Variance & Standard Deviation | Introduction to Statistics and Data Analysis | Probability & Statistics for Engineers & Scientists | Walpole | Problem 1.9


Exercise 1.3 on page 13 showed samples of tensile strength data, one for specimens that were exposed to an aging process and one in which there was no aging of the specimens. Calculate the sample variance as well as standard deviation in tensile strength for both samples. 


Solution:


For the samples with NO AGING:

We know, based on our answer in Exercise 1.3, that the sample mean for samples with no aging is \displaystyle \overline{x}_{no\:aging}=222.10.

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

\displaystyle \left(s^2\right)_{no\:aging}=\sum _{i=1}^{10}\:\frac{\left(x_i-\overline{x}\right)^2}{n-1}

\displaystyle =\frac{1}{10-1}\left[\left(227-222.10\right)^2+\left(222-222.10\right)^2+...+\left(221-222.10\right)^2\right]

\displaystyle =42.12

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

\displaystyle s_{no\:aging}=\sqrt{s^2}=\sqrt{42.12}=6.49


For the samples with AGING:

We know, based on our answer in Exercise 1.3, that the sample mean for samples with aging is \displaystyle \overline{x}_{no\:aging}=209.90.

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

\displaystyle \left(s^2\right)_{aging}=\sum _{i=1}^{10}\:\frac{\left(x_i-\overline{x}\right)^2}{n-1}

\displaystyle =\frac{1}{10-1}\left[\left(219-209.90\right)^2+\left(214-209.90\right)^2+...+\left(205-209.90\right)^2\right]

\displaystyle =23.62

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

\displaystyle s_{aging}=\sqrt{s^2}=\sqrt{23.62}=4.86


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