Download Week 2 - Seminar

Chapter 2
Descriptive Statistics
1
Larson/Farber 4th ed.
Useful screencast/videos:
Video on creating a frequency distribution
by hand:
http://screencast.com/t/OGY3ZjJj
 Video on using Excel 2007 to create
frequency distributions:
http://screencast.com/t/tkMv2FMWhJe
 Video on using Excel 2007 to create a
histogram
http://screencast.com/t/L0u9UI2eI

Larson/Farber 4th ed.
2
Section 2.1
Frequency Distributions
and Their Graphs
3
Larson/Farber 4th ed.
Frequency Distribution Terminology
Frequency
Distribution
 A table that shows
classes or intervals
of data with a count of
the number of entries
in each class.
 The frequency, f, of a
class is the number of
data entries in the
class.
Class
Frequency, f
1–5
5
6 – 10
8
11 – 15
6
16 – 20
8
21 – 25
5
26 – 30
4
4
Larson/Farber 4th ed.
Determining the Relative Frequency
Relative Frequency of a class
 Portion or percentage of the data that falls in a
particular class.
class frequency
f

• relative frequency 
Sample size
n
Class
Frequency, f
7 – 18
6
19 – 30
10
31 – 42
13
Relative Frequency
6
 0.12
50
10
 0.20
50
13
 0.26
50
5
Larson/Farber 4th ed.
Example: Constructing a Frequency
Distribution
The following sample data set lists the number of minutes
50 Internet subscribers spent on the Internet during their
most recent session. Construct a frequency distribution
that has seven classes.
50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86
41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20
18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44
Video on computing frequency distribution using this
data: http://screencast.com/t/OGY3ZjJj
6
Larson/Farber 4th ed.
Expanded Frequency Distribution
Class
Frequency, f
Midpoint
Relative
frequency
7 – 18
6
12.5
0.12
6
19 – 30
10
24.5
0.20
16
31 – 42
13
36.5
0.26
29
43 – 54
8
48.5
0.16
37
55 – 66
5
60.5
0.10
42
67 – 78
6
72.5
0.12
48
79 – 90
2
84.5
0.04
f
 1
n
50
Σf = 50
Cumulative
frequency
7
Larson/Farber 4th ed.
Graphs of Frequency Distributions
frequency
Frequency Histogram
 A bar graph that represents the frequency
distribution.
 The horizontal scale is quantitative and
measures the data values.
 The vertical scale measures the
frequencies of the classes.
 Consecutive bars must touch.
data values
Larson/Farber 4th ed.
8
Solution: Frequency Histogram
(using Midpoints)
Larson/Farber 4th ed.
9
Graphs of Frequency Distributions
relative
frequency
Relative Frequency Histogram
 Has the same shape and the same horizontal
scale as the corresponding frequency histogram.
 The vertical scale measures the relative
frequencies, not frequencies.
data values
Larson/Farber 4th ed.
10
Solution: Relative Frequency
Histogram
6.5
18.5
30.5
42.5
54.5
66.5
78.5
90.5
From this graph you can see that 20% of Internet subscribers
spent between 18.5 minutes and 30.5 minutes online.
Larson/Farber 4th ed.
11
Section 2.2
More Graphs and Displays
Larson/Farber 4th ed.
12
Graphing Quantitative Data Sets
Stem-and-leaf plot
 Each number is separated into a stem and a
leaf.
26
 Similar to a histogram.
 Still contains original data values.
Data: 21, 25, 25, 26, 27, 28,
30, 36, 36, 45
2
3
1 5 5 6 7 8
0 6 6
4
5
Larson/Farber 4th ed.
13
Graphing Qualitative Data Sets
Pie Chart
 A circle is divided into sectors that represent
categories.
 The area of each sector is proportional to the
frequency of each category.
Larson/Farber 4th ed.
14
Section 2.3
Measures of Central Tendency
Larson/Farber 4th ed.
15
Measures of Central Tendency
Measure of central tendency
 A value that represents a typical, or
central, entry of a data set.
 Most common measures of central
tendency:
◦ Mean
◦ Median
◦ Mode
Larson/Farber 4th ed.
16
Measure of Central Tendency: Mean
Mean (average)
 The sum of all the data entries divided by
the number of entries.
 Sigma notation: Σx = add all of the
data entries (x)in
the data set.
x

