Question

Consider the following data and corresponding weights.

xi3.222.55Weight(wi)6328

a. Compute the weighted mean.
b. Compute the sample mean of the four data values without weighting. Note the difference
in the results provided by the two computations.

Answer 1

a) Weighted Mean = 3.014

b) Sample Mean = 24.1925

Explanation:

We are given with the following data and corresponding weights;

`x_i` Weight (
`w_i`)
`x_i*w_i`

3.22 63 202.86

2.55 28 71.4

`\sum w_i` = 91
`\sum w_i*x_i` = 274.26

a) Weighted Mean is given by the formula;

Weighted Mean =
`(\sum x_i*w_i)/(\sum w_i)`

=
`(274.26)/(91)` = 3.014 .

b) Now, the four data values without weighting are;

`X_i` = 3.22, 2.55, 63, 28

Sample Mean =
`(\sum X_i)/(n)` =
`(3.22 + 2.55 + 63+28)/(4)` = 24.1925 .

We see that without weighting sample mean is higher than weighted mean because in sample mean data values there comes two higher extreme values because of which mean shifted towards them and is higher than weighted mean.