The Standard Deviation

Measures of Variance

Learning Objectives

Calculate and understand the sample and population standard deviations.

The Standard Deviation (St. Dev.):

Is the average distance that the numbers are away from the mean.
I.e., the average of $(x_i-\bar{x})$ .
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:

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

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

Name	Homework	Term Test#1	Term Test#2	Term Test#3	Concept Quizzes
Sunita	80	80	80	82	83
Sanjay	48	80	80	98	99

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 $\bar{x}$ and $s$ .

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

$x_i$	$\bar{x}$	$(x_i -\bar{x})$	$(x_i-\bar{x})^2$
80	81	-1	1
80	81	-1	1
80	81	-1	1
82	81	1	1
83	81	2	4
		SUM	$\Sigma = 8$

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

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

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