Chapter 2

Descriptive Statistics Larson/Farber 4th ed.

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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.

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Section 2.1

Frequency Distributions and Their Graphs Larson/Farber 4th ed.

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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 1 – 5 6 – 10 11 – 15 16 – 20 21 – 25 26 – 30 Frequency, f 5 8 6 8 5 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. • relative frequency  class frequency Sample size 

f n

Class 7 – 18 19 – 30 31 – 42 Frequency, f 6 10 13 Relative Frequency 6 50  0.12

10 50  0.20

13 50  0.26

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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 7 – 18 19 – 30 31 – 42 43 – 54 55 – 66 67 – 78 79 – 90 Frequency, f 6 10 13 8 5 6 2 Σ

f

= 50 Midpoint 12.5

24.5

36.5

48.5

60.5

72.5

84.5

Relative frequency 0.12

0.20

0.26

0.16

0.10

0.12

0.04



f n

 1 Cumulative frequency 6 16 29 37 42 48 50 Larson/Farber 4th ed.

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Graphs of Frequency Distributions

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.

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Solution: Frequency Histogram (using Midpoints) Larson/Farber 4th ed.

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Graphs of Frequency Distributions

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.

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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.

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Section 2.2

More Graphs and Displays Larson/Farber 4th ed.

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Graphing Quantitative Data Sets

Stem-and-leaf plot

 Each number is separated into a stem and a leaf.

 Similar to a histogram.

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 Still contains original data values.

Data: 21, 25, 25,

26

, 27, 28, 2 1 5 5 6 7 8 30, 36, 36, 45 3 0 6 6 4 5 Larson/Farber 4th ed.

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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.

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Section 2.3

Measures of Central Tendency Larson/Farber 4th ed.

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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.

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Measure of Central Tendency: Mean

Mean

   (average) The sum of all the data entries divided by the number of entries.

Sigma notation: Σ x = add all of the data entries (x) in the data set.

 

N x



x

 Larson/Farber 4th ed.

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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.

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Solution: Finding a Sample Mean

872 432 397 427 388 782 397 • The sum of the flight prices is Σ

x

= 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.

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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.

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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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 Frequency, f

Democrat Republican Other Did not respond 34 56 21 9 Larson/Farber 4th ed.

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Solution: Finding the Mode

Political Party

Democrat Republican Other Did not respond

Frequency, f

34 56 21 9 The mode is Republican affiliation.

(the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single Larson/Farber 4th ed.

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Section 2.4

Measures of Variation Larson/Farber 4th ed.

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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.

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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 N

 415 10  41.5

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Solution: Finding the Deviation

• Determine the deviation for each data entry.

Salary ($1000s), x

41 38 39 45 47 41 44 41 37 42 Σ

x

= 415 Larson/Farber 4th ed.

Deviation: x – μ

41 – 41.5 = –0.5

38 – 41.5 = –3.5

39 – 41.5 = –2.5

45 – 41.5 = 3.5

47 – 41.5 = 5.5

41 – 41.5 = –0.5

44 – 41.5 = 2.5

41 – 41.5 = –0.5

37 – 41.5 = –4.5

42 – 41.5 = 0.5

Σ(

x

– μ) = 0 33

Deviation, Variance, and Standard Deviation

Population Variance

 2  

x N

 ) 2 Sum of squares, SS

x



Population Standard Deviation

   2 

x N

 ) 2 Larson/Farber 4th ed.

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Deviation, Variance, and Standard Deviation

Sample Variance

s

2  

x x

) 2

n

 1

Sample Standard Deviation

s

 

s

2 

x x

) 2

n

 1 Larson/Farber 4th ed.

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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

36.50

39.25

37.75

26.50

40.00

37.50

37.25

31.25

32.00

34.75

36.75

27.00

37.00

24.50

35.75

29.00

33.00

26.00

40.50

38.00

Larson/Farber 4th ed.

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Solution: Using Technology to Find the Standard Deviation Sample Mean Sample Standard Deviation Larson/Farber 4th ed.

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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.

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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: 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.

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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

x

 3

s

2.35%

x

 2

s

13.5%

x



s

34% 34%

x

2.35%

x



s

13.5%

x

 2

s x

 3

s

Larson/Farber 4th ed.

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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.

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•

Solution: Using the Empirical Rule

Because the distribution is bell-shaped, you can use the Empirical Rule. 34% 13.5% 55.87

x

 3

s

58.58

x

 2

s

61.29

x



s

64

x

66.71

x



s

69.42

x

 2

s

72.13

x

 3

s

34% + 13.5% = 47.5% of women are between 64 and 69.42 inches tall.

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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 $120 thousand is 1 standard deviation below the mean and $135 thousand is 2 standard deviations above the mean.

130 135 140 145 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: Class

1 – 5 6 – 10 11 – 15 16 – 20

Frequency, f

21 16 28 13

A. 3 B. 4 C. 5 D. 19

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slid e 2 48

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 2 49

1.

2.

3.

4.

81.5% have a value between $120 and $135 thousand. xbar = 63, median = 3, mode = 2. This is a sample, so these are all sample statistics. (C) 5 (C) 95% Larson/Farber 4th ed.

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