Variation
Measures of variation quantify how spread out the data is.
Variation is one of the core ideas in Statistics

Super-simple measure of variation
Range = highest value – lowest value
Not good for much, but gives us some idea how spread out the data is.

Standard Deviation
Standard Deviation is a measure of variation based on the mean
Because of this, it can be strongly influenced by outliers, just like the mean.
Standard Deviation is always positive or 0 (zero only if all the data are the same)
The standard deviation has the same units as the data

Calculating Standard Deviation
Definitional formula s


( x n


1 x )
2
Notice we are measuring variation of the data from the mean.
This formula is for the sample standard deviation , and is based on the sample mean and sample size

Calculating Standard Deviation
Shortcut Formula s
 n
  x n (
2 n



 
1 )
2
The advantage: No need to calculate the mean first
The disadvantage: Doesn’t make as much sense

Example: Definitional Form x

12 .
3
Data x
7
8
10
11
13
25 x
 x
7-12.3 = -5.3
8-12.3 = -4.3
10-12.3 = -2.3
11-12.3 = -1.3
13-12.3 = 0.7
 x
 x

2
28.09
18.49
5.29
1.69
.49
25-12.3 = 12.7
161.29
s


( x n


1 x )
2

215 .
34
5

6 .
6

Data x
7
8
10
11
13
25
Sums: 74
Example: Shortcut Form x
2
49
64
100
121
169
625
1128 s
 s
 n
  x n (
2 n



 
1 )
2
6

1128
  
2
6 ( 6

1 ) s

1292
30 s

6 .
6

Population Standard Deviation
If we have the population data, we can calculate the population standard deviation.
To distinguish it, we use a different symbol.
 

( x
 
)
2
N

Variance
Sample Variance: s
2
Population Variance:

2

Understanding Standard Deviation
Main idea:
Bigger value, data is more spread out.
Smaller value, data is closer together.

Rule of Thumb
To very roughly approximate s , s
 range
4
Rough interpretation:
“Most” data will be within two standard deviations of the mean. In other words,
Approximate highest value
 x

2 s
Approximate lowest value
 x

2 s

Empirical Rule
For data sets with a bell-shaped distribution ,

Example
For a particular fast-food store, the time people have to wait at the drive-through has a bell-shaped distribution with x

3 .
5 min s

0 .
7 min
Then about 68% of people wait between x
 s

2 .
8 min and x
 s

4 .
2 min
About 95% of people wait between x

2 s

2 .
1 min and x

2 s

4 .
9 min
Almost everyone (99.7%) of people wait between x

3 s

1 .
4 min and x

3 s

5 .
6 min

Homework
2.5: 3, 9, 21, 23, 25, 33