) and sample mean(s).
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)).)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.
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.
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 °).
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.
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:
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 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.
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.
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:
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.
Calculating this value in a situation where three numbers are selected will not differ much from the previous option:
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.
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:
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.
Performing this operation will require a certain algorithm of actions.
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.
Reviewing various formula options to calculate the arithmetic mean, you can pay attention to the fact that they have a 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 more reasonable to use a universal formula and not 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.
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.
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.
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.
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:
Three children went into the forest to pick berries. The eldest daughter found 18 berries, the middle one - 15, and the 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 the total number of berries by the 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 called 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 asked 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 the practical application of this concept.
Is it possible to register an employee for the position of financial director - chief accountant? The chief accountant claims that yes, but...
The head of a small business can easily manage the budget independently. CHECKED! If you're budgeting...
The creation of new projects involves a preliminary economic justification for their feasibility, subsequent...
Reporting is generated by the RM, is agreed upon (approved) by the Risk Committee under the Management Board and transmitted to...
At the edge of a large, very large meadow, on a long emerald blade of grass lived a tiny Ladybug. Little Ladybug...
Nowadays, it is quite common for people to turn to the stars. With the help of a horoscope a person can learn a lot about...
Business or friendly. If you were pursuing a stranger, it means your level of trust in the female sex...
according to Freud's dream book If you dreamed about how you were fishing, it means that in real life you can hardly switch off...
The New Year's whirlwind will spin us around so quickly and rapidly that it's time to thoroughly think through the New Year's...
The article offers you some of the most delicious recipes for making vinaigrette. Vinaigrette is a delicious salad...
July 4th, 2015 , 08:33 pm Lately our Nastena has been completely giving up sweets (and thank God), but...
Fragrant, very tasty chocolate cake, it’s impossible to stop eating! The Negro’s Kiss cake is very easy to prepare,...
Dream interpretation is an ancient art that allows you to at least assess the psychological status...
As soon as the first mirrors were born, people immediately endowed them with all sorts of mystical abilities....
The head of a small business can easily manage the budget independently. CHECKED! If you manage...
The creation of new projects involves a preliminary economic justification for their feasibility, subsequent...