 Population mean:
N
x
x
 Sample mean:
n
Larson/Farber 4th ed.
17
Example: Finding a Sample Mean
The prices (in dollars) for a sample of roundtrip
flights from Chicago, Illinois to Cancun, Mexico
are listed. What is the mean price of the flights?
872 432 397 427 388 782 397
Larson/Farber 4th ed.
18
Solution: Finding a Sample Mean
872 432 397 427 388 782 397
• The sum of the flight prices is
Σx = 872 + 432 + 397 + 427 + 388 + 782 + 397 = 3695
• To find the mean price, divide the sum of the prices
by the number of prices in the sample
x 3695
x

 527.9
n
7
The mean price of the flights is about $527.90.
Larson/Farber 4th ed.
19
Measure of Central Tendency:
Median
Median
 The value that lies in the middle of the data
when the data set is ordered.
 Measures the center of an ordered data set by
dividing it into two equal parts.
 If the data set has an
◦ odd number of entries: median is the middle
data entry.
◦ even number of entries: median is the mean of
the two middle data entries.
Larson/Farber 4th ed.
20
Example: Finding the Median
The prices (in dollars) for a sample of roundtrip
flights from Chicago, Illinois to Cancun, Mexico
are listed. Find the median of the flight prices.
872 432 397 427 388 782 397
Larson/Farber 4th ed.
21
Solution: Finding the Median
872 432 397 427 388 782 397
• First order the data.
388 397 397 427 432 782 872
• There are seven entries (an odd number), the median
is the middle, or fourth, data entry.
The median price of the flights is $427.
Larson/Farber 4th ed.
22
Example: Finding the Median
The flight priced at $432 is no longer available.
What is the median price of the remaining
flights?
872 397 427 388 782 397
Larson/Farber 4th ed.
23
Solution: Finding the Median
872 397 427 388 782 397
• First order the data.
388 397 397 427 782 872
• There are six entries (an even number), the median is
the mean of the two middle entries.
397  427
Median 
 412
2
The median price of the flights is $412.
Larson/Farber 4th ed.
24
Measure of Central Tendency: Mode
Mode
 The data entry that occurs with the greatest
frequency.
 If no entry is repeated the data set has no
mode.
 If two entries occur with the same greatest
frequency, each entry is a mode (bimodal).
Larson/Farber 4th ed.
25
Example: Finding the Mode
The prices (in dollars) for a sample of roundtrip
flights from Chicago, Illinois to Cancun, Mexico
are listed. Find the mode of the flight prices.
872 432 397 427 388 782 397
Larson/Farber 4th ed.
26
Solution: Finding the Mode
872 432 397 427 388 782 397
• Ordering the data helps to find the mode.
388 397 397 427 432 782 872
• The entry of 397 occurs twice, whereas the other
data entries occur only once.
The mode of the flight prices is $397.
Larson/Farber 4th ed.
27
Example: Finding the Mode
At a political debate a sample of audience
members was asked to name the political party
to which they belong. Their responses are shown
in the table. What is the mode of the responses?
Political Party
Democrat
Frequency, f
34
Republican
Other
56
21
Did not respond
9
Larson/Farber 4th ed.
28
Solution: Finding the Mode
Political Party
Democrat
Frequency, f
34
Republican
Other
Did not respond
56
21
9
The mode is Republican (the response occurring with
the greatest frequency). In this sample there were
more Republicans than people of any other single
affiliation.
Larson/Farber 4th ed.
29
Section 2.4
Measures of Variation
Larson/Farber 4th ed.
30
Deviation,Variance, and Standard
Deviation
Deviation
 The difference between the data entry, x,
and the mean of the data set.
 Population data set:
◦ Deviation of x = x – μ

Sample data set:
◦ Deviation of x = x – x
Larson/Farber 4th ed.
31
Example: Finding the Deviation
A corporation hired 10 graduates. The starting
salaries for each graduate are shown. Find the
deviation of the starting salaries.
Starting salaries (1000s of dollars)
41 38 39 45 47 41 44 41 37 42
Solution:
• First determine the mean starting salary.
x 415


 41.5
N
10
Larson/Farber 4th ed.
32
Solution: Finding the Deviation
•
Determine the
deviation for
each data entry.
Salary ($1000s), x Deviation: x – μ
41
41 – 41.5 = –0.5
38
38 – 41.5 = –3.5
39
39 – 41.5 = –2.5
45
45 – 41.5 = 3.5
47
47 – 41.5 = 5.5
41
41 – 41.5 = –0.5
44
44 – 41.5 = 2.5
41
41 – 41.5 = –0.5
37
37 – 41.5 = –4.5
42
42 – 41.5 = 0.5
Σx = 415
Σ(x – μ) = 0
Larson/Farber 4th ed.
33
Deviation,Variance, and Standard
Deviation
Population Variance
( x   )
 
