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.
Please note
If you need to find the geometric mean for just two numbers, then you don’t need an engineering calculator: take the second root ( square root) from any number can be done using the most ordinary calculator.
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
- average geometric formula
Average value is one of the characteristics of a set of numbers. Represents a number that cannot be outside the range determined by the largest and lowest values in this 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 last value sets and print 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 a table editor for the same purpose. Microsoft Excel. 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 presented decimals, the solution is carried out using the method of calculating the arithmetic mean of integers, but the result is reduced according to the requirements of the problem for the accuracy of the answer.When working with natural fractions they should be brought to 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 the number equal to the sum 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.
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 consider brief instructions, value 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.
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 the series random numbers in the line:
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 three last 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 average arithmetic numbers, which 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 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.
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 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 the calculations given 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:
- 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 for finding 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
Conducting a review various options formulas 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:
- 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 you can see, 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).
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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 up and divide them by 3:
- For four numbers you need to add them up and divide them 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. An arithmetic mean can be misinterpreted as a median, leading to the conclusion that people with big income more than it actually is. “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 the residents, will surprisingly yield large 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, 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 °).