1. Unit 6B
Measures of Variation

2. VARIATION
Variation describes how widely data values are spread out about the center of a distribution.

3. WAITING TIMES AT DIFFERENT BANKS
The table below list the waiting times (in minutes) for two different banks.
Jefferson Valley Bank (single waiting line)
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
Bank of Providence (multiple waiting lines)
4.2
5.4
5.8
6.2
8.5
9.3
10.0
All the measures of center are equal for both banks.
Mean = 7.15 min
Median = 7.20 min
Mode = 7.7 min

4. range = highest value (max) − lowest value (min)
The range of a set of data is the difference between the highest and the lowest values:
range = highest value (max) − lowest value (min)
EXAMPLE:
Jefferson Valley Bank range = 7.7 − 6.5 = 1.2 min
Bank of Providence range = 10.0 − 4.2 = 5.8 min

5. QUARTILES
The lower quartile (or first quartile) divides the lowest fourth of a data set from the upper three-fourths.
The middle quartile (or second quartile) is the median.
The upper quartile (or third quartile) divides the lowest three-fourths of the data set from the upper fourth.
NOTE: There is no universal agreement on how to calculate quartiles. We will use the results of the Texas Instruments™ calculators.

6. THE FIVE NUMBER SUMMARY
The five-number summary for a data set consists of the following five numbers.
low
(min) value
lower
(first) quartile
median
upper (third) quartile
high (max) value

7. FINDING THE FIVE-NUMBER SUMMARY ON THE TI-81/84
Press STAT; select 1:Edit….
Enter your data values in L1. (You may enter the values in any of the lists.)
Press 2ND, MODE (for QUIT).
Press STAT; arrow over to CALC. Select 1:1-Var Stats.
Enter L1 by pressing 2ND, 1.
Press ENTER.
Scroll down to see the five-number summary. The five numbers are labeled: minX, Q1, Med, Q3, maxX.

8. EXAMPLE
Find the five-number summary for the Jefferson Valley Bank and the Bank of Providence.
Jefferson Valley Bank (single waiting line)
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
Bank of Providence (multiple waiting lines)
4.2
5.4
5.8
6.2
8.5
9.3
10.0

9. BOXPLOT
A boxplot show the five-number summary visually, with a rectangular box enclosing the lower (first) and upper (third) quartiles, a line marking the median, and whiskers extending to the low and high values.
We always write the values for the quartiles, low value, and high value on the boxplot.

10. EXAMPLE
Draw a boxplot (on the same axis) for Jefferson Valley Bank and the Bank of Providence.

11. DRAWING A BOXPLOT ON THE TI-81/84
Press STAT; select 1:Edit….
Enter your data values in L1. (Note: You could enter them in a different list.)
Press 2ND, Y= (for STATPLOT). Select 1:Plot1.
Turn the plot ON. For Type, select the boxplot (middle one on second row).
For Xlist, put L1 by pressing 2ND, 1.
For Freq, enter the number 1.
Press ZOOM. Select 9:ZoomStat.

12. THE STANDARD DEVIATION
The standard deviation is a measure of the average of all the deviations of data values from the mean of a data set.

13. CALCULATING THE STANDARD DEVIATION
Step 1: Compute the mean of the data set. Then find the deviation from the mean for every data value using the formula
deviation from mean = data value − mean
Step 2: Find the squares of all the deviations from the man.
Step 3: Add all the squares of the deviations from the mean.
Step 4: Divide this sum by the total number of
data values minus 1.
Step 5: The standard deviation is the square root of the number from Step 4.

14. STANDARD DEVIATION FORMULA
All of the steps from the previous slide can be summarized by the formula:

15. EXAMPLE
Find the standard deviation of the following.

16. FINDING THE STANDARD DEVIATION ON THE TI-81/84
Press STAT; select 1:Edit….
Enter your data values in L1. (You may enter the values in any of the lists.)
Press 2ND, MODE (for QUIT).
Press STAT; arrow over to CALC. Select 1:1-Var Stats.
Enter L1 by pressing 2ND, 1.
Press ENTER.
The standard deviation is given by Sx.

17. EXAMPLE
Find the standard deviation for the Jefferson Valley Bank and the standard deviation for the Bank of Providence.
Jefferson Valley Bank (single waiting line)
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
Bank of Providence (multiple waiting lines)
4.2
5.4
5.8
6.2
8.5
9.3
10.0

18. RANGE RULE OF THUMB
The standard deviation is approximately related to the range of a data set by the range rule of thumb:
If we know the range of a data set (range = high − low), we can use this rule to estimate the standard deviation.
Alternately, if we know the standard deviation for a data set, we estimate the low and high values as follows:

19. EXAMPLE
Use the Range Rule of Thumb to estimate the standard deviations for the Jefferson Valley Bank and the Bank of Providence.
Jefferson Valley Bank (single waiting line)
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
Bank of Providence (multiple waiting lines)
4.2
5.4
5.8
6.2
8.5
9.3
10.0

20. EXAMPLE
Dr. Fuller kept track of the gas mileage of his Honda Civic during the Fall Semester of The mean gas mileage was 40.5 miles per gallon and the standard deviation was 1.3 miles per gallon. Estimate the minimum and maximum gas mileage that Dr. Fuller can expect under normal driving conditions.