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 the average value, 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 the numbers 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. Consider a brief instruction on how to find the arithmetic mean 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 the drop down list.
- 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.
Average
This term has other meanings, see the average meaning.Average(in mathematics and statistics) sets of numbers - the sum of all numbers divided by their number. It is one of the most common measures of central tendency.
It was proposed (along with the geometric mean and harmonic mean) by the Pythagoreans.
Special cases of the arithmetic mean are the mean (of the general population) and the sample mean (of samples).
Introduction
Denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually denoted by a horizontal bar over the variable (x ¯ (\displaystyle (\bar (x))) , pronounced " x with a dash").
The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which a 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))) is that μ is a typical variable because you can see the sample rather than the entire population. Therefore, if the sample is represented 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)).)
If X is a random variable, then the mathematical expectation X can be considered as the arithmetic mean of the values in repeated measurements of the quantity X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown mathematical expectation.
In elementary algebra, it is proved that the mean n+ 1 numbers above average n numbers if and only if the new number is greater than the old average, less if and only if the new number is less than the average, and does not change if and only if the new number is equal to the average. The more n, the smaller the difference between the new and old averages.
Note that there are several other "averages" available, including power-law mean, Kolmogorov mean, harmonic mean, arithmetic-geometric mean, and various weighted means (eg, arithmetic-weighted mean, geometric-weighted mean, harmonic-weighted mean).
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:
Or easier 5+5=10, 10:2. Because we added 2 numbers, which means that how many numbers we add, we divide by that much.
Continuous random variable
For a continuously distributed value f (x) (\displaystyle f(x)) the arithmetic mean on the interval [ a ; b ] (\displaystyle ) is defined via 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)
Some problems of using the average
Lack of robustness
Main article: Robustness in statisticsAlthough 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 a 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 high 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
Main article: ROIIf 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% = 117% , i.e. a total increase of 17%, and the average annual compound interest is 117% ≈ 108.2% (\displaystyle (\sqrt (117\%))\approx 108.2\%) , that is, an average annual increase of 8.2%.
Directions
Main article: Destination statisticsWhen 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 1° and 359° would be 1 ∘ + 359 ∘ 2 = (\displaystyle (\frac (1^(\circ )+359^(\circ ))(2))=) 180°. This number is incorrect for two reasons.
- First, angular measures are only defined for the range from 0° to 360° (or from 0 to 2π when measured in radians). Thus, the same pair of numbers could be written as (1° and −1°) or as (1° and 719°). The averages of each pair will be different: 1 ∘ + (− 1 ∘) 2 = 0 ∘ (\displaystyle (\frac (1^(\circ )+(-1^(\circ )))(2))=0 ^(\circ )) , 1 ∘ + 719 ∘ 2 = 360 ∘ (\displaystyle (\frac (1^(\circ )+719^(\circ ))(2))=360^(\circ )) .
- Second, in this case, a value of 0° (equivalent to 360°) would be the geometrically best mean, since the numbers deviate less from 0° than from any other value (value 0° has the smallest variance). Compare:
- the number 1° deviates from 0° by only 1°;
- the number 1° deviates from the calculated average of 180° by 179°.
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 (the center point) is chosen as the average. 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 °).
Weighted average - what is it and how to calculate it?
In the process of studying mathematics, students get acquainted with the concept of the arithmetic mean. In the future, in statistics and some other sciences, students are also faced with the calculation of other averages. What can they be and how do they differ from each other?
Averages: Meaning and Differences
Not always accurate indicators give an understanding of the situation. In order to assess this or that situation, it is sometimes necessary to analyze a huge number of figures. And then averages come to the rescue. They allow you to assess the situation in general.
Since school days, many adults remember the existence of the arithmetic mean. It is very easy to calculate - the sum of a sequence of n terms is divisible by n. That is, if you need to calculate the arithmetic mean in the sequence of values 27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values \u200b\u200bare used in the calculations. In this case, the desired value will be equal to 30.
