How to find cf arithmetic. Entertaining mathematics. Average value

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:

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 total number 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. Define average 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 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 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 mean arithmetic 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.

) and sample mean (samples).

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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 (, pronounced " x with a dash").
The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random quantity , for which the mean value is determined, μ is probability mean or 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 mathematical 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:
x 1 + x 2 + x 3 3 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3))(3)).)
- For four numbers, you need to add them and divide by 4:
x 1 + x 2 + x 3 + x 4 4 . (\displaystyle (\frac (x_(1)+x_(2)+x_(3)+x_(4))(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
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

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 coefficient of 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 payback investments 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%. 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 °).

What is the arithmetic mean?
1. The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms
2. divide
3. Number Average (Mean), Arithmetic Mean (Arithmetic Mean) - the average value characterizing any group of observations; is calculated by adding the numbers from this series and then dividing the resulting sum by the number of summed numbers. If one or more numbers included in the group differ significantly from the rest, then this can lead to a distortion of the resulting arithmetic mean. Therefore, in this case, it is preferable to use the geometric mean (geometric mean) (it is calculated in a similar way, but here the arithmetic mean of the logarithms of the values of the observations is determined, and then its antilogarithm is found) or - which is most often used - to find the median (average value from a series of values arranged in ascending order). Another method for obtaining the average value of any value from a group of observations is to determine the mode (mode) - an indicator (or set of indicators) that evaluates the most frequent manifestations of a variable; more often this method is used to determine the average value in several series of experiments.
  For example: the numbers 1 and 99, add and divide by two:
  (1+99)/2=50 - arithmetic mean
  If we take the numbers (1,2,3,15,59) / 5 \u003d 16 - the arithmetic mean, etc., etc.
4. The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, which is the sum of all recorded values divided by their number.
  This term has other meanings, see the average meaning.
  The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, which is the sum of all recorded values divided by their number.
  It was proposed (along with the geometric mean and the harmonic mean) by the Pythagoreans 1.
  Special cases of the arithmetic mean are the mean (of the general population) and the sample mean (of samples).
  The Greek letter is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is defined, there is a probabilistic mean or mathematical expectation of the random variable. If the set X is a collection of random numbers with a probability mean, then for any sample xi from this population = E(xi) is the expectation of this sample.
  In practice, the difference between and bar(x) is what 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 bar(x) , (but not) can be treated as a random variable that has a probability distribution on the sample (probability distribution of the mean).
  Both of these quantities are calculated in the same way:
  
  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 expectation of X can be thought of as the arithmetic mean of the values in repeated measurements of 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 of n + 1 numbers is greater than the mean of n numbers if and only if the new number is greater than the old mean, less if and only if the new number is less than the mean, and does not change if and only if the new the number is the average. The larger n, the smaller the difference between the new and old averages.
  Note that there are several other means, including the power mean, Kolmogorov mean, harmonic mean, arithmetic geometric mean, and various weighted mean.
  Examples edit wiki text
  For three numbers, you need to add them and divide by 3:
  frac(x_1 + x_2 + x_3)(3).
  For four numbers, you need to add them and divide by 4:
  frac(x_1 + x_2 + x_3 + x_4)(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 value edit edit wiki text
  For a continuously distributed value f(x), the arithmetic mean over the interval a;b is defined by the definite integral: Some problems in the application of the mean Lack of robustness robust statistics, which means that the arithmetic mean is strongly influenced by large deviations. It is noteworthy that for distributions with large skewness, the arithmetic mean
5. You add up the numbers and divide how many of them it was like this 33 + 66 + 99 = add up 33 + 66 + 99 = 198 and divide how many were read out for us 3 numbers are 33 66 and 99 and we need what we managed to divide like this: 33+ 66+99=198:3=66 is the orphmetic mean
6. well, it's like 2+8=10 and the average is 5
7. The arithmetic mean of a set of numbers is defined as their sum divided by their number. That is, the sum of all the numbers in a set is divisible by the number of numbers in that set.
  The simplest case is to find the arithmetic mean of two numbers x1 and x2. Then their arithmetic mean X = (x1+x2)/2. For example, X = (6+2)/2 = 4 is the arithmetic mean of the numbers 6 and 2.
  2
  The general formula for finding the arithmetic mean of n numbers will look like this: X = (x1+x2+...+xn)/n. It can also be written as: X = (1/n)xi, where the summation is over the index i from i = 1 to i = n.
  For example, the arithmetic mean of three numbers X = (x1+x2+x3)/3, five numbers - (x1+x2+x3+x4+x5)/5.
  3
  Of interest is the situation where the set of numbers are members of an arithmetic progression. As you know, the members of an arithmetic progression are equal to a1+(n-1)d, where d is the step of the progression, and n is the number of the progression member.
  Let a1, a1+d, a1+2d,...a1+(n-1)d be members of an arithmetic progression. Their arithmetic mean is S = (a1+a1+d+a1+2d+...+a1+(n-1)d)/n = (na1+d+2d+...+(n-1)d)/n = a1+(d+2d+...+(n-2)d+(n-1)d)/n = a1+(d+2d+...+dn-d+dn-2d)/n = a1+(n* d*(n-1)/2)/n = a1+dn/2 = (2a1+d(n-1))/2 = (a1+an)/2. Thus, the arithmetic mean of the members of an arithmetic progression is equal to the arithmetic mean of its first and last members.
  4
  The property is also true that each member of an arithmetic progression is equal to the arithmetic mean of the previous and subsequent members of the progression: an = (a(n-1)+a(n+1))/2, where a(n-1), an, a( n+1) are consecutive members of the sequence.
8. Divide the sum of the numbers by their number
9. when you add and divide everything
10. If I'm not mistaken, this is when you add the sum of numbers and divide by the number of numbers themselves ...
11. this is when you have several numbers, you add them up, and then divide by their number! let's say 25 24 65 76, add: 25+24+65+76:4=arithmetic mean!
12. Vyachaslav Bogdanov answered incorrectly!!! !
  Do with your words!
  The arithmetic mean is the average value between two values .... It is found as the sum of numbers divided by their number ... . Or simply, if two numbers are around some number (or rather, there is some number between them in order), then this number will be cf. are. !
  6 + 8... cf ar = 7
13. divisor gygygygygygygy
14. The average between the maximum and minimum (all numerical indicators are added up and divided by their number
  )
15. when you add the numbers and divide by the number of numbers

How to find cf arithmetic. Entertaining mathematics. Average value

simple arithmetic mean

Arithmetic weighted average

Arithmetic mean for an interval series

The arithmetic mean has a number of properties that more fully reveal its essence and simplify the calculation:

How to find the arithmetic mean of numbers?

Average by condition

How to calculate the weighted average price in Excel?

Standard deviation: formula in Excel

Encyclopedic YouTube

Examples

Continuous random variable

Some problems of using the average

Lack of robustness

Compound interest