) and sample mean(s).
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Let us denote the set of data X = (x 1 , x 2 , …, x n), then the sample mean is usually indicated by a horizontal bar over the variable (pronounced " x with a line").
The Greek letter μ is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is determined, μ is probabilistic average or mathematical expectation of a random variable. If the set X is a collection of random numbers with a probabilistic mean μ, then for any sample x i from this set μ = E( x i) is the mathematical 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 a sample rather than the entire population. Therefore, if the sample is random (in terms of probability theory), then x ¯ (\displaystyle (\bar (x)))(but not μ) can be treated as a random variable having a probability distribution over 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:
Or simpler 5+5=10, 10:2. Because we were adding 2 numbers, which means how many numbers we add, we divide by that many.
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 arithmetic means are often used as averages or central tendencies, this concept is not a robust statistic, meaning 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 “mean”, and the values of the mean from robust statistics (for example, the median) may better describe the central tendency.
A classic example is calculating average income. The arithmetic mean can be misinterpreted as a median, which may lead to the conclusion that there are more people with higher incomes than there actually are. “Average” income is interpreted to mean that most people have incomes around this number. This “average” (in the sense of the arithmetic mean) income is higher than the incomes of most people, since a high income with a large deviation from the average makes the arithmetic mean highly skewed (in contrast, the average income at the median “resists” such 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 you take the concepts of “average” and “most people” lightly, you can draw the incorrect conclusion that most people have incomes higher than they actually are. For example, a report of the "average" net income in Medina, Washington, calculated as the arithmetic average of all annual net incomes of residents, would yield 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 out of six values are below this mean.
Compound interest
If the numbers multiply, but not fold, you need to use the geometric mean, not the arithmetic mean. Most often this incident occurs when calculating the return on investment in finance.
For example, if a stock fell 10% in the first year and rose 30% in the second, then it is incorrect to calculate the “average” increase over those two years as the arithmetic mean (−10% + 30%) / 2 = 10%; the correct average in this case is given by the compound annual growth rate, which gives an annual growth rate of 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 a stock started out at $30 and fell 10%, it is worth $27 at the start of the second year. If the stock rose 30%, it would be worth $35.1 at the end of the second year. The arithmetic average of this growth is 10%, but since the stock has only risen by $5.1 over 2 years, the average growth 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 average 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 2 years: 90% * 130% = 117%, that is, the total increase is 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 using the above formula will be artificially shifted relative to the real average towards 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 selected as the average value. Also, instead of subtraction, the modular distance (that is, the circumferential distance) is used. For example, the modular distance between 1° and 359° is 2°, not 358° (on the circle between 359° and 360°==0° - one degree, between 0° and 1° - also 1°, in total - 2 °).
What is the arithmetic mean? How to find the arithmetic mean? Where and what is this value used for?
To fully understand the essence of the problem, you need to study algebra for several years at school, and then at the institute. But in everyday life, in order to know how to find the arithmetic mean of numbers, it is not necessary to know everything about it thoroughly. In simple terms, it is the sum of numbers divided by the number of those numbers added.
Since it is not always possible to calculate the arithmetic mean without a remainder, the value may even turn out to be fractional, even when calculating the average number of people. This is due to the fact that the arithmetic mean is an abstract concept.
This abstract value affects many areas of modern life. It is used in mathematics, business, statistics, often even in sports.
For example, many are interested in all the members of a group or the average number of foods eaten per month in terms of one day. And data about how much was spent on average on any expensive event can be found in all media sources. Most often, of course, such data is used in statistics: to know exactly which phenomenon has declined and which has increased; which product is most in demand and in what period; to easily eliminate unwanted indicators.
In sports, we can come across the concept of an average, when, for example, we are told the average age of athletes or goals scored in football. How is the average score earned during competitions or at our beloved KVN calculated? Yes, for this you don’t need to do anything else but find the arithmetic mean of all the marks given by the judges!
By the way, often in school life some teachers resort to a similar method, giving quarterly and annual grades to their students. It is also often used in higher educational institutions, often in schools, to calculate the average score of students, to determine the effectiveness of the teacher or to distribute students according to their capabilities. There are still many areas of life in which this formula is used, but the goal is basically the same - to find out and control.
In business, the arithmetic average can be used to calculate and control income and losses, salaries and other expenses. For example, when submitting income certificates to some organizations, the monthly average for the last six months is required. It is surprising that some employees whose duties include collecting such information, having received a certificate not with the average monthly salary, but simply about income for six months, do not know how to find the arithmetic average, that is, calculate the average monthly salary.
