Arithmetic mean, also known asmean valueIs the most basic and commonly used average index in statistics, which is divided into simple arithmetic mean and weighted arithmetic mean.It is mainly applicable toNumerical data, not applicable to quality data.According to the different forms of expression, the arithmetic mean has different calculation forms and formulas.
The arithmetic mean isWeighted averageA special form of (special in terms of equal weight).In practical problems, when the weights are not equal, the weighted average should be used when calculating the average;When the weights of each item are equal, the arithmetic mean shall be used to calculate the average.[1]
(1) Applicable: Mainly used for raw data not grouped.Set a group of data as Xone，Xtwo，...，Xn, the simple arithmetic mean calculation formula is:
(2) For example, a sales team has five salesmen. The sales on New Year's Day are 520 yuan, 600 yuan, 480 yuan, 750 yuan and 500 yuan respectively. Calculate the average sales on that day.
Average sales=(520+600+480+750+500)/5=570 yuan
The calculation results show that the average turnover of five salesmen on New Year's Day is 570 yuan.
(3) Expand: a set of data xone.xtwo.xthreeWave up and down around the number a in order to get the arithmetic mean
First, subtract a from each original data to get a set of new data recorded as xone' xtwo' .....xn'
X1'=X1-a X2'=X2-a ..... Xn'=Xn-a
Namely: X1=X1'+aX2=X2'+a.....Xn=Xn'+a
Then, calculate the average
Formula (1)
take
，
，....，..
Substitution formula (1)
Get:
I.e. arithmetic mean=
+ a
So:
=
+ a
Weighted arithmetic mean
Applicable: It is mainly used to process grouped data.Let the original data be divided into K groups, the values of each group are X1, X2,..., Xk, and the frequency of each group is fone，ftwo，...，fk, the calculation formula of weighted arithmetic mean is:[2]
Special instructions
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1. The weighted arithmetic mean is affected by two factors at the same time, one is the value of each group, the other is the number of distribution frequencies of each group.When the value is constant, the more frequencies of a group, the greater the effect of the group's value on the average, and vice versa.
Frequency atweightingThe arithmetic mean plays a role in weighing, which is also the meaning of weighted arithmetic mean.
2. The arithmetic mean is vulnerable to extreme values.For example, there are the following data: 5, 7, 5, 4, 6, 7, 8, 5, 4, 7, 8, 6, 20. The average value of all data is 7.1. In fact, most data (10) do not exceed 7. If 20 is removed, the average value of the remaining 12 numbers is 6.
It can be seen that the appearance of extreme values will interfere with the authenticity of the average.
characteristic
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① The arithmetic mean is a good oneConcentrated quantityIt has the advantages of sensitive response, rigorous determination, concise and easy solution, simple calculation, suitable for further calculation and less affected by sampling changes.
② The arithmetic mean is vulnerable to extreme data becauseaverageThe response is sensitive, and the final result will be affected by the large or small changes of each data.[1]
