Name _________________________ Period ________ Date ____________

*Lesson 2 - Exploring Standard Deviation : *

Calculating It and Understanding It

### The standard deviation reveals how a set of numbers vary in relation to their mean. A set of numbers can
- cluster about the mean,
- be widely separated from the mean, or
- evenly or unevenly arrange around the mean.

Into this wildernesss of variation comes the standard deviation.

In a single statistic, the standard deviation tells the average amount a set of numbers differ from their mean. In other words, it tells us if the mean fairly represents the numbers in its set.
The range showed variation, but only between the highest and lowest numbers. The standard deviation shows the variation between all the numbers.

*1. The steps below will show how this statistic is calculated. The set used for this illustration is*
{ 2, 3, 5, 10 }

### Step 1

** Find the mean of the set.
Mean = ______

### Step 2

** Subtract each number from the mean.
__5__ - __2__ = __3__

_____ - _____ = _____

_____ - _____ = _____

_____ - _____ = _____

### Step 3

** Square each of those differences.
__3__ X __3__ = __9__

_____ X _____ = _____

_____ X _____ = _____

_____ X _____ = _____

### Step 4

** Find the average of those squared differences.
_____ + _____ + _____ + _____ = _____

_____ / _____ = _____

### Step 5

** Take the square root of that average ____________

*This number is the standard deviation!*

2. If the standard deviation is the average of the differences from the mean, why was it necessary to square the differences in Step 3 and then undo that work by taking the square root in Step 5? Why not just find the average of the differences in Step 2?
** To answer that, take the differences in Step 2 and add them together. What is the answer ? ______

The problem will always happen and thus prevent averaging at this point. This is why it was necessary to square and square root in the calculation.

You may wonder why we did not take the absolute value of the differences. That way of solving, called the mean deviation, causes problems for important methods of statistical inference, so is not much used.

3. Not only does the standard deviation tell how much a set of numbers vary from the mean, it also reveals how much they vary from each other. The larger the standard deviation, the more widely separated the numbers are from each other. The smaller it is, the closer the numbers.

*Look at these sets from Lesson 1: *

A : { 3, 3, 4, 4, 4, 5, 5 }

B : { 3, 3, 3, 4, 5, 5, 5 }

C : { 1, 2, 3, 4, 5, 6, 7 }

D : { 1, 1, 3, 4, 5, 7, 7 }

E : { 1. 1. 3. 3. 6. 7. 7 }

F : { 1, 1, 1, 4, 7, 7, 7 }

** Which set would you expect to have the greatest standard deviation? _____

** Which set should have the least standard deviation? _____

** Which sets might have a standard deviation midway between these two? ______

4. ** Calculate the standard deviations for these sets of numbers to see if the answers above were correct.

A : _____

B : _____

C : _____

D : _____

E : _____

F : _____

####
URL http://cs.rice.edu/~bchristo/lessons/standev/printcalc.html

##### Copyright January 1997 Barbara Christopher

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