Name _________________________ Period ________ Date ____________

*Exploring Standard Deviation : *

Why We Need It

### Some statistics, such as the mean and the range, are easily calculated, but hide the true pattern of the numbers that produce them.

The mean describes a set of numbers with a single central number. In contrast, the range reveals how the numbers in a set vary from each other. But often these two statistics do not fairly represent the numbers in their sets.

*The exercises below will illustrate this problem, and show why a visual aide and a third statistic, the standard deviation, is needed*.

1. ** Find the range and mean of these 5 sets of numbers. Refer to the supplement The Mean, the Range, and Line Plots if you need help in calculating the range and mean.

A : { 3, 3, 4, 4, 4, 5, 5 } Range = _____ Mean = _____

B : { 3, 3, 3, 4, 5, 5, 5 } Range = _____ Mean = _____

C : { 1, 2, 3, 4, 5, 6, 7 } Range = _____ Mean = _____

D : { 1, 1, 3, 4, 5, 7, 7 } Range = _____ Mean = _____

E : { 1. 1. 3. 3. 6. 7. 7 } Range = _____ Mean = _____

F : { 1, 1, 1, 4, 7, 7, 7 } Range = _____ Mean = _____

Although the sets are quite different, their means and ranges are almost identical!

** Mathematically, why was the mean the same for all the sets?

** Mathematically, why was the range the same?

2. There is another tool that will help. A line plot visually shows how a set of numbers are distibuted.
** Make line plots for the sets A though F above. Refer to the supplement The Mean, the Range, and Line Plots if you need help in making a line plot.

A : ---- | ---- | ---- | ---- | ---- | ---- | --- | ----
1 2 3 4 5 6 7
B : ---- | ---- | ---- | ---- | ---- | ---- | --- | ----
1 2 3 4 5 6 7
C : ---- | ---- | ---- | ---- | ---- | ---- | --- | ----
1 2 3 4 5 6 7
D : ---- | ---- | ---- | ---- | ---- | ---- | --- | ----
1 2 3 4 5 6 7
E : ---- | ---- | ---- | ---- | ---- | ---- | --- | ----
1 2 3 4 5 6 7
F : ---- | ---- | ---- | ---- | ---- | ---- | --- | ----
1 2 3 4 5 6 7

** Study the line plots and the pattern of their distributions. How do the patterns help to explain why all the sets had a mean of 4?

** How do the patterns explain why sets A and B had a range of 2 and sets C, D, E, and F all had a range of 6?

Although the line plot is helpful in showing how the numbers in a data set relate to the mean, it is not as convenient as a single statistic would be. There is such a statistic -- the standard deviation. The next lesson will show you how to calculate that statistic.
####
URL http://cs.rice.edu/~bchristo/lessons/standev/printwhy.html

##### Copyright January 1997 Barbara Christopher

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