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

In the process of studying mathematics students are introduced to the concept of the arithmetic mean. In the future, in statistics and other Sciences students faced with the calculation of other averages. What they can be and differ from one another?

Averages: meaning and differences

Not always accurate indicators provide an understanding of the situation. In order to assess a particular situation, sometimes need to analyze a huge number of digits. And then come to the aid of the average values. They allow you to assess the situation in General.

Since high school, many adults remember the existence of the arithmetic mean. It's very easy to calculate the sum of a sequence of n terms divided by n. That is, if you want to calculate the arithmetic mean of the sequence values 27, 22, 34 and 37, it is necessary to solve the expression (27+22+34+37)/4, since in the calculations we use 4 values. In this case, the searched value will be 30.

As part of school course of study and the geometric mean. The calculation of this value is based on the root n-th degree of the product of n members. If you take the same numbers: 27, 22, 34 and 37, the result is equal to 29.4.

Harmonic mean of secondary 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 the quotient of the n - number of values and the sum of 1/a₁+1/a₂+...+1/a_N. If again we take the same series of numbers for calculation, the harmonic will be 29.6.

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Weighted average: features

However, all of these values can be used everywhere. For example, in the statistics when calculating some average values, an important role has the "weight" of each number used in the calculations. The results are more revealing and correct with respect to more information. This group of values is generally called "weighted average". They do not go to school, so they should stay in more detail.

First of all, it should tell you what is meant by "weight" of a value. The easiest way to explain this with a concrete example. Twice a day in the hospital going on measure the body temperature of each patient. Of the 100 patients in different hospital 44 would be the normal temperature - 36.6 degrees. Another 30 will be the increased value of 37.2, 14 - 38, 7 - 38,5, 3 - 39, and the remaining two - 40. And if you take the arithmetic mean, this value is in General in the hospital will be more than 38 degrees! But almost half of the patients absolutely normal temperature. And here more correct to use a weighted average, the "weight" of each variable will be number of people. In this case, the calculation result will 37,25 degrees. The difference is obvious.

In the case of weighted average calculations for the "weight" can be taken a number of shipments, number of employees in any given day people in General, anything that can be measured and affect the final result.

Varieties

The average value correlates with the average discussed in 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 the weighted geometric and harmonic value.

There is another interesting variation used in the series. We are talking about weighted moving average. It calculated trends. In addition to the values and their weight there is also used frequently. And when calculating the average value at some point in time are also taken into account value for the previous time periods.

The Calculation of these values is not so complicated, but in practice there are typically only a normal weighted average.

Methods of payment

In an age of rampant computerization is not necessary to calculate the average value. However, it is useful to know the calculation formula to be able to check and, if necessary, correct the results obtained.

The easiest way is to consider the calculation of a concrete example.

Need to know what the average wage at the enterprise taking into account the number of workers receiving a particular wage.

So, the weighted average calculation is performed using this formula:

X = (a₁*w₁+a₂*w₂+...+a_N*w_N)/(w₁+w₂+...+w_N)

For example, the same 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's no special difficulty to manually compute a weighted average. The formula to calculate this value in one of the most popular applications with formulas - Excel - looks like a function AMPREISV (a series of numbers; the number of weights)/SUM (number of scales).