Unit 6B

Measures of Variation

VARIATION

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

WAITING TIMES ATDIFFERENT BANKS

Jefferson Valley Bank (single waiting line) 6.5 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7

Bank of Providence (multiple waiting lines) 4.2 5.4 5.8 6.2 6.7 7.7 7.7 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

The table below list the waiting times (in minutes) for two different banks.

RANGE

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

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.

THE FIVE NUMBER SUMMARY

The five-number summary for a data set consists of the following five numbers.

low

(min) value

lower

(first) quartile

medianupper (third)

quartile

high (max) value

FINDING THE FIVE-NUMBER SUMMARY ON THE TI-81/84

1. Press STAT; select 1:Edit….2. Enter your data values in L1. (You may enter the

values in any of the lists.)3. Press 2ND, MODE (for QUIT).4. Press STAT; arrow over to CALC. Select 1:1-Var

Stats.5. Enter L1 by pressing 2ND, 1.6. Press ENTER.7. Scroll down to see the five-number summary. The

five numbers are labeled: minX, Q1, Med, Q3, maxX.

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 7.7 7.7

Bank of Providence (multiple waiting lines) 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0

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.

EXAMPLE

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

DRAWING A BOXPLOTON THE TI-81/84

1. Press STAT; select 1:Edit….

2. Enter your data values in L1. (Note: You could enter them in a different list.)

3. Press 2ND, Y= (for STATPLOT). Select 1:Plot1.

4. Turn the plot ON. For Type, select the boxplot (middle one on second row).

5. For Xlist, put L1 by pressing 2ND, 1.

6. For Freq, enter the number 1.

7. Press ZOOM. Select 9:ZoomStat.

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.

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.

STANDARD DEVIATION FORMULA

All of the steps from the previous slide can be summarized by the formula:

1valuesdataofnumbertotal

mean) thefroms(deviationofsum

deviationstandard

2

EXAMPLE

Find the standard deviation of the following.

3 7 4 2

FINDING THE STANDARD DEVIATION ON THE TI-81/84

1. Press STAT; select 1:Edit….

2. Enter your data values in L1. (You may enter the values in any of the lists.)

3. Press 2ND, MODE (for QUIT).

4. Press STAT; arrow over to CALC. Select 1:1-Var Stats.

5. Enter L1 by pressing 2ND, 1.

6. Press ENTER.

7. The standard deviation is given by Sx.

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 7.7 7.7

Bank of Providence (multiple waiting lines) 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0

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:

RANGE RULE OF THUMB

4

rangedeviationstandard

deviation)(standard2meanvaluehigh

deviation)(standard2meanvaluelow

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 7.7 7.7

Bank of Providence (multiple waiting lines) 4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0

EXAMPLEDr. Fuller kept track of the gas mileage of his Honda Civic during the Fall Semester of 2005. 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.