© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter Three

Frequency Distributions and

Percentiles

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Raw Scores (Data)

• Dr. Peabody gave a statistics exam to students in his Introduction to Statistics course:

• 55, 57.5, 65, 67.5, 67.5, 72.5, 75, 75, 75, 77.5, 77.5, 77.5, 77.5, 82.5, 82.5, 82.5, 82.5, 82.5, 82.5, 85, 85, 85, 85, 85, 85, 85, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 87.5, 92.5, 92.5, 95, 95, 95, 97.5

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 3

New Statistical Notation

• The number of times a score occurs is the score’s frequency, which is symbolized by f

• A distribution is the general name for any organized set of data

• N is the total number of scores in the data

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 4

SimpleFrequency Distributions

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Chapter 3 - 5

Simple Frequency Distribution

• A simple frequency distribution shows the number of times each score occurs in a set of data

• The symbol for a score’s simple frequency is simply f

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 6

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Raw Scores

• Following is a data set of raw scores. We will use these raw scores to construct a simple frequency distribution table.

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 7

Simple FrequencyDistribution Table

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 8

Graphing a SimpleFrequency Distribution

• A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis

• The type of measurement scale (nominal, ordinal, interval, or ratio) determines whether we use– A bar graph– A histogram– A polygon

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 9

A Simple Frequency Bar Graph Is Used for Nominal and Ordinal Data

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 10

A Histogram Is Used for a Small Range of Different Interval or Ratio Scores

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 11

A Frequency Polygon Is Used for a Large Number of Different Scores

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 12

Types of Simple Frequency Distributions

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Chapter 3 - 13

The Normal Distribution

• A bell-shaped curve

• Called the normal curve or a normal distribution

• It is symmetrical

• The far left and right portions containing the low-frequency extreme scores are called the tails of the distribution

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 14

An Ideal Normal Distribution

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 15

Skewed Distributions

• A skewed distribution is not symmetrical as it has only one pronounced tail

• A distribution may be either negatively skewed or positively skewed

• Whether a skewed distribution is negative or positive corresponds to whether the distinct tail slopes toward or away from zero

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 16

Negatively Skewed Distribution

A negatively

skewed distribution

contains extreme

low scores that have a

low frequency, but

does not contain low

frequency extreme high

scores

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 17

Positively Skewed Distribution

A positively

skewed distribution

contains extreme

high scores that have a

low frequency, but

does not contain low

frequency extreme low

scores

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 18

Bimodal Distribution

A bimodal

distribution is a

symmetrical

distribution

containing two

distinct humps

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 19

Rectangular Distribution

A rectangular

distribution is a

symmetrical

distribution shaped

like a rectangle

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 20

Relative Frequency and the Normal Curve

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 21

N

frel. f

Relative Frequency

• Relative frequency is the proportion of time the score occurs

• The symbol for relative frequency is

rel. f

• The formula for a score’s relative frequency is

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 22

A RelativeFrequency Distribution

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 23

The proportion of the total area under the normal curve that is occupied by a group of scores corresponds to the combined relative frequency of those scores.

Finding Relative Frequency Using the Normal Curve

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 24

Computing CumulativeFrequency and Percentile

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Chapter 3 - 25

Cumulative Frequency

• Cumulative frequency is the frequency of all scores at or below a particular score

• The symbol for cumulative frequency is cf

• To compute a score’s cumulative frequency, we add the simple frequencies for all scores below the score with the frequency for the score

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 26

A Cumulative Frequency Distribution

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Chapter 3 - 27

Percentile

• A percentile is the percent of all scores in the data that are at or below the score

• If the scores cumulative frequency is known, the formula for finding the percentile is

Score’s Percentile = 100

N

cf

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 28

Finding Percentiles

The percentile for a given score corresponds

to the percent of the total area under the

curve that is to the left of the score.

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 29

Percentiles

Normal distribution showing the area under the curve to the left of selected scores.

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 30

Grouped FrequencyDistributions

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Chapter 3 - 31

Grouped Distributions

In a grouped distribution, scores are combined to form small groups, and then we report the total f, rel. f, or cf of each group

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 3 - 32

A Grouped Distribution

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Chapter 3 - 33

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example 1

• Using the following data set, find the relative frequency of the score 12

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Chapter 3 - 34

Example 1

• The simple frequency table for this set of data is as follows.

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Chapter 3 - 35

06.018

1.

N

ffrel

Example 1

• The frequency for the score of 12 is 1. N = 18.

• Therefore, the rel. f of 12 is

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Chapter 3 - 36

Example 2

• What is the cumulative frequency for the score of 14?

• Answer: The cumulative frequency of 14 is the frequency of all scores at or below 14 in this data set

• cf = 1 + 1 + 1 + 4 + 6 = 13

• The cf for the score of 14 in this data set is 13

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Chapter 3 - 37

72.018

13

N

cf

Example 3

• What is the percentile for the score of 14?

• Answer: The percentile of 14 is the percentage of all scores at or below 14 in this data set

0.056 + 0.056 + 0.056 + 0.222 + 0.333 = 0.72

• Another way to calculate this percentile is

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Key Terms

• bar graph• bimodal distribution• cumulative frequency• distribution• frequency• frequency polygon• grouped distribution• histogram

Chapter 3 - 38

• negatively skewed distribution

• normal curve• normal distribution• percentile• positively skewed

distribution• proportion of the

total area under the curve

© 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Key Terms (Cont’d)

Chapter 3 - 39

• rectangular distribution

• relative frequency• relative frequency

distribution• simple frequency

• simple frequency distribution

• tail• ungrouped

distribution