Exploring Standard Deviation
Answer Keys



Answer Key for * Why We Need It *

1.

A : { 3, 3, 4, 4, 4, 5, 5 } Range = __2__ Mean = __4__

B : { 3, 3, 3, 4, 5, 5, 5 } Range = __2__ Mean = __4__

C : { 1, 2, 3, 4, 5, 6, 7 } Range = __6__ Mean = __4__

D : { 1, 1, 3, 4, 5, 7, 7 } Range = __6__ Mean = __4__

E : { 1, 1, 3, 3, 6, 7, 7 } Range = __6__ Mean = __4__

F : { 1, 1, 1, 4, 7, 7, 7 } Range = __6__ Mean = __4__

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

The sum of the numbers was always 28 and the divisor was always 7.

** Mathematically, why was the range the same?

In sets A and B the maximum and minimum were 5 and 3, so the range was 2 for both sets. Similarly, the maximum and minimum of sets C through F were 7 and 1, so the range was always 6.















2.

                                     X
                              X      X      X
                              X      X      X
A : ---- | ---- | ---- | ---- | ---- | ---- | --- | ---- 
         1      2      3      4      5      6     7

                              X             X
                              X             X
                              X      X      X
B : ---- | ---- | ---- | ---- | ---- | ---- | --- | ---- 
         1      2      3      4      5      6     7



         X      X      X      X      X      X     X
C : ---- | ---- | ---- | ---- | ---- | ---- | --- | ---- 
         1      2      3      4      5      6     7


         X                                        X
         X             X      X      X            X
D : ---- | ---- | ---- | ---- | ---- | ---- | --- | ---- 
         1      2      3      4      5      6     7


         X             X                          X
         X             X                    X     X
E : ---- | ---- | ---- | ---- | ---- | ---- | --- | ---- 
         1      2      3      4      5      6     7

         X                                        X
         X                                        X
         X                    X                   X
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?

The X's seem to balance out on either side of the 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?

The beginning and ending X's are over the same numbers for A and B (4 and 6). They are also over the same for C, D, E, and F (1 and 7).





Answer Key for * Calculating It and Understanding It *

1.

Step 1



Step 2



Step 3



Step 4




Step 5




2.

** To answer that, take the differences in Step 2 and add them together. What is the answer ? __0__

3.

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

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

** Which sets might have a standard deviation midway between these two?
__C or D__



4.

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

A : __0.8__
B : __0.9__
C : __2.0__
D : __2.3__
E : __2.4__
F : __2.8__


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

Copyright January 1997 Barbara Christopher
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