Often, as part of the school course, the geometric mean is also studied. The calculation of this value is based on extracting the root of the nth degree from the product of n terms. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be 29.4.
The harmonic mean in a general education school is usually not the subject of study. However, it is used quite often. This value is the reciprocal of the arithmetic mean and is calculated as a quotient of n - the number of values and the sum 1/a 1 +1/a 2 +...+1/a n . If we again take the same series of numbers for calculation, then the harmonic will be 29.6.
Weighted Average: Features
However, all of the above values may not be used everywhere. For example, in statistics, when calculating some average values, the "weight" of each number used in calculations plays an important role. The results are more revealing and correct because they take into account more information. This group of values is collectively referred to as the "weighted average". They are not passed at school, so it is worth dwelling on them in more detail.
First of all, it is worth explaining what is meant by the "weight" of a particular value. The easiest way to explain this is with a concrete example. The body temperature of each patient is measured twice a day in the hospital. Of the 100 patients in different departments of the hospital, 44 will have a normal temperature - 36.6 degrees. Another 30 will have an increased value - 37.2, 14 - 38, 7 - 38.5, 3 - 39, and the remaining two - 40. And if we take the arithmetic mean, then this value in general for the hospital will be over 38 degrees! But almost half of the patients have a completely normal temperature. And here it would be more correct to use the weighted average, and the "weight" of each value will be the number of people. In this case, the result of the calculation will be 37.25 degrees. The difference is obvious.
In the case of weighted average calculations, the "weight" can be taken as the number of shipments, the number of people working on a given day, in general, anything that can be measured and affect the final result.
Varieties
The weighted average corresponds to the arithmetic average discussed at the beginning of the article. However, the first value, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also weighted geometric and harmonic values.
There is another interesting variety used in series of numbers. This is a weighted moving average. It is on its basis that trends are calculated. In addition to the values themselves and their weight, periodicity is also used there. And when calculating the average value at some point in time, values for previous time periods are also taken into account.
Calculating all these values is not that difficult, but in practice only the usual weighted average is usually used.
Calculation methods
In the age of computerization, there is no need to manually calculate the weighted average. However, it would be useful to know the calculation formula so that you can check and, if necessary, correct the results obtained.
It will be easiest to consider the calculation on a specific example.
It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers receiving a particular salary.
So, the calculation of the weighted average is carried out using the following formula:
x = (a 1 *w 1 +a 2 *w 2 +...+a n *w n)/(w 1 +w 2 +...+w n)
For example, the calculation would be:
x = (32*20+33*35+34*14+40*6)/(20+35+14+6) = (640+1155+476+240)/75 = 33.48
Obviously, there is no particular difficulty in manually calculating the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT (series of numbers; series of weights) / SUM (series of weights) function.
How to find average value in excel?
how to find arithmetic mean in excel?
Vladimir09854
As easy as pie. In order to find the average value in excel, you only need 3 cells. In the first we write one number, in the second - another. And in the third cell, we will score a formula that will give us the average value between these two numbers from the first and second cells. If cell No. 1 is called A1, cell No. 2 is called B1, then in the cell with the formula you need to write like this:
This formula calculates the arithmetic mean of two numbers.
For the beauty of our calculations, we can highlight the cells with lines, in the form of a plate.
There is also a function in Excel itself to determine the average value, but I use the old-fashioned method and enter the formula I need. Thus, I am sure that Excel will calculate exactly as I need, and will not come up with some kind of rounding of its own.
M3sergey
This is very easy if the data is already entered into the cells. If you are just interested in a number, just select the desired range / ranges, and the value of the sum of these numbers, their arithmetic mean and their number will appear in the status bar at the bottom right.
You can select an empty cell, click on the triangle (drop-down list) "Autosum" and select "Average" there, after which you will agree with the proposed range for calculation, or choose your own.
