Weighted average value - what is it and how to calculate it?

In the process of studying mathematics, students get acquainted with the concept of the arithmetic mean. Later on, in statistics and some other sciences, students are faced with the calculation of other mean values. What they can be and how 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 average values come to the rescue. They allow you to assess the situation in general.

Since school times, many adults remember the existence of the arithmetic mean. It is very simple to calculate - the sum of a sequence of n terms is divisible by n. That is, if you want to calculate the arithmetic mean in a sequence of values 27, 22, 34, and 37, 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 is 30.

Often within the school course, the geometric mean is also studied. The calculation of this value is based on extracting the nth root from the product of n-terms. If we take the same numbers: 27, 22, 34 and 37, the result of the calculations will be 29.4.

The average harmonic in a general education 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 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 to calculate, then the harmonic will be 29.6.

Weighted average: features

However, all of the above values can not be used everywhere. For example, in the calculation of some mean values, the "weight" of each number used in calculations plays an important role in statistics. The results are more revealing and correct, since they take more information into account. This group of values is collectively referred to as a "weighted average". They do not pass in the school, so it's worth to stop in detail.

First of all, it is worthwhile to tell what is meant by the "weight" of this or that value. The easiest way to explain this is with a specific example. Twice a day in the hospital, body temperature is measured in each patient. 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 of 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 More than 38 degrees! But almost half of the patients have completely normal temperature. And here it will be more correct to use a 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 number of shipments, the number of people working on one day or another, in general, anything that can be measured and affect the final result can be taken as "weight".

Varieties

The weighted average value is related to the arithmetic mean, 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 is also a weighted average geometric and harmonic value.

There is another interesting variety used in the 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 mean value at some point in time, the values for the previous time intervals are also taken into account.

Calculating all these values is not so complicated, but in practice, usually only the usual weighted average is used.

Methods of calculation

In the century of generalized computerization, there is no need to calculate the weighted average value manually. However, it will be superfluous to know the calculation formula, so that you can check and, if necessary, adjust the results.

The easiest way is to look at the calculation using a specific example.

It is necessary to find out what is the average salary at this enterprise, taking into account the number of workers receiving one or another earnings.

So, the calculation of the weighted average value is made using the following formula:

X = (a ₁ * w ₁ + a ₂ * w ₂ + ... + a _n * w _n ) / (w ₁ + w ₂ + ... + w _n )

For an example, the calculation will be:

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 manually calculating the weighted average value. The formula for calculating this value in one of the most popular applications with formulas - Excel - looks like a function SUMPROVIS (series of numbers, a series of weights) / SUM (a number of weights).