What is the arithmetic mean of numbers examples. How to find the arithmetic mean in Excel
- first add these numbers (1+3+5+7) and get 16
- We need to divide the resulting result by 4 (quantity): 16/4 and get the result 4.
- are looking for total weight of all apples (the sum of all indicators) - it is equal to 1080 grams,
- divide the total weight by the number of apples 1080:5 = 216 grams. This is the arithmetic mean.
- N.Ya. Vilenkin. Mathematics: textbook. for 5th grade. general education uchr. - Ed. 17th. - M.: Mnemosyne, 2005. )
- Igor had 45 rubles with him, Andrey had 28, and Denis had 17.
- With all their money they bought 3 movie tickets. How much did one ticket cost?
- Initially, you need to carry out the operation of adding the selected numbers. This can often be done, as they say, manually, without using electronic equipment.
- After addition is performed and its result is obtained, division must be performed. 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 obtain the required value.
- To do this, select the numbers needed in the calculation and add them to get the total.
- After this amount three will be found, you need 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.
- Four digits are selected for which the arithmetic mean must be calculated. Next, summation is performed and the final result of this procedure is found.
- 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.
- 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, just need to be added 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.
- The number of numbers that are specified to obtain the required value is counted. This operation can be carried out either manually with a small number of numbers or using 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.
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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, will surprisingly yield big number because of 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 °).
The arithmetic mean is the sum of numbers divided by the number of these same numbers. And finding the arithmetic mean is very simple.
As follows from the definition, we must take the numbers, add them and divide by their number.
Let's give an example: we are given the numbers 1, 3, 5, 7 and we need to find the arithmetic mean of these numbers.
So, the arithmetic mean of the numbers 1, 3, 5 and 7 is 4.
Arithmetic mean - the average value among the given indicators.
It is found by dividing the sum of all indicators by their number.
For example, I have 5 apples weighing 200, 250, 180, 220 and 230 grams.
We find the average weight of 1 apple as follows:
This is the most commonly used indicator in statistics.
The arithmetic mean is numbers added together and divided by their number, the resulting answer is the arithmetic mean.
For example: Katya put 50 rubles in the piggy bank, Maxim 100 rubles, and Sasha put 150 rubles in the piggy bank. 50 + 100 + 150 = 300 rubles in the piggy bank, now we divide this amount by three (three people put money in). So 300: 3 = 100 rubles. These 100 rubles will be the arithmetically average, each of them put in the piggy bank.
There is such a simple example: one person eats meat, another person eats cabbage, and the arithmetically average they both eat cabbage rolls.
The average salary is calculated in the same way...
The arithmetic mean is the sum of all values and divided by their number.
For example the numbers 2, 3, 5, 6. You need to add them up 2+ 3+ 5 + 6 = 16
We divide 16 by 4 and get the answer 4.
4 is the arithmetic mean of these numbers.
The arithmetic mean of several numbers is the sum of these numbers divided by their number.
x avg arithmetic mean
S sum of numbers
n number of numbers.
For example, we need to find the arithmetic mean of the numbers 3, 4, 5 and 6.
To do this, we need to add them up and divide the resulting sum by 4:
(3 + 4 + 5 + 6) : 4 = 18: 4 = 4,5.
I remember taking the final test in mathematics
So there it was necessary to find the arithmetic mean.
It’s good that kind people suggested what to do, otherwise there would be trouble.
For example, we have 4 numbers.
Add up the numbers and divide by their number (in this case 4)
For example the numbers 2,6,1,1. Add 2+6+1+1 and divide by 4 = 2.5
As you can see, nothing complicated. So the arithmetic mean is the average of all numbers.
We know this from school. Who had good teacher in mathematics, it was possible to remember this simple action the first time.
When finding the arithmetic mean, you need to add up all the available numbers and divide by their number.
For example, I bought 1 kg of apples, 2 kg of bananas, 3 kg of oranges and 1 kg of kiwi at the store. How many kilograms of fruit did I buy on average?
7/4= 1.8 kilograms. This will be the arithmetic mean.
The arithmetic mean is the average number between several numbers.
For example, between the numbers 2 and 4, the average number is 3.
The formula for finding the arithmetic mean is:
You need to add up all the numbers and divide by the number of these numbers:
For example, we have 3 numbers: 2, 5 and 8.
Finding the arithmetic mean:
X=(2+5+8)/3=15/3=5
The scope of application of the arithmetic mean is quite wide.
For example, knowing the coordinates of two points on a segment, you can find the coordinates of the middle of this segment.
For example, the coordinates of the segment: (X1,Y1,Z1)-(X2,Y2,Z2).
Let us denote the middle of this segment by coordinates X3,Y3,Z3.
We separately find the midpoint for each coordinate:
The arithmetic mean is the average of the given...
Those. Simply, we have a number of sticks of different lengths and want to find out their average value..
It is logical that for this we bring them together, getting a long stick, and then divide it into the required number of parts..
Here comes the arithmetic mean...
This is how the formula is derived: Sa=(S(1)+..S(n))/n..
Arithmetic is considered the most elementary branch of mathematics and studies simple steps with numbers. Therefore, the arithmetic mean is also very easy to find. Let's start with a definition. The arithmetic mean is a value that shows which number is closest to the truth after several successive operations of the same type. For example, when running a hundred meters, a person shows every time different time, but the average value will be within, for example, 12 seconds. Finding the arithmetic mean in this way comes down to sequentially summing all the numbers in a certain series (race results) and dividing this sum by the number of these races (attempts, numbers). In formula form it looks like this:
Sarif = (Х1+Х2+..+Хn)/n
As a mathematician, I am interested in questions on this subject.
I'll start with the history of the issue. Average values have been thought about since ancient times. Arithmetic mean, geometric mean, harmonic mean. These concepts are proposed in ancient Greece Pythagoreans.
And now the question that interests us. What is meant by arithmetic mean of several numbers:
So, to find the arithmetic mean of numbers, you need to add all the numbers and divide the resulting sum by the number of terms.
The formula is:
Example. Find the arithmetic mean of the numbers: 100, 175, 325.
Let's use the formula for finding the arithmetic mean of three numbers (that is, instead of n there will be 3; you need to add up all 3 numbers and divide the resulting sum by their number, i.e. by 3). We have: x=(100+175+325)/3=600/3=200.
Three children went into the forest to pick berries. The eldest daughter found 18 berries, the middle one - 15, and younger brother- 3 berries (see Fig. 1). They brought the berries to mom, who decided to divide the berries equally. How many berries did each child receive?
Rice. 1. Illustration for the problem
Solution
(Yag.) - children collected everything
2) Divide total berries per number of children:
(Yag.) went to every child
Answer: Each child will receive 12 berries.
In problem 1, the number obtained in the answer is the arithmetic mean.
Arithmetic mean several numbers is the quotient of dividing the sum of these numbers by their number.
Example 1
We have two numbers: 10 and 12. Find their arithmetic mean.
Solution
1) Let's determine the sum of these numbers: .
2) The number of these numbers is 2, therefore, the arithmetic mean of these numbers is: .
Answer: The arithmetic mean of the numbers 10 and 12 is the number 11.
Example 2
We have five numbers: 1, 2, 3, 4 and 5. Find their arithmetic mean.
Solution
1) The sum of these numbers is equal to: .
2) By definition, the arithmetic mean is the quotient of dividing the sum of numbers by their number. We have five numbers, so the arithmetic mean is:
Answer: the arithmetic mean of the data in the numbers condition is 3.
In addition to the fact that it is constantly suggested to be found in lessons, finding the arithmetic mean is very useful in Everyday life. For example, let's say we want to go on holiday to Greece. To choose suitable clothing, we look at what the temperature is in this country at the moment. However, we will not know the overall weather picture. Therefore, it is necessary to find out the air temperature in Greece, for example, for a week, and find the arithmetic average of these temperatures.
Example 3
Temperature in Greece for the week: Monday - ; Tuesday - ; Wednesday - ; Thursday - ; Friday - ; Saturday - ; Sunday - . Calculate the average temperature for the week.
Solution
1) Let's calculate the sum of temperatures: .
2) Divide the resulting amount by the number of days: .