Finally, you can use the formulas directly - click "Insert Function" next to the formula bar and cell address. The AVERAGE function is in the "Statistical" category, and takes as arguments both numbers and cell references, etc. There you can also choose more complex options, for example, AVERAGEIF - calculation of the average by condition.
Find average in excel is a fairly simple task. Here you need to understand whether you want to use this average value in some formulas or not.
If you need to get only the value, then it is enough to select the required range of numbers, after which excel will automatically calculate the average value - it will be displayed in the status bar, the heading "Average".
In the case when you want to use the result in formulas, you can do this:
1) Sum the cells using the SUM function and divide it all by the number of numbers.
2) A more correct option is to use a special function called AVERAGE. The arguments to this function can be numbers given sequentially, or a range of numbers.
Vladimir Tikhonov
circle the values that will be involved in the calculation, click the "Formulas" tab, there you will see "AutoSum" on the left and next to it a triangle pointing down. click on this triangle and choose "Average". Voila, done) at the bottom of the column you will see the average value :)
Ekaterina Mutalapova
Let's start at the beginning and in order. What does average mean?
The mean value is the value that is the arithmetic mean, i.e. is calculated by adding a set of numbers and then dividing the total sum of numbers by their number. For example, for the numbers 2, 3, 6, 7, 2 it will be 4 (the sum of the numbers 20 is divided by their number 5)
In an Excel spreadsheet, for me personally, the easiest way was to use the formula =AVERAGE. To calculate the average value, you need to enter data into the table, write the function =AVERAGE() under the data column, and in brackets indicate the range of numbers in the cells, highlighting the column with the data. After that, press ENTER, or simply left-click on any cell. The result will be displayed in the cell below the column. On the face of it, the description is incomprehensible, but in fact it is a matter of minutes.
Adventurer 2000
The Excel program is multi-faceted, so there are several options that will allow you to find the average:
First option. You simply sum all the cells and divide by their number;
Second option. Use a special command, write in the required cell the formula "=AVERAGE (and here specify the range of cells)";
Third option. If you select the required range, then note that on the page below, the average value in these cells is also displayed.
Thus, there are a lot of ways to find the average value, you just need to choose the best one for you and use it constantly.
In Excel, using the AVERAGE function, you can calculate the simple arithmetic mean. To do this, you need to enter a number of values. Press equals and select in the Statistical category, among which select the AVERAGE function
Also, using statistical formulas, you can calculate the arithmetic weighted average, which is considered more accurate. To calculate it, we need the values of the indicator and the frequency.
How to find the average in Excel?
The situation is this. There is the following table:
The columns shaded in red contain the numerical values of the grades for the subjects. In the "Average" column, you need to calculate their average value.
The problem is this: there are 60-70 objects in total and some of them are on another sheet.
I looked in another document, the average has already been calculated, and in the cell there is a formula like
="sheet name"!|E12
but this was done by some programmer who got fired.
Tell me, please, who understands this.
Hector
In the line of functions, you insert "AVERAGE" from the proposed functions and choose from where they need to be calculated (B6: N6) for Ivanov, for example. I don’t know for sure about neighboring sheets, but for sure this is contained in the standard Windows help
Tell me how to calculate the average value in Word
Please tell me how to calculate the average value in Word. Namely, the average value of the ratings, and not the number of people who received ratings. 
Yulia pavlova
Word can do a lot with macros. Press ALT+F11 and write a macro program..
In addition, Insert-Object... will allow you to use other programs, even Excel, to create a sheet with a table inside a Word document.
But in this case, you need to write down your numbers in the table column, and put the average in the bottom cell of the same column, right?
To do this, insert a field into the bottom cell.
Insert-Field...-Formula
Field content
[=AVERAGE(ABOVE)]
returns the average of the sum of the cells above.
If the field is selected and the right mouse button is pressed, then it can be Updated if the numbers have changed,
view the code or field value, change the code directly in the field.
