In the process of studying mathematics, students get acquainted with the concept of the arithmetic mean. In the future, in statistics and some other sciences, students are also faced with the calculation of other averages. What can they be and how do they differ from each other?
Mean values: 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 figures. And then averages come to the rescue. They allow you to assess the situation in general.
Since school days, many adults remember the existence of the arithmetic mean. It is very easy to calculate - the sum of a sequence of n terms is divisible by n. That is, if you need to calculate the arithmetic mean in the sequence of values 27, 22, 34 and 37, then you need to solve the expression (27 + 22 + 34 + 37) / 4, since 4 values \u200b\u200bare used in the calculations. In this case, the desired value will be equal to 30.
The geometric mean is often studied as part of the school course. The calculation of this value is based on extracting the root of the nth degree from the productn-members. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be 29, 4.
The harmonic mean in a comprehensive school is usually not a subject of study. However, it is used quite often. This value is the reciprocal of the arithmetic mean and is calculated as a quotient of n - the number of values and the sum 1/a1+1/a2+…+1/a. If we again take the same series of numbers for 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 plays an important role. The results are more revealing and correct because they take into account more information. This group of values is collectively referred to as the "weighted average". They are not passed at school, so it is worth dwelling on them in more detail.
First of all, it is worth explaining what is meant by the "weight" of a value. The easiest way to explain this is with a specific example. The body temperature of each patient 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 remaining two - 40. And if we take the arithmetic mean, then this value in general for the hospital will be over 38degrees! But almost half of the patients have a completely normal temperature. And here it would be more correct to use the weighted average, 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, the "weight" can be taken as 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.
Varieties
The weighted average corresponds to the arithmetic average discussed 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 geometric and harmonic weighted averages.
There is another interesting variation used in series of numbers. This is a weighted moving average. It is on its basis that trends are calculated. In addition to the values themselves and their weight, periodicity is also used there. And when calculating the average value at some point in time, the values for previous time periods are also taken into account.
The calculation of all these values is not that difficult, but in practice only the usual weighted average is usually used.
Calculation methods
In the age of computerization, there is no need to manually calculate the weighted average. However, it would be useful to know the calculation formula so that you cancheck and, if necessary, correct the results obtained.
It will be easiest to consider the calculation on a specific example.
| Salary (thousand rubles) | Number of workers (persons) |
| 32 | 20 |
| 33 | 35 |
| 34 | 14 |
| 40 | 6 |
It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers receiving this or that income.
So, the weighted average is calculated using the following formula:
x=(a1w1+a2w 2+…+a w)/(w1+w 2+…+w)
For an example, the calculation will be as follows:
x=(3220+3335+3414+406)/(20+35+14+6)=(640+1155+476+240)/75=33, 48
Obviously, it's not too hard to manually calculate the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT (series of numbers; series of weights) / SUM (series of weights) function.
