Download Unit 6B - Gordon State College

Unit 6B
Measures of Variation
VARIATION
Variation describes how widely data values are
spread out about the center of a distribution.
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 7.7
7.7
Bank of Providence
4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
(multiple waiting lines)
All the measures of center are equal for both banks.
Mean = 7.15 min
Median = 7.20 min
Mode = 7.7 min
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 threefourths.
• 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
median
upper
(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
4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
(multiple waiting lines)
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 BOXPLOT
ON 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:
standard deviation
sum of (deviation s from the mean)

total number of data values  1
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
4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
(multiple waiting lines)
RANGE RULE OF THUMB
The standard deviation is approximately related to the
range of a data set by the range rule of thumb:
range
standard deviation 
4
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:
low value  mean  2  (standard deviation)
high value  mean  2  (standard deviation)
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
4.2 5.4 5.8 6.2 6.7 7.7 7.7 8.5 9.3 10.0
(multiple waiting lines)
EXAMPLE
Dr. 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.