How to find the arithmetic mean of prime numbers. How to find and calculate the arithmetic mean for two

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:

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:

Weighted 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.

) 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:
  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 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 topic of arithmetic mean and geometric mean is included in the mathematics program for grades 6-7. Since the paragraph is quite easy to understand, it is quickly passed over, and by the end of the school year, students have forgotten it. But knowledge in basic statistics is needed to pass the Unified State Exam, as well as for international SAT exams. And for everyday life, developed analytical thinking never hurts.

  How to calculate the arithmetic mean and geometric mean of numbers

  Let's say there is a series of numbers: 11, 4, and 3. The arithmetic mean is the sum of all numbers divided by the number of given numbers. That is, in the case of the numbers 11, 4, 3, the answer will be 6. How do you get 6?

  Solution: (11 + 4 + 3) / 3 = 6

  The denominator must contain a number equal to the number of numbers whose average needs to be found. The sum is divisible by 3, since there are three terms.

  Now we need to figure out the geometric mean. Let's say there is a series of numbers: 4, 2 and 8.

  The geometric mean of numbers is the product of all given numbers, located under the root with a power equal to the number of given numbers. That is, in the case of numbers 4, 2 and 8, the answer will be 4. Here's how it turned out:

  Solution: ∛(4 × 2 × 8) = 4

  In both options, we got whole answers, since special numbers were taken for the example. This does not always happen. In most cases, the answer has to be rounded or left at the root. For example, for the numbers 11, 7 and 20, the arithmetic mean is ≈ 12.67, and the geometric mean is ∛1540. And for the numbers 6 and 5, the answers will be 5.5 and √30, respectively.

  Could it happen that the arithmetic mean becomes equal to the geometric mean?

  Of course it can. But only in two cases. If there is a series of numbers consisting only of either ones or zeros. It is also noteworthy that the answer does not depend on their number.

  Proof with units: (1 + 1 + 1) / 3 = 3 / 3 = 1 (arithmetic mean).

  ∛(1 × 1 × 1) = ∛1 = 1(geometric mean).

  

  Proof with zeros: (0 + 0) / 2=0 (arithmetic mean).

  √(0 × 0) = 0 (geometric mean).

  There is no other option and cannot be.

  Answer: everyone got one 4 pears.

  Example 2. 15 people came to English courses on Monday, 10 on Tuesday, 12 on Wednesday, 11 on Thursday, 7 on Friday, 14 on Saturday, 8 on Sunday. Find the average course attendance for the week.
  Solution: Let's find the arithmetic mean:

  15 + 10 + 12 + 11 + 7 + 14 + 8	=	77	= 11
  7		7

  Answer: On average, people attended English language courses 11 person per day.

  Example 3. A racer rode for two hours at 120 km/h and an hour at 90 km/h. Find the average speed of the car during the race.
  Solution: Let's find the arithmetic average of the car speeds for each hour of travel:

  120 + 120 + 90	=	330	= 110
  3		3

  Answer: the average speed of the car during the race was 110 km/h

  Example 4. The arithmetic mean of 3 numbers is 6, and the arithmetic mean of 7 other numbers is 3. What is the arithmetic mean of these ten numbers?
  Solution: Since the arithmetic mean of 3 numbers is 6, their sum is 6 3 = 18, similarly, the sum of the remaining 7 numbers is 7 3 = 21.
  This means the sum of all 10 numbers will be 18 + 21 = 39, and the arithmetic mean is equal to

  39	= 3.9
  10

  Answer: the arithmetic mean of 10 numbers is 3.9 .

  It gets lost in calculating the average.

  Average meaning set of numbers is equal to the sum of numbers S divided by the number of these numbers. That is, it turns out that average meaning equals: 19/4 = 4.75.

  note

  If you need to find the geometric mean for just two numbers, then you don’t need an engineering calculator: you can extract the second root (square root) of any number using the most ordinary calculator.

  Helpful advice

  Unlike the arithmetic mean, the geometric mean is not as strongly affected by large deviations and fluctuations between individual values ​​in the set of indicators under study.

  Sources:

  • Online calculator that calculates the geometric mean
  • geometric mean formula

  Average value is one of the characteristics of a set of numbers. Represents a number that cannot fall outside the range defined by the largest and smallest values ​​in that set of numbers. Average arithmetic value is the most commonly used type of average.

  

  Instructions

  Add up all the numbers in the set and divide them by the number of terms to get the arithmetic mean. Depending on the specific calculation conditions, it is sometimes easier to divide each of the numbers by the number of values ​​in the set and sum the result.

  Use, for example, included in the Windows OS if it is not possible to calculate the arithmetic average in your head. You can open it using the program launch dialog. To do this, press the hot keys WIN + R or click the Start button and select Run from the main menu. Then type calc in the input field and press Enter or click the OK button. The same can be done through the main menu - open it, go to the “All programs” section and in the “Standard” section and select the “Calculator” line.

