The Standard Deviation (St. Dev.):

• Is the average distance that the numbers are away from the mean.
• I.e., the average of [latex](x_i-\bar{x})[/latex].
• It takes into account ALL the numbers when calculating the ‘spread’ of the data.
• Is a good measure of dispersion of data.

The SAMPLE standard deviation is defined as:

\[s = \sqrt{\frac{\Sigma(x_i-\bar{x})^2}{n-1}}\]

The POPULATION standard deviation (indicated by σ, the LOWER case SIGMA) is given by:

\[\sigma= \sqrt{\frac{\Sigma(x_i-\mu)^2}{N}}\]

Example 6.1.1 – Sunita’s Grades

Let us go back to look at Sunita’s and Sanjay’s grades from a couple of sections back:

• Sunita’s range is small because ALL the numbers are close to the mean of 81.
• We will regard this data as a sample, and therefore we will talk about [latex]\bar{x}[/latex]and [latex]s[/latex].

Let us work Sunita’s standard deviation of her grades. It is easiest if we put it in the following table:

This gives the following standard deviation for Sunita’s grades:

\[s = \sqrt{\frac{\Sigma (x_i-\bar{x})^2}{n-1}} = \sqrt{\frac{8}{5-1}}=\sqrt{2}=1.41\]

Example 6.1.2 – Sanjay’s Vs. Sunita’s Grades (Excel)

Now, let us go back to look at Sanjay’s grades:

• For Sanjay, the range= [latex]99 - 48 = 51[/latex].
• Sanjay’s range is large because SOME of the numbers are not so very close to the mean of 81.

Let us calculate Sanjay’s standard deviation in Excel and compare it to Sunita’s standard deviation:

• We see that the standard deviation of Sanjay’s grades is much higher.
• Ie: Sanjay’s grades have a larger deviation, on average, from the average grade of 81.
• Click here to download the spreadsheet shown above

Key Takeaways

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