N
2
2
Sum of squares, SSx
Population Standard Deviation
2

(
x


)
  2 

N
Larson/Farber 4th ed.
34
Deviation,Variance, and Standard
Deviation
Sample Variance
( x  x )
s 
n 1
2
2
Sample Standard Deviation
2

(
x

x
)
s  s2 

n 1
Larson/Farber 4th ed.
35
Example: Using Technology to Find
the Standard Deviation
Sample office rental rates (in
dollars per square foot per
year) for Miami’s central
business district are shown in
the table. Use a calculator or
a computer to find the mean
rental rate and the sample
standard deviation. (Adapted from:
Cushman & Wakefield Inc.)
Office Rental Rates
35.00
33.50
37.00
23.75
26.50
31.25
36.50
40.00
32.00
39.25
37.50
34.75
37.75
37.25
36.75
27.00
35.75
26.00
37.00
29.00
40.50
24.50
33.00
38.00
Larson/Farber 4th ed.
36
Solution: Using Technology to Find
the Standard Deviation
Sample Mean
Sample Standard
Deviation
Larson/Farber 4th ed.
37
Interpreting Standard Deviation
Standard deviation is a measure of the
typical amount an entry deviates from the
mean.
 The more the entries are spread out, the
greater the standard deviation.

Larson/Farber 4th ed.
38
Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
For data with a (symmetric) bell-shaped
distribution, the standard deviation has the
following
characteristics:
• About 68%
of the data lie within one standard
deviation of the mean.
• About 95% of the data lie within two standard
deviations of the mean.
• About 99.7% of the data lie within three standard
deviations of the mean.
Larson/Farber 4th ed.
39
Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations
95% within 2 standard deviations
68% within 1
standard deviation
34%
2.35%
x  3s
34%
13.5%
x  2s
2.35%
13.5%
x s
x
xs
x  2s
Larson/Farber 4th ed.
x  3s
40
Example: Using the Empirical
Rule
In a survey conducted by the National Center
for Health Statistics, the sample mean height of
women in the United States (ages 20-29) was 64
inches, with a sample standard deviation of 2.71
inches. Estimate the percent of the women
whose heights are between 64 inches and 69.42
inches.
Larson/Farber 4th ed.
41
•
Solution: Using the Empirical
Rule
Because the distribution is bell-shaped, you can use
the Empirical Rule.
34%
13.5%
55.87
x  3s
58.58
x  2s
61.29
x s
64
x
66.71
xs
69.42
x  2s
72.13
x  3s
34% + 13.5% = 47.5% of women are between 64 and
69.42 inches tall.
Larson/Farber 4th ed.
42
Larson/Farber 4th ed.
43
Larson/Farber 4th ed.
44
Important Formulas
Range = Maximum value – Minimum value
Population Variance
Population Standard Deviation
Sample Variance
Sample Standard Deviation
Using the Empirical Rule
1. The mean value of homes on a street is $125 thousand with a standard
deviation of $5 thousand.The data set has a bell shaped distribution.
Estimate the percent of homes between $120 and $135 thousand.
105
110
115
120
125
130
135
140
145
$120 thousand is 1 standard deviation below
the mean and $135 thousand is 2 standard
deviations above the mean.
68% + 13.5% = 81.5%
2. An instructor recorded the average number
of absences for his students in one semester.
For a random sample the data are:
2
4
2
0
40
2
4
3
6
Calculate the mean, the median, and the mode, using the
appropriate notation. [Hint: is this a sample or a population?]
3. Find the class width:
A. 3
Class
1– 5
Frequency, f
21
6 – 10
11 – 15
16 – 20
16
28
13
B. 4
C. 5
D. 19
Copyright © 2007 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Slid
e 248
4. The mean annual automobile insurance
premium is $950, with a standard deviation of
$175. The data set has a bell-shaped distribution.
Estimate the percent of premiums that are
between $600 and $1300.
A. 68%
B. 75%
C. 95%
D. 99.7%
Copyright © 2007 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley
Slid
e 249
1.
81.5% have a value between $120 and $135 thousand.
2.
xbar = 63, median = 3, mode = 2. This is a sample, so these are all
sample statistics.
3.
(C) 5
4.
(C) 95%
Larson/Farber 4th ed.
50