*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 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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