If something goes wrong, delete the entire field in the cell and re-create it.
AVERAGE means average, ABOVE - about, that is, a row of cells above.
I did not know all this myself, but I easily found it in HELP, of course, thinking a little.
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.
Most of all in eq. In practice, one has to use the arithmetic mean, which can be calculated as the simple and weighted arithmetic mean.
Arithmetic mean (CA)-n the most common type of medium. It is used in cases where the volume of a variable attribute for the entire population is the sum of the values of the attributes of its individual units. Social phenomena are characterized by the additivity (summation) of the volumes of the varying attribute, this determines the scope of the SA and explains its prevalence as a generalizing indicator, for example: the general salary fund is the sum of the salary 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.
Consider first the simple arithmetic mean.
1-CA simple (initial, defining form) is equal to the simple sum of the individual values of the averaged feature, divided by the total number of these values (used when there are ungrouped index values of the feature):
The calculations made can be summarized in the following formula:
(1)
Where 
- the average value of the variable attribute, i.e., the simple arithmetic mean;
means summation, i.e., the addition of individual features;
x- individual values of a variable attribute, which are called variants;
n - number of population units
Example1, it is required to find the average output of one worker (locksmith), if it is known how many parts each of the 15 workers produced, i.e. given a number of ind. trait values, pcs.: 21; 20; 20; 19; 21; 19; 18; 22; 19; 20; 21; 20; 18; 19; 20.
SA simple is calculated by the formula (1), pcs.:
Example2. Let us calculate SA based on conditional data for 20 stores that are part of a trading company (Table 1). Table 1
Distribution of shops of the trading company "Vesna" by trading area, sq. M
|
store number |
store number | ||
To calculate the average store area ( 
) it is necessary to add up the areas of all stores and divide the result by the number of stores:
Thus, the average store area for this group of trade enterprises is 71 sq.m.
Therefore, in order to determine the SA is simple, it is necessary to divide the sum of all values of a given attribute by the number of units that have this attribute.
2
Where f 1
,
f 2
,
… ,f n
–
weight (frequency of repetition of the same features); is the sum of the products of the magnitude of features and their frequencies; is the total number of population units.
(2)
Where X- options;
f- frequency (weight).
SA weighted is the quotient of dividing the sum of the products of the variants 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 the weighted SA.
We will illustrate the technique for calculating weighted SA using the example 1 considered above. To do this, we group the initial data and place them in Table.
The average of the grouped data is determined as follows: first, the variants 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, pcs.: 
P
Distribution of Vesna stores by retail space, sq. m
Thus, the result is the same. However, this will already be the arithmetic weighted average.
In the previous example, we computed the arithmetic average, provided that the absolute frequencies (number of stores) are known. However, in some cases there are no absolute frequencies, but relative frequencies are known, or, as they are commonly called, frequencies that show the proportion or the proportion of frequencies in the entire population.
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 is 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 groups in the total number of stores of the company "Spring". So, for the first group, the specific gravity corresponds to 10% 
. We get the following data Table3
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 production output per worker is such a value of the volume of production 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:
Example 1 . A team of 6 workers receives 3 3.2 3.3 3.5 3.8 3.1 thousand rubles per month.simple arithmetic mean— Equal to the ratio of the sum of individual values of a feature to the number of features in the aggregate
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:
Example 2 . Find the average wages of shop workers per monthWeighted 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.
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 limits, 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 values is equal to the sum of the arithmetic means of these values:
3. The algebraic sum of the deviations of the individual values of the attribute from the average is zero:
4. The sum of the squared deviations of the options from the mean is less than the sum of the squared deviations from any other arbitrary value, i.e.
In mathematics and statistics average arithmetic (or easily average) of a set of numbers is the sum of all the numbers in that set divided by their number. The arithmetic mean is a particularly general and most common representation of the average.
You will need
- Knowledge in mathematics.
