Chapter Three
Frequency
Distributions and
Percentiles
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 sample size indicating the
number of scores
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Chapter 3 - 2
Simple
Frequency Distributions
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Chapter 3 - 3
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
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Chapter 3 - 4
Raw Scores
• Following is a data set of raw scores.
We will use these raw scores to
construct a simple frequency distribution
table.
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
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Chapter 3 - 5
Simple Frequency
Distribution Table
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Chapter 3 - 6
Graphing a Simple
Frequency 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
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Chapter 3 - 7
A Simple Frequency Bar Graph Is Used for
Nominal and Ordinal Data.
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Chapter 3 - 8
A Histogram Is Used for a Small Range of
Different Interval or Ratio Scores.
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Chapter 3 - 9
A Frequency Polygon Is Used for a
Large Range of Different Scores
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Chapter 3 - 10
Types of Simple
Frequency Distributions
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Chapter 3 - 11
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
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Chapter 3 - 12
An Ideal Normal Distribution
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Chapter 3 - 13
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
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Chapter 3 - 14
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
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Chapter 3 - 15
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
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Chapter 3 - 16
Bimodal Distribution
A bimodal
distribution is a
symmetrical
distribution
containing two
distinct humps
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Chapter 3 - 17
Rectangular Distribution
A rectangular
distribution is a
symmetrical
distribution shaped
like a rectangle
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Chapter 3 - 18
Relative Frequency and
the Normal Curve
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Chapter 3 - 19
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
f
rel. f =
N
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Chapter 3 - 20
A Relative
Frequency Distribution
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Chapter 3 - 21
Finding Relative Frequency Using the
Normal Curve
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.
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Chapter 3 - 22
Computing Cumulative
Frequency and Percentile
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Chapter 3 - 23
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
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Chapter 3 - 24
A Cumulative
Frequency Distribution
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Chapter 3 - 25
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
 cf 
Score’s Percentile =  100 
N
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Chapter 3 - 26
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.
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Chapter 3 - 27
Percentiles
Normal distribution showing the area under the
curve to the left of selected scores.
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Chapter 3 - 28
Grouped Frequency
Distributions
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Chapter 3 - 29
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
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Chapter 3 - 30
A Grouped Distribution
[Insert new Table 3.6 here.]
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Chapter 3 - 31
Example 1
• Using the following data set, find the
relative frequency of the score 12
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
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Chapter 3 - 32
Example 1
• The simple
frequency table
for this set of
data is as
follows.
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Chapter 3 - 33
Example 1
• The frequency for the score of 12 is 1.
N = 18.
• Therefore, the rel. f of 12 is
f
1
rel. f 

 0.06
N 18
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Chapter 3 - 34
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 - 35
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
cf 13

 0.72
N 18
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Chapter 3 - 36