Sample Variance & Standard Deviation | Introduction to Statistics and Data Analysis | Probability & Statistics for Engineers & Scientists | Walpole | Problem 1.8


Compute the sample variance and standard deviation for the water absorbency data of Exercise 1.2 on page 13.


Solution:

We know, based on our answer in Exercise 1.2, that the sample mean is \displaystyle \overline{x}=20.768.

To compute for the sample variance, we shall use the formula

\displaystyle s^2=\sum _{i=1}^n\:\frac{\left(x_i-\overline{x}\right)^2}{n-1}

The formula states that we need to get the sum of \displaystyle \left(x_i-\overline{x}\right)^2, so we can use a table to solve \displaystyle \left(x_i-\overline{x}\right)^2 for every sample.

Note: You can refer to the solution of Exercise 1.7 on how to use a table to solve for the variance. 

The variance is

s^2=\frac{1}{20-1}[\left(18.71-20.768\right)^2+\left(21.41-20.768\right)^2

\displaystyle +...+\left(21.12-20.768\right)^2]

\displaystyle s^2=2.5345


The standard deviation is just the square root of the variance. That is

\displaystyle s=\sqrt{s^2}=\sqrt{2.5345}=1.592