Instruction
1. Let a set of four numbers be given. Need to discover average meaning this kit. To do this, we first find the sum of all these numbers. These numbers are possible 1, 3, 8, 7. Their sum is equal to S = 1 + 3 + 8 + 7 = 19. The set of numbers must consist of numbers of the same sign, otherwise the sense in calculating the average value is lost.
2. Average meaning set of numbers is equal to the sum of the numbers S divided by the number of these numbers. That is, it turns out that average meaning equals: 19/4 = 4.75.
3. For a set of numbers, it is also possible to detect not only average arithmetic, but average geometric. The geometric mean of several regular real numbers is a number that is allowed to replace any of these numbers so that their product does not change. The geometric mean G is sought by the formula: the root of the Nth degree of the product of a set of numbers, where N is the number of the number in the set. Let's look at the same set of numbers: 1, 3, 8, 7. Let's find them average geometric. To do this, we calculate the product: 1 * 3 * 8 * 7 = 168. Now from the number 168 you need to extract the root of the 4th degree: G = (168) ^ 1/4 = 3.61. Thus average the geometric set of numbers is 3.61.
Average the geometric mean is used less frequently than the arithmetic mean, but it can be useful in calculating the average value of indicators that change over time (the salary of an individual employee, the dynamics of academic performance, etc.).
You will need
- Engineering Calculator
Instruction
1. In order to find the geometric mean of a series of numbers, you first need to multiply all these numbers. Let's say you are given a set of five indicators: 12, 3, 6, 9 and 4. Let's multiply all these numbers: 12x3x6x9x4 = 7776.
2. Now from the resulting number it is necessary to extract the root of the degree equal to the number of elements of the series. In our case, from the number 7776, it will be necessary to extract the fifth root using an engineering calculator. The number obtained after this operation - in this case, the number 6 - will be the geometric mean for the initial group of numbers.
3. If you don’t have an engineering calculator at hand, then you can calculate the geometric mean of a series of numbers with support for the CPGEOM function in Excel or using one of the online calculators that are deliberately prepared for calculating geometric mean values.
Note!
If you need to find the geometric mean of each for 2 numbers, then you do not need an engineering calculator: you can extract the 2nd degree root (square root) from any number using the most ordinary calculator.
Helpful advice
In contrast to the arithmetic mean, the geometric mean is not so powerfully influenced by huge deviations and fluctuations between individual values in the studied set of indicators.
Average value is one of the collations of a set of numbers. Represents a number that cannot be outside the range defined by the largest and smallest values in this set of numbers. Average an arithmetic value is a particularly commonly used variety of averages.
Instruction
1. Add all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on certain calculation conditions, it is sometimes easier to divide any of the numbers by the number of values of the set and sum up the total.
2. Use, say, the calculator included with Windows OS, if calculating the arithmetic mean in your head is not possible. It can be opened with the support of the program launch dialog. To do this, press the "burning keys" WIN + R or click the "Start" button and select the "Run" command from the main menu. After that, type in the input field calc and press Enter on the keyboard or click the "OK" button. The same can be done through the main menu - open it, go to the "All Programs" section and to the "Typical" segments and select the "Calculator" line.
3. Enter all the numbers in the set in steps by pressing the Plus key on the keyboard after all of them (besides the last one) or by clicking the corresponding button in the calculator interface. Entering numbers is also allowed both from the keyboard and by clicking the corresponding interface buttons.
4. Press the slash key or click this icon 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.
5. It is allowed to use the spreadsheet editor Microsoft Excel for the same purpose. In this case, start the editor and enter all the values of the sequence of numbers into adjacent cells. If, after entering the entire number, you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.
6. Select all the entered values and in the lower left corner of the editor window (in the status bar) you will see the arithmetic mean for the selected cells.
