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.
Geometric mean is applied in cases where individual values of a characteristic represent relative dynamics values, constructed in the form of chain values, as a ratio to the previous level of each level in a series of dynamics, i.e., characterizes the average growth coefficient.
Mode and median are very often calculated in statistics problems and they are additional to the average characteristics of the population and are used in mathematical statistics to analyze the type of distribution series, which can be normal, asymmetric, symmetric, etc.
Just like the median, the values of a characteristic that divides the population into four equal parts are calculated - quartels, into five parts - quintels, into ten equal parts - decels, into one hundred equal parts - percentels. Using the distribution of the considered characteristics in statistics when analyzing variation series allows us to characterize the population under study in more depth and detail.
Unlike the arithmetic mean, the geometric mean allows you to estimate the degree of change in a variable over time. The geometric mean is the nth root of the product of n values (in Excel, the =SRGEOM function is used):
G = (X 1 * X 2 * … * X n) 1/n
A similar parameter - the geometric mean value of the rate of profit - is determined by the formula:
G = [(1 + R 1) * (1 + R 2) * … * (1 + R n)] 1/n - 1,
where R i is the rate of profit for the i-th period of time.
For example, suppose the initial investment is $100,000. By the end of the first year, it falls to $50,000, and by the end of the second year it recovers to the initial level of $100,000. The rate of return of this investment over a two-year period equals 0, since the initial and final amounts of funds are equal to each other. However, the arithmetic average of the annual rates of return is = (-0.5 + 1) / 2 = 0.25 or 25%, since the rate of return in the first year R 1 = (50,000 - 100,000) / 100,000 = -0.5 , and in the second R 2 = (100,000 - 50,000) / 50,000 = 1. At the same time, the geometric mean value of the rate of profit for two years is equal to: G = [(1-0.5) * (1+1 )] 1/2 - 1 = S - 1 = 1 - 1 = 0. Thus, the geometric mean more accurately reflects the change (more precisely, the absence of changes) in the volume of investments over a two-year period than the arithmetic mean.
Interesting Facts. Firstly, the geometric mean will always be less than the arithmetic mean of the same numbers. Except for the case when all the numbers taken are equal to each other. Secondly, by considering the properties of a right triangle, you can understand why the mean is called geometric. The height of a right triangle, lowered to the hypotenuse, is the average proportional between the projections of the legs onto the hypotenuse, and each leg is the average proportional between the hypotenuse and its projection onto the hypotenuse. This gives a geometric way to construct the geometric mean of two (lengths) segments: you need to construct a circle on the sum of these two segments as a diameter, then the height restored from the point of their connection to the intersection with the circle will give the desired value:
Rice. 4.
The second important property of numerical data is their variation, which characterizes the degree of dispersion of the data. Two different samples may differ in both means and variances.
There are five estimates of data variation:
interquartile range,
dispersion,
standard deviation,
the coefficient of variation.
The range is the difference between the largest and smallest elements of the sample:
Range = X Max - X Min
The range of a sample containing the average annual returns of 15 very high-risk mutual funds can be calculated using an ordered array: Range = 18.5 - (-6.1) = 24.6. This means that the difference between the highest and lowest average annual returns of very high-risk funds is 24.6%.
Range measures the overall spread of data. Although sample range is a very simple estimate of the overall spread of the data, its weakness is that it does not take into account exactly how the data are distributed between the minimum and maximum elements. Scale B demonstrates that if a sample contains at least one extreme value, the sample range is a very imprecise estimate of the spread of the data.
Average values in statistics play an important role because... they allow us to obtain a general characteristic of the phenomenon being analyzed. The most common average is, of course, . It occurs when an aggregating indicator is formed using the sum of elements. For example, the mass of several apples, the total revenue for each day of sales, etc. But this doesn't always happen. Sometimes an aggregate indicator is formed not as a result of summation, but as a result of other mathematical operations.
Consider the following example. Monthly inflation is the change in the price level of one month compared to the previous month. If the inflation rates for each month are known, how to obtain the annual value? From a statistical point of view, this is a chain index, so the correct answer is: by multiplying monthly inflation rates. That is, the overall inflation rate is not a sum, but a product. Now how can you find out the average inflation for a month if there is an annual value? No, not divide by 12, but take the 12th root (the degree depends on the number of factors). In general, the geometric mean is calculated using the formula:
That is, it is the root of the product of the original data, where the degree is determined by the number of factors. For example, the geometric mean of two numbers is the square root of their product
of three numbers - the cube root of the product
etc.
If each original number is replaced by their geometric mean, then the product will give the same result.
To better understand what geometric mean is and how it differs from arithmetic mean, consider the following figure. There is a right triangle inscribed in a circle.
The median is omitted from a right angle a(to the middle of the hypotenuse). Also from the right angle the height is lowered b, which is at the point P divides the hypotenuse into two parts m And n. Because The hypotenuse is the diameter of the circumscribed circle, and the median is the radius, then it is obvious that the length of the median a is the arithmetic mean of m And n.
Let's calculate what the height is b. Due to the similarity of triangles ABP And BCP equality is true
That is, the height of a right triangle is the geometric mean of the segments into which it divides the hypotenuse. Such a clear difference.
In MS Excel, the geometric mean can be found using the SRGEOM function.
Everything is very simple: call the function, specify the range and you're done.
In practice, this indicator is not used as often as the arithmetic average, but it still occurs. For example, there is this human development index, which is used to compare living standards in different countries. It is calculated as the geometric mean of several indices.
There are other averages. About them another time.
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 a 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 the 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 power of 1/3, 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.