  Enter all the numbers in the set sequentially by pressing the Plus key after each of them (except the last one) or clicking the corresponding button in the calculator interface. You can also enter numbers either from the keyboard or by clicking the corresponding interface buttons.

  Press the slash key or click this in the calculator interface after entering the last set value and type the number of numbers in the sequence. Then press the equal sign and the calculator will calculate and display the arithmetic mean.

  You can use the Microsoft Excel spreadsheet editor for the same purpose. In this case, launch the editor and enter all the values ​​of the sequence of numbers into the adjacent cells. If, after entering each number, you press Enter or the down or right arrow key, the editor itself will move the input focus to the adjacent cell.

  Click the cell next to the last number entered if you don't want to just see the average. Expand the Greek sigma (Σ) drop-down menu for the Edit commands on the Home tab. Select the line " Average" and the editor will insert the desired formula for calculating the arithmetic mean into the selected cell. Press the Enter key and the value will be calculated.

  The arithmetic mean is one of the measures of central tendency, widely used in mathematics and statistical calculations. Finding the arithmetic average for several values ​​is very simple, but each task has its own nuances, which are simply necessary to know in order to perform correct calculations.

  

  What is an arithmetic mean

  The arithmetic mean determines the average value for the entire original array of numbers. In other words, from a certain set of numbers a value common to all elements is selected, the mathematical comparison of which with all elements is approximately equal. The arithmetic average is used primarily in the preparation of financial and statistical reports or for calculating the results of similar experiments.

  How to find the arithmetic mean

  Finding the arithmetic mean for an array of numbers should begin by determining the algebraic sum of these values. For example, if the array contains the numbers 23, 43, 10, 74 and 34, then their algebraic sum will be equal to 184. When writing, the arithmetic mean is denoted by the letter μ (mu) or x (x with a bar). Next, the algebraic sum should be divided by the number of numbers in the array. In the example under consideration there were five numbers, so the arithmetic mean will be equal to 184/5 and will be 36.8.

  Features of working with negative numbers

  If the array contains negative numbers, then the arithmetic mean is found using a similar algorithm. The difference only exists when calculating in the programming environment, or if the problem has additional conditions. In these cases, finding the arithmetic mean of numbers with different signs comes down to three steps:

  1. Finding the general arithmetic average using the standard method;
  2. Finding the arithmetic mean of negative numbers.
  3. Calculation of the arithmetic mean of positive numbers.

  The responses for each action are written separated by commas.

  Natural and decimal fractions

  If an array of numbers is represented by decimal fractions, the solution is carried out using the method of calculating the arithmetic mean of integers, but the result is reduced according to the task’s requirements for the accuracy of the answer.

  When working with natural fractions, they should be reduced to a common denominator, which is multiplied by the number of numbers in the array. The numerator of the answer will be the sum of the given numerators of the original fractional elements.

  • Engineering calculator.

  Instructions

  Keep in mind that in general, the geometric mean of numbers is found by multiplying these numbers and taking the root of the power from them, which corresponds to the number of numbers. For example, if you need to find the geometric mean of five numbers, then you will need to extract the root of the power from the product.

  To find the geometric mean of two numbers, use the basic rule. Find their product, then take the square root of it, since the number is two, which corresponds to the power of the root. For example, in order to find the geometric mean of the numbers 16 and 4, find their product 16 4=64. From the resulting number, extract the square root √64=8. This will be the desired value. Please note that the arithmetic mean of these two numbers is greater than and equal to 10. If the entire root is not extracted, round the result to the desired order.

  To find the geometric mean of more than two numbers, also use the basic rule. To do this, find the product of all numbers for which you need to find the geometric mean. From the resulting product, extract the root of the power equal to the number of numbers. For example, to find the geometric mean of the numbers 2, 4, and 64, find their product. 2 4 64=512. Since you need to find the result of the geometric mean of three numbers, take the third root from the product. It is difficult to do this verbally, so use an engineering calculator. For this purpose it has a button "x^y". Dial the number 512, press the "x^y" button, then dial the number 3 and press the "1/x" button, to find the value of 1/3, press the "=" button. We get the result of raising 512 to the 1/3 power, which corresponds to the third root. Get 512^1/3=8. This is the geometric mean of the numbers 2.4 and 64.

  Using an engineering calculator, you can find the geometric mean in another way. Find the log button on your keyboard. After that, take the logarithm for each of the numbers, find their sum and divide it by the number of numbers. Take the antilogarithm from the resulting number. This will be the geometric mean of the numbers. For example, in order to find the geometric mean of the same numbers 2, 4 and 64, perform a set of operations on the calculator. Dial the number 2, then press the log button, press the "+" button, dial the number 4 and press log and "+" again, dial 64, press log and "=". The result will be a number equal to the sum of the decimal logarithms of the numbers 2, 4 and 64. Divide the resulting number by 3, since this is the number of numbers for which the geometric mean is sought. From the result, take the antilogarithm by switching the case button and use the same log key. The result will be the number 8, this is the desired geometric mean.