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Putting Statistics to Work
Discussion Paragraph 6A(write at least 5 sentences)
1 web
Salary Data
New York Marathon
Tax Statistics
Education Statistics
1 world
Averages in the News
Daily Averages
Distributions in the News
Copyright © 2011 Pearson Education, Inc.
Unit 6B
Measures of Variation
Copyright © 2011 Pearson Education, Inc.
Slide 6-3
6-B
Why Variation Matters
Consider the following waiting times for 11 customers
at 2 banks.
Big Bank (three lines):
4.1 5.2 5.6 6.2 6.7 7.2
7.7 7.7 8.5 9.3 11.0
Best Bank (one line):
6.6 6.7 6.7 6.9 7.1 7.2
7.3 7.4 7.7 7.8 7.8
Which bank is likely to have more unhappy customers?
→ Big Bank, due to more surprise long waits
Copyright © 2011 Pearson Education, Inc.
Slide 6-4
6-B
Range
The range of a data set is the difference between its
highest and lowest data values.
range = highest value (max) – lowest value (min)
Copyright © 2011 Pearson Education, Inc.
Slide 6-5
6-B
Misleading Range
CN (1)

Consider the following two sets of quiz scores for
nine students.

Quiz 1: 1 10 10 10 10 10 10 10 10
Quiz 2: 2 3 4 5 6 7 8 9 10


1. Which set has the greater range? Would you
also say that the scores in this set are more
varied?
Copyright © 2011 Pearson Education, Inc.
Slide 6-6
6-B
Quartiles

The lower quartile (or first quartile) divides the
lowest fourth of a data set from the upper threefourths. It is the median of the data values in the
lower half of a data set.

The middle quartile (or second quartile) is the
overall median.

The upper quartile (or third quartile) divides the
lower three-fourths of a data set from the upper
fourth. It is the median of the data values in the
upper half of a data set.
Copyright © 2011 Pearson Education, Inc.
Slide 6-7
6-B
The Five-Number Summary

The five-number summary for a data set
consists of the following five numbers:
low value

lower quartile
median
upper quartile
high value
A boxplot shows the five-number summary
visually, with a rectangular box enclosing the
lower and upper quartiles, a line marking the
median, and whiskers extending to the low and
high values.
Copyright © 2011 Pearson Education, Inc.
Slide 6-8
6-B
The Five-Number Summary
Five-number summary of the waiting times at each bank:
Big Bank
Best Bank
low value (min) = 4.1
lower quartile = 5.6
median = 7.2
upper quartile = 8.5
high value (max) = 11.0
low value (min) = 6.6
lower quartile = 6.7
median = 7.2
upper quartile = 7.7
high value (max) = 7.8
The corresponding boxplot:
Copyright © 2011 Pearson Education, Inc.
Slide 6-9
6-B
Race Times
CN (2a-b)


Consider the following two sets of twenty 100meter running times (in seconds):
Set 1: 9.92 9.97 9.99 10.01 10.06 10.07 10.08 10.10
10.13 10.13 10.14 10.15 10.17 10.17 10.18 10.21
10.24 10.26 10.31 10.38

Set 2: 9.89 9.90 9.98 10.05 10.35 10.41 10.54 10.76
10.93 10.98 11.05 11.21 11.30 11.46 11.55 11.76
11.81 11.85 11.87 12.00

Compare the variation in the two data sets with
a)five-number summaries and b)box plots.
Copyright © 2011 Pearson Education, Inc.
Slide 6-10
6-B
Standard Deviation
The standard deviation is the single number most
commonly used to describe variation.
sum of (deviation s from the mean) 2
standard deviation 
total number of data values  1
Copyright © 2011 Pearson Education, Inc.
Slide 6-11
6-B
Calculating the Standard Deviation
The standard deviation is calculated by completing
the following steps:
1. Compute the mean of the data set. Then find the
deviation from the mean for every data value.
deviation from the mean = data value – mean
2. Find the squares of all the deviations from the mean.
3. Add all the squares of the deviations from the mean.
4. Divide this sum by the total number of data values
minus 1.
5. The standard deviation is the square root of this
quotient.
Copyright © 2011 Pearson Education, Inc.
Slide 6-12
6-B
Standard Deviation
Let A = {2, 8, 9, 12, 19} with a mean of 10. Find the sample
standard deviation of the data set A.
x (data value)
2
8
9
12
19
x – mean
(deviation)
2 – 10 = –8
8 – 10 = –2
9 – 10 = –1
12 – 10 = 2
19 – 10 = 9
Total
(deviation)2
(-8)2 = 64
(-2)2 = 4
(-1)2 = 1
(2)2 = 4
(9)2 = 81
154
sum of (deviation s from the mean) 2
standard deviation 
total number of data values  1
154

 6.2
5 1
Copyright © 2011 Pearson Education, Inc.
Slide 6-13
6-B
Calculating Standard Deviation
CN (3a-b)

3. Calculate the standard deviations for the
waiting times at a)Big Bank and b)Best Bank.

BigB sum 38.46

BestB sum 1.98
Copyright © 2011 Pearson Education, Inc.
Slide 6-14
6-B
The 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 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
Copyright © 2011 Pearson Education, Inc.
Slide 6-15
6-B
Using the Range Rule of Thumb
CN (4a-b)

4. Use the range rule of thumb to estimate the
standard deviations for the waiting times at a)Big
Bank and b)Best Bank.

*hint, find range of times
BigB
BestB



Compare the estimates to the actual standard
deviation you found in the last problem
Copyright © 2011 Pearson Education, Inc.
Slide 6-16
6-B
Estimating a Range
CN (5a-b)

Studies of the gas mileage of a Prius under
varying driving conditions show that it gets a
mean of 45 miles per gallon with a standard
deviation of 4 miles per gallon.

5. Estimate the a)minimum and b)maximum gas
mileage that you can expect under ordinary
driving conditions.
Copyright © 2011 Pearson Education, Inc.
Slide 6-17
6-B
Quick Quiz
CN (6)

Answer the 10 multiple choice questions of the
quick quiz on p.388.
Copyright © 2011 Pearson Education, Inc.
Slide 6-18
6-B
6B Homework




Discussion Paragraph 6A
p.389: 1-12
1 web
 26. Web Data Sets
1 world
 27. Ranges in the News
 28. Summarizing a News Data Set
 29. Range Rule in the News
Class Notes 1-6
Copyright © 2011 Pearson Education, Inc.
Slide 6-19