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

2. 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.

3. 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

4. Calculating Standard Deviation
Definitional formula
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

5. Calculating Standard Deviation
Shortcut Formula
The advantage: No need to calculate the mean first
The disadvantage: Doesn’t make as much sense

6. Example: Definitional Form
Data x 7
= -5.3
28.09 8
= -4.3
18.49 10
= -2.3
5.29 11
= -1.3
1.69 13
= 0.7
.49 25
= 12.7
161.29

7. Example: Shortcut Form
Data x 7 49 8 64 10
100 11
121 13
169 25
625 74
1128
Sums:

8. Population Standard Deviation
If we have the population data, we can calculate the population standard deviation. To distinguish it, we use a different symbol.

9. Variance
Sample Variance:
Population Variance:

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

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

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

13. Example
For a particular fast-food store, the time people have to wait at the drive-through has a bell-shaped distribution with
Then about 68% of people wait between
About 95% of people wait between
Almost everyone (99.7%) of people wait between

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