to it.
Let's find the average value of the numbers by 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 the 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 do we know the weighted average price?
Formula: =SUMPRODUCT(C2:C12,B2:B12)/SUM(C2:C12).
Using the SUMPRODUCT formula, we find out the total revenue after the sale of the entire quantity of goods. And the SUM function - sums up the quantity of goods. By dividing the total revenue from the sale of goods by the total number of 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
Distinguish between the standard deviation 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 taken from it. But in Excel there is a ready-made function for finding the standard deviation.
The standard deviation is linked to the scale of the source data. This is not enough for a figurative representation of the variation of the analyzed range. To get the relative level of scatter in the 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 concept of arithmetic mean means the result of a simple sequence of calculations of the average value for a series of numbers determined in advance. It should be noted that this value is currently widely used by specialists in a number of industries. For example, formulas are known when performing calculations by economists or employees of the statistical industry, where it is required to have a value of this type. In addition, this indicator is actively used in a number of other industries that are related to the above.
One of the features of calculating this value is the simplicity of the procedure. Carry out calculations anyone can. You don't need any special education for this. Often there is no need to use computer technology.
As an answer to the question of how to find the arithmetic mean, consider a number of situations.
The simplest way to calculate this value is to calculate it for two numbers. The calculation procedure in this case is very simple:
- Initially, it is required to carry out the operation of adding the selected numbers. This can often be done, as they say, manually, without using electronic equipment.
- After the addition is made and its result is obtained, it is necessary to divide. 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 get the required value.
Formula
Thus, the formula for calculating the required value in the case of two will look like this:
(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 a value for three
The calculation of 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 the total.
- After this sum of three is found, it is required 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 required when calculating the arithmetic three will look like this:
(A+B+C)/3
In this formula the following notation has been adopted:
A, B and C are the numbers to which it will be necessary to find the arithmetic mean.
Calculating the arithmetic mean of four
As already seen by analogy with the previous options, the calculation of this value for a quantity equal to four will be of the following order:
- Four digits are selected for which the arithmetic mean is to be calculated. Next, the summation and finding the final result of this procedure is carried out.
- 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 get the following formula:
(A+B+C+E)/4
In this formula variables have the following meaning:
A, B, C and E are those for which you need to find the value of the arithmetic mean.
Using this formula, it will always be possible to calculate the required value for a given number of 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, you just need to add up 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 have the following notation:
A, B, C, E and P are the numbers for which you want to get the arithmetic mean.
Universal Calculation Formula
Carrying out consideration of various variants of 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 apply the general formula for finding the arithmetic mean. After all, there are situations when the number and size of calculations can be very large. Therefore, it would be wiser to use a universal formula and not deduce an individual technology each time to calculate this value.
The main thing in determining the formula is the principle of calculating the arithmetic mean about.
This principle, as it was seen from the above examples, looks like this:
- The number of numbers that are specified to obtain the required value is counted. This operation can be carried out both manually with a small number of numbers, and with the help of 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, the general formula for calculating the arithmetic mean of a series of selected numbers will look like this:
(А+В+…+N)/N
This formula contains the following variables:
A and B are numbers that are chosen in advance to calculate their arithmetic mean.
N is the number of numbers that were taken in order to calculate the required value.
Substituting the selected numbers into this formula each time, we can always get the required value of the arithmetic mean.
As seen, finding the arithmetic mean is an easy procedure. However, one must be attentive to the calculations and check the result obtained. This approach is explained by the fact that even in the simplest situations, there is a possibility of getting an error, which can then affect further calculations. In this regard, it is recommended to use computer technology that is capable of making calculations of any complexity.
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:
