- 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.
- we are looking for the 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.
-
1 / 5
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 °).
The concept of arithmetic average of numbers means the result of a simple sequence of calculations of the average value for a number of numbers determined in advance. It should be noted that this value is currently widely used by specialists in a number of industries. For example, formulas are known for calculations by economists or workers in the statistical industry, where a value of this type is required. 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. You don't need any special education to do this. 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 simplest option for calculating this value is to calculate it for two numbers. The calculation procedure in this case is very simple:
- 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.
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:
- To do this, select the numbers needed in the calculation and add them to get the total.
- After this sum of three has been found, the division procedure must be performed again. In this case, the resulting amount must be divided by three, which corresponds to the number of selected numbers.
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:
- 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.
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.
Using this formula, it will always be possible to calculate the required value for a given number of numbers.
Calculating the arithmetic mean of five
Performing this operation will require a certain algorithm of actions.
- First of all, you need to select five numbers for which the arithmetic mean will be calculated. After this selection, these numbers, as in the previous options, 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.
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
Reviewing various formula options to calculate the arithmetic mean, you can pay attention to the fact that they have a common pattern.
Therefore, it will be more practical to 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 more reasonable to use a 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:
- 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.
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.
What is the arithmetic mean
The arithmetic mean of several quantities is the ratio of the sum of these quantities to their number.
The arithmetic mean of a certain series of numbers is the sum of all these numbers divided by the number of terms. Thus, the arithmetic mean is the average value of a number series.
What is the arithmetic mean of several numbers? And they are equal to the sum of these numbers, which is divided by the number of terms in this sum.
How to find the arithmetic mean
There is nothing complicated in calculating or finding the arithmetic mean of several numbers; it is enough to add all the numbers presented and divide the resulting sum by the number of terms. The result obtained will be the arithmetic mean of these numbers.
Let's look at this process in more detail. What do we need to do to calculate the arithmetic mean and obtain the final result of this number.
First, to calculate it, you need to determine a set of numbers or their number. This set can include large and small numbers, and their number can be anything.
Secondly, all these numbers need to be added and their sum is obtained. Naturally, if the numbers are simple and there are a small number of them, then the calculations can be made by writing them by hand. But if the set of numbers is impressive, then it is better to use a calculator or spreadsheet.
And fourthly, the amount obtained from addition must be divided by the number of numbers. As a result, we will get a result, which will be the arithmetic mean of this series.
Why do you need the arithmetic mean?
The arithmetic mean can be useful not only for solving examples and problems in mathematics lessons, but for other purposes necessary in a person’s everyday life. Such goals can be calculating the arithmetic average to calculate the average financial expenditure per month, or to calculate the time you spend on the road, also in order to find out attendance, productivity, speed of movement, yield and much more.
So, for example, let's try to calculate how much time you spend traveling to school. When going to school or returning home, you spend different time on the road each time, because when you are in a hurry, you walk faster, and therefore the road takes less time. But when returning home, you can walk slowly, communicating with classmates, admiring nature, and therefore the journey will take more time.
Therefore, you will not be able to accurately determine the time spent on the road, but thanks to the arithmetic average, you can approximately find out the time you spend on the road.
Let's assume that on the first day after the weekend you spent fifteen minutes on the way from home to school, on the second day your journey took twenty minutes, on Wednesday you covered the distance in twenty-five minutes, and your journey took the same amount of time on Thursday, and on Friday you were in no hurry and returned for a whole half an hour.
Let's find the arithmetic mean, adding time, for all five days. So,
15 + 20 + 25 + 25 + 30 = 115
Now divide this amount by the number of days
Thanks to this method, you learned that the journey from home to school takes approximately twenty-three minutes of your time.
Homework
1.Using simple calculations, find the arithmetic average of the attendance of students in your class for the week.
2. Find the arithmetic mean:
3. Solve the problem:
The most common type of average is the arithmetic mean.
Simple arithmetic mean
A 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 the given population. Thus, the average annual output per employee is the amount of output that would be produced by each employee if the entire volume of output were equally distributed among all employees of the organization. The arithmetic mean simple value is calculated using 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 average— Equal to the ratio of the sum of individual values of a characteristic to the number of characteristics in the aggregate
Find average salary
Solution: (3 + 3.2 + 3.3 +3.5 + 3.8 + 3.1) / 6 = 3.32 thousand rubles.Arithmetic average weighted
If the volume of the data set is large and represents a distribution series, then the 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 by the price of a unit of production) is divided by the total quantity of production.
Let's imagine this in the form of the following formula:
Example 2 . Find the average salary of workshop workers per monthWeighted arithmetic average— equal to the ratio of (the sum of the products of the value of a feature to the frequency of repetition of this feature) to (the sum of the frequencies of all features). It is used when variants of the population under study occur an unequal number of times.
Average wages can be obtained by dividing the total wages by the total number of workers:
Answer: 3.35 thousand rubles.
Arithmetic mean for interval series
When calculating the arithmetic mean for an interval variation series, first determine the mean for each interval as the half-sum of the upper and lower limits, and then the mean of the entire series. In the case of open intervals, the value of the lower or upper interval is determined by the size of the intervals adjacent to them.
Averages calculated from interval series are approximate.
Example 3. Determine the average age of 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 distribution.
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 calculations:
1. The product of the average by the sum of frequencies is always equal to the sum of the products of the variant by frequencies, i.e.
2. The arithmetic mean of the sum of varying quantities is equal to the sum of the arithmetic means of these quantities:
3. The algebraic sum of deviations of individual values of a characteristic from the average is equal to zero:
4. The sum of squared deviations of options from the average is less than the sum of squared deviations from any other arbitrary value, i.e.
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 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. Anyone who had a good math teacher could 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 middle 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 operations 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 a different time each 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 were proposed in ancient Greece by the 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.
) and sample mean(s).