Answer: Average temperature for the week is approx.
The ability to find the arithmetic mean may also be needed to determine the average age of the players on a football team, that is, in order to determine whether the team is experienced or not. It is necessary to sum up the ages of all players and divide by their number.
Problem 2
The merchant was selling apples. At first he sold them at a price of 85 rubles per 1 kg. So he sold 12 kg. Then he reduced the price to 65 rubles and sold the remaining 4 kg of apples. What was the average price for apples?
Solution
1) Let's calculate how much money the merchant earned in total. He sold 12 kilograms at a price of 85 rubles per 1 kg: 
(rub.).
He sold 4 kilograms at a price of 65 rubles per 1 kg: (rubles).
Therefore, the total amount of money earned is equal to: (rub.).
2) The total weight of apples sold is equal to: .
3) Divide the received amount of money by the total weight of apples sold and get the average price for 1 kg of apples: (rubles).
Answer: the average price of 1 kg of apples sold is 80 rubles.
The arithmetic mean helps evaluate the data as a whole, without taking each value separately.
However, it is not always possible to use the concept of arithmetic mean.
Example 4
The shooter fired two shots at the target (see Fig. 2): the first time he hit a meter above the target, and the second time he hit a meter below. The arithmetic average will show that he hit the center exactly, although he missed both times.
Rice. 2. Illustration for example
In this lesson we learned about the concept of arithmetic mean. We learned the definition of this concept, learned how to calculate the arithmetic mean for several numbers. We also learned practical use this concept.
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 get next 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 total revenue after the entire quantity of goods has been sold. 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 ready function to find 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 / average arithmetic value
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 average size for a series of numbers determined in advance. It should be noted that this value in given time widely used by specialists in a number of industries. For example, formulas are known when carrying out calculations by economists or workers in 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 do it. To do this you don't need to have special education. Often there is no need to use computer technology.
To answer the question of how to find the arithmetic mean, consider a number of situations.
The most simple option calculating a given value is calculating it for two numbers. The calculation procedure in this case is very simple:
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 the value for three
Calculating this value in a situation where three numbers are selected will not differ much from the previous option:
Formula
Thus, the formula necessary for calculating the arithmetic three will look like this:
(A+B+C)/3
In this formula The following notation is accepted:
A, B and C are the numbers for which you will need to find the arithmetic mean.
Calculating the arithmetic mean of four
As can already be seen by analogy with the previous options, the calculation of this value for a quantity equal to four will be in the following order:
Formula
From the sequence of actions described above to find the arithmetic mean for four, you can obtain the following formula:
(A+B+C+E)/4
In this formula the variables have the following meaning:
A, B, C and E are those for which it is necessary to find the value of the arithmetic mean.
Applying this formula, it will always be possible to calculate the required value for given quantity numbers.
Calculating the arithmetic mean of five
Performing this operation will require a certain algorithm of actions.
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 are designated as follows:
A, B, C, E and P are numbers for which it is necessary to obtain the arithmetic mean.
Universal calculation formula
Conducting a review various options formulas to calculate the arithmetic mean, you can pay attention to what they have general pattern.
Therefore, it will be more practical to use a general formula to find the arithmetic mean. After all, there are situations when the number and magnitude of calculations can be very large. Therefore it would be wiser to use universal formula and not to develop an individual technology each time to calculate this value.
The main thing when determining the formula is principle of calculating the arithmetic mean O.
This principle, as can be seen from the examples given, looks like this:
Thus, the general formula for calculating the arithmetic mean of a series of selected numbers will look like this:
(A+B+…+N)/N
This formula contains the following variables:
A and B are numbers that are selected in advance to calculate their arithmetic mean.
N is the number of numbers that were taken to calculate the required value.
By substituting the selected numbers into this formula each time, we can always obtain the required value of the arithmetic mean.
As seen, finding the arithmetic mean is a simple procedure. However, you must be careful about the calculations performed and check the results obtained. This approach is explained by the fact that even in the simplest situations there is a possibility of receiving an error, which can then affect further calculations. In this regard, it is recommended to use computer technology that is capable of performing calculations of any complexity.
) and sample mean(s).