In the process of studying mathematics, students get acquainted with the concept of arithmetic mean. In the future, in statistics and some other sciences, students are faced with the calculation of other averages. What can they be and how do they differ from each other?
Average values: meaning and differences
Not always accurate indicators give an understanding of the situation. In order to evaluate a particular situation, it is sometimes necessary to analyze a huge number of numbers. And then averages come to the rescue. It is they that make it possible to assess the situation as a whole.
Since school times, many adults remember the existence of an arithmetic mean. It is very simple to calculate - the sum of a sequence of n members is divided by n. That is, if you need to calculate the arithmetic mean in a sequence of values 27, 22, 34 and 37, then you need 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 30.
Often in the framework of the school course, geometric mean is also studied. The calculation of this value is based on extracting the root of the nth degree from the product of n-members. If we take the same numbers: 27, 22, 34 and 37, then the result of the calculations will be 29.4.
Harmonic secondary in a comprehensive school is usually not the subject of study. Nevertheless, it is used quite often. This value is the inverse of the arithmetic mean and is calculated as the quotient from 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 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 the calculations plays an important role. The results are more indicative and correct, since they take into account more information. This group of quantities is collectively called the "weighted average value". They do not go to school, so you should dwell on them in more detail.
First of all, it is worth telling what is meant by the "weight" of a particular value. The easiest way to explain this is with a specific example. Twice a day, the patient’s body temperature is measured in the hospital. Of the 100 patients in different departments of the hospital, 44 will have a normal temperature of 36.6 degrees. Another 30 will have an increased value - 37.2, for 14 - 38, for 7 - 38.5, for 3 - 39, and for the remaining two - 40. And if you take the arithmetic mean, this value in the hospital 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, 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 value correlates with the arithmetic average, considered at the beginning of the article. However, the first quantity, as already mentioned, also takes into account the weight of each number used in the calculations. In addition, there are also weighted average geometric and harmonic values.
There is another interesting variation used in rows 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 so complicated, but in practice, only the usual average weighted value is usually used.
Calculation Methods
In the age of rampant computerization, there is no need to manually calculate the weighted average. However, it will be useful to know the calculation formula so that you can check and, if necessary, adjust the results.
The easiest way to consider the calculation is with a specific example.
| Salary (thousand rubles) | Number of workers (people) |
| 32 | twenty |
| 33 | 35 |
| 34 | fourteen |
| 40 | 6 |
It is necessary to find out what is the average wage at this enterprise, taking into account the number of workers receiving a particular income.
So, the calculation of the weighted average value is performed 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 like this:
x = (32 * 20 + 33 * 35 + 34 * 14 + 40 * 6) / (20 + 35 + 14 + 6) = (640 + 1155 + 476 + 240) / 75 = 33.48
Obviously, there is no particular difficulty in manually calculating the weighted average. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like the SUMPRODUCT function (a series of numbers; a number of weights) / SUMM (a number of weights).