7. Click the cell next to the last number you entered if you'd rather just see the arithmetic mean. Expand the drop-down list with the image of the Greek letter sigma (Σ) in the "Editing" group of commands on the "Basic" tab. Select the line " Average” and the editor will insert the necessary formula for calculating the arithmetic mean in the selected cell. Press the Enter key and the value will be calculated.
The arithmetic mean is one of the measures of central propensity widely used in mathematics and statistical calculations. Finding the arithmetic mean for several values is very easy, but every task has its own nuances, which you need to know in order to perform correct calculations.
What is the arithmetic mean
The arithmetic mean determines the average value for each initial array of numbers. In other words, from a certain set of numbers, a value that is universal for all elements is selected, the mathematical comparison of which with all elements is approximately equal. The arithmetic mean is preferably used when compiling financial and statistical reports or for calculating the quantitative results of similar skills performed.
How to find the arithmetic mean
The search for the arithmetic mean for an array of numbers should begin with 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 184. When writing, the arithmetic mean is denoted by the letter? (mu) or x (x with a dash). Next, the algebraic sum should be divided by the number of numbers in the array. In this example, there were five numbers, so the arithmetic mean will be 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. There is a difference only when calculating in the programming environment, or if there is additional data in the task. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps: 1. Finding the general arithmetic mean in the standard way; 2. Finding the arithmetic mean of negative numbers.3. Calculation of the arithmetic mean of positive numbers. The results of any of the actions are written separated by commas.
Natural and decimal fractions
If the array of numbers is represented by decimal fractions, the solution occurs according to the method of calculating the arithmetic mean of integers, but the total is reduced according to the requirements of the problem for the accuracy of the result. When working with natural fractions, they should be reduced to a common denominator, the one that is multiplied by the number of numbers in the array. The numerator of the result will be the sum of the reduced numerators of the initial fractional elements.
The geometric mean of numbers depends not only on the absolute value of the numbers themselves, but also on their number. It is impossible to confuse the geometric mean and the arithmetic mean of numbers, because they are found according to different methodologies. The geometric mean is invariably less than or equal to the arithmetic mean.
You will need
- Engineering calculator.
Instruction
1. Consider that in the general case the geometric mean of numbers is found by multiplying these numbers and extracting from them the root of the degree that corresponds to the number of numbers. Say, if you need to find the geometric mean of five numbers, then from the product it will be necessary to extract the root of the fifth degree.
2. To find the geometric mean of 2 numbers, use the basic rule. Find their product, then extract the square root from it, from the fact that the number is two, which corresponds to the degree of the root. Let's say, 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 2 numbers is larger and equals 10. If the root is not taken completely, round the total to the desired order.
3. In order to find the geometric mean of more than 2 numbers, also use the basic rule. To do this, find the product of all the numbers for which you need to find the geometric mean. From the resulting product, extract the root of the degree equal to the number of numbers. Let's say, in order to find the geometric mean of the numbers 2, 4 and 64, find their product. 2 4 64=512. From the fact that it is necessary to find the total of the geometric mean of 3 numbers, that extract the root of the third degree from the product. It is difficult to do this verbally, so use an engineering calculator. To do this, 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 1/3, press the “=” button. We get the result of raising 512 to the power of 1/3, which corresponds to the root of the third degree. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.
4. With the support of an engineering calculator, it is possible to detect the geometric mean using a different method. Find the log button on the keyboard. After that, take the logarithm of all of the numbers, find their sum, and divide it by the number of numbers. From the resulting number, take the antilogarithm. This will be the geometric mean of the numbers. Let's say, in order to find the geometric mean of the same numbers 2, 4 and 64, make 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 of the decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, from the fact that this is the number of numbers by which the geometric mean is sought. From the total, take the antilogarithm by toggling the register button and use the same log key. The result will be the number 8, this is the desired geometric mean.
Note!
The average value cannot be larger than the largest number in the set and smaller than the smallest.
Helpful advice
In mathematical statistics, the average value of a quantity is called the mathematical expectation.
