COURSE: JUST 3900
INTRODUCTORY STATISTICS
FOR CRIMINAL JUSTICE
Chapter 2: Frequency Distributions
Peer Tutor Slides
Instructor:
Mr. Ethan W. Cooper, Lead Tutor
© 2013 - - PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE
WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS,
PLEASE EMAIL MR. COOPER AT THE FOLLWING: coopere07@students.ecu.edu
Key Terms: Don’t Forget
Notecards
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Frequency Distribution (p. 39)
ΣX (p. 40)
Proportions (p. 41)
Percentages (p. 41)
Grouped Frequency Distribution (p. 42)
Class Intervals (p. 42)
Histogram (p. 46)
Polygon (p. 47)
Bar Graphs (p. 48)
More Key Terms: Think Notecