In the process of studying mathematics, schoolchildren become acquainted with the concept of arithmetic mean. In the future, in statistics and some other sciences, students are faced with the calculation of other average values. What they can be and how they differ from each other?
Averages: Meaning and Differences
Not always accurate indicators give an understanding of the situation. In order to assess this or that situation, it is sometimes necessary to analyze a huge number of numbers. And then average values come to the rescue. They allow to assess the situation in general and in general.
Since school times, many adults remember the existence of the arithmetic mean. It is very easy to calculate it - the sum of a sequence of n members is divided by n. That is, if it is necessary to calculate the arithmetic mean in the sequence of values 27, 22, 34 and 37, then it is necessary to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values are used in the calculations. In this case, the desired value will be equal to 30.
Often, the geometric course is also studied as part of a school course. The calculation of this value is based on the extraction of the nth root of the product of n-terms. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be equal to 29.4.
Secondary harmonic in a general education school is usually not a subject of study. However, it is used quite often. This value is inverse to the arithmetic mean and is calculated as a quotient of n - the number of values and the sum 1 / a 1 + 1 / a 2 + … + 1 / a n . If we again take the same series of numbers for the calculation, then the harmonic will be 29.6.
Weighted average: features
However, all of the above values may not be used everywhere. For example, in statistics, when calculating some average values, the "weight" of each number used in calculations has an important role. The results are more revealing and correct, because they take into account more information. This group of quantities has the general name "weighted average". They are not in school, so they should stay in more detail.
First of all, it is necessary to tell what is meant by the "weight" of a value. The easiest way to explain this is with a concrete example. The patient's body temperature is measured twice a day in the hospital. Of the 100 patients in different departments of the hospital, 44 will have a normal temperature - 36.6 degrees. Another 30 will have an increased value - 37.2, 14 - 38, 7 - 38.5, 3 - 39, and the other two - 40. And if we take the arithmetic average, then this value will be more than 38 degrees! But almost half of the patients have a completely normal temperature. And here it will be more correct to use the weighted average value, and the “weight” of each value will be the number of people. In this case, the result of the calculation will be 37.25 degrees. The difference is obvious.
In the case of weighted average calculations for the "weight" can be taken the number of shipments, the number of people working on a given day, in general, anything that can be measured and affect the final result.
The weighted average value is related to the arithmetic average considered at the beginning of the article. However, the first value, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also weighted geometric mean and harmonic values.
There is another interesting variation used in series of numbers. This is a weighted moving average. It is based on it that trends are calculated. In addition to the values themselves and their weights, frequency is also used there. And when calculating the average value at some point in time, values for previous time periods are also taken into account.
The calculation of all these values is not so complicated, but in practice only the usual weighted average is usually used.
In this age of computerization there is no need to calculate the weighted average manually. However, it will be useful to know the calculation formula so that you can verify and, if necessary, correct the results obtained.
The easiest way to consider the calculation is with a concrete example.
|Salary (thousand rubles)||Number of workers (people)|
It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers who receive a certain income.
So, the weighted average is calculated using the following formula:
x = (a 1 * w 1 + a 2 * w 2 + … + a n * w n ) / (w 1 + w 2 + … + w n )
For example, the calculation will be as follows:
x = (32 * 20 + 33 * 35 + 34 * 14 + 40 * 6) / (20 + 35 + 14 + 6) = (640 + 1155 + 476 + 240) / 75 = 33.48
Obviously, there are no special difficulties in order to manually calculate the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like a function SUMMPRODUCT (a series of numbers; a series of weights) / SUM (a series of weights).