An arithmetic average is a characteristic (price, salary, population, etc.), the volume of which does not change during calculation. In simple words, when the average number of apples eaten by Petya and Masha is calculated, the result will be a number that will be equal to half of the total number of apples. Even if Masha ate ten, and Petya only got one, then when we divide their total quantity in half, then we will get the arithmetic average.
Today, many joke about Putin’s statement that the average salary of those living in Russia is 27 thousand rubles. The jokes of wits basically sound like this: “Or am I not a Russian? Or am I no longer living? And the whole question is that these wits also apparently don’t know how to find the arithmetic average of the salaries of Russian residents.
You just need to add up the incomes of oligarchs, business executives, businessmen on the one hand and the salaries of cleaners, janitors, salesmen and conductors on the other. And then divide the resulting amount by the number of people whose income included this amount. So we get an amazing figure, which is expressed as 27,000 rubles.
In order to find the average value in Excel (no matter whether it is a numeric, text, percentage or other value), there are many functions. And each of them has its own characteristics and advantages. Indeed, in this task certain conditions may be set.
For example, the average values of a series of numbers in Excel are calculated using statistical functions. You can also manually enter your own formula. Let's consider various options.
How to find the arithmetic mean of numbers?
To find the arithmetic mean, you need to add up all the numbers in the set and divide the sum by the quantity. For example, a student’s grades in computer science: 3, 4, 3, 5, 5. What is included in the quarter: 4. We found the arithmetic mean using the formula: =(3+4+3+5+5)/5.
How to quickly do this using Excel functions? Let's take for example a series of random numbers in a string:
Or: make the active cell and simply enter the formula manually: =AVERAGE(A1:A8).
Now let's see what else the AVERAGE function can do.
Let's find the arithmetic mean of the first two and last three numbers. Formula: =AVERAGE(A1:B1,F1:H1). Result:
Condition average
The condition for finding the arithmetic mean can be a numerical criterion or a text one. We will use the function: =AVERAGEIF().
Find the arithmetic mean of numbers that are greater than or equal to 10.
Function: =AVERAGEIF(A1:A8,">=10")
The result of using the AVERAGEIF function under the condition ">=10": The third argument – “Averaging range” – is omitted. First of all, it is not required. Secondly, the range analyzed by the program contains ONLY numeric values. The cells specified in the first argument will be searched according to the condition specified in the second argument.
Attention! The search criterion can be specified in the cell. And make a link to it in the formula.
Let's find the average value of the numbers using the text criterion. For example, the average sales of the product “tables”.
The function will look like this: =AVERAGEIF($A$2:$A$12,A7,$B$2:$B$12). Range – a column with product names. The search criterion is a link to a cell with the word “tables” (you can insert the word “tables” instead of link A7). Averaging range – those cells from which data will be taken to calculate the average value.
As a result of calculating the function, we obtain the following value:
Attention! For a text criterion (condition), the averaging range must be specified.
How to calculate the weighted average price in Excel?
How did we find out the weighted average price?
Formula: =SUMPRODUCT(C2:C12,B2:B12)/SUM(C2:C12).
Using the SUMPRODUCT formula, we find out the total revenue after selling the entire quantity of goods. And the SUM function sums up the quantity of goods. 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
There are standard deviations for the general population and for the sample. In the first case, this is the root of the general variance. In the second, from the sample variance.
To calculate this statistical indicator, a dispersion formula is compiled. The root is extracted from it. But in Excel there is a ready-made function for finding the standard deviation.
The standard deviation is tied to the scale of the source data. This is not enough for a figurative representation of the variation of the analyzed range. To obtain the relative level of data scatter, 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.
In mathematics, the arithmetic mean of numbers (or simply the mean) is the sum of all the numbers in a given set divided by the number of numbers. This is the most generalized and widespread concept of average value. As you already understood, to find you need to sum up all the numbers given to you, and divide the resulting result by the number of terms.
What is the arithmetic mean?
Let's look at an example.
Example 1. Given numbers: 6, 7, 11. You need to find their average value.
Solution.
First, let's find the sum of all these numbers.
Now divide the resulting sum by the number of terms. Since we have three terms, we will therefore 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 can be clearly seen in the illustration.
The average is a bit like “evening out” a series of numbers. As you can see, the piles of pencils have become the same level.
Let's look at another example to consolidate the knowledge gained.
Example 2. Given numbers: 3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29. You need to find their arithmetic mean.
Solution.
Find the amount.
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 let's look at negative numbers. Let's remember how to summarize them. For example, you have two numbers 1 and -4. Let's find their sum.
1 + (-4) = 1 - 4 = -3
Knowing this, let's look at another example.
Example 3. Find the average value of a series of numbers: 3, -7, 5, 13, -2.
Solution.
Find the sum of numbers.
3 + (-7) + 5 + 13 + (-2) = 12
Since there are 5 terms, 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 Microsoft Office software package. Let's look at a brief instruction, the value of 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 refer to ranges and arrays).To make it more clear, let’s try out the knowledge we have 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 this, 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, you should have the answer in cell C7 - 13.7. When you click on cell C7, the function (=Average(C1:C6)) will appear in the formula bar.
This feature is very useful for accounting, invoices, or when you just need to find the average of a very long series of numbers. Therefore, it is often used in offices and large companies. This allows you to maintain order in your records and makes it possible to quickly calculate something (for example, average monthly income). You can also use Excel to find the average value of a function.
What is the arithmetic mean?
- The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms
- divide
- Number Mean (Mean), Arithmetic Mean (Arithmetic Mean) - an average value characterizing a 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 in a group differ significantly from the rest, this may distort the resulting arithmetic mean. Therefore, in this case, it is preferable to use the geometric mean (it is calculated in a similar way, but here the arithmetic mean of the logarithms of the observation values is determined, and then its antilogarithm is found) or - which is used most often - to find the mean value (median). from a series of quantities arranged in ascending order). Another method of 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 any variable; More often, this method is used to determine the average value in several series of experiments.
For example: numbers 1 and 99, add and divide by two:
(1+99)/2=50 - arithmetic mean
If you take the numbers (1,2,3,15,59)/5=16 - the arithmetic mean, etc., etc. - The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, representing the sum of all recorded values divided by their number.
This term has other meanings, see average meaning.
The arithmetic mean (in mathematics and statistics) is one of the most common measures of central tendency, representing the sum of all recorded values divided by their number.Proposed (along with the geometric mean and harmonic mean) by the Pythagoreans 1.
Special cases of the arithmetic mean are the mean (general population) and the sample mean (sample).
A Greek letter is used to denote the arithmetic mean of the entire population. For a random variable for which the mean value is determined, there is a probabilistic mean or mathematical expectation of the random variable. If the set X is a collection of random numbers with a probabilistic mean, then for any sample xi from this population = E(xi) is the mathematical expectation of this sample.
In practice, the difference between and bar(x) is that it is a typical variable, because you can see a sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then bar(x) , (but not) 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:
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 expected value of X can be viewed as the arithmetic mean of repeated measurements of X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown expected value.In elementary algebra, it is proven that the average of n + 1 numbers is greater than the average of 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 the number is equal to the average. The larger n, the smaller the difference between the new and old averages.
Note that there are several other averages, including the power average, the Kolmogorov average, the harmonic average, the arithmetic-geometric average, and various weighted averages.
Examples edit 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 simpler 5+5=10, 10:2. Because we were adding 2 numbers, which means how many numbers we add, we divide by that many.Continuous random variable edit edit wiki text
For a continuously distributed quantity f(x), the arithmetic mean on the segment a;b is determined through a definite integral: Some problems of using the mean Lack of robustness edit Main article: Robustness in statistics Although the arithmetic mean is often used as average values or central tendencies, this concept does not apply to Robust statistics, which means that the arithmetic mean is strongly influenced by large deviations. It is noteworthy that for distributions with a large skewness coefficient, the arithmetic mean - This is adding up the numbers and dividing them, how many were like this 33+66+99= adding up 33+66+99= 198 and dividing how many were read out, we have 3 numbers that are 33 66 and 99 and we need to divide what we got like this: 33+ 66+99=198:3=66 is the average orethmetic
- well it’s like 2+8=10 and the average is 5
- 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 divided by the number of numbers in this set.
The simplest case is to find the arithmetic mean of two numbers x1 and x2. Then their arithmetic mean is 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 in the form: X = (1/n)xi, where the summation is carried out over 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 when a set of numbers represents members of an arithmetic progression. As is known, the terms 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 term.Let a1, a1+d, a1+2d,...a1+(n-1)d be terms of an arithmetic progression. Their arithmetic mean is equal to 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. - Divide the sum of the numbers by their number
- this is when you add everything up and divide it
- If I'm not mistaken, this is when you add up the sum of numbers and divide by the number of numbers themselves...
- 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!
- Vyachaslav Bogdanov answered incorrectly!!! !
In your own words!
The arithmetic mean is the average value between two values.... It is found as the sum of numbers divided by the number... Or simply, if two numbers are around someone’s number (or rather, there is some number in order between them), then this number will be the average. ar. !6 + 8... av ar = 7
- divider gygygygygygyggy
- The average between maximum and minimum (all numerical indicators are added up and divided by their number
) - this is when you add up numbers and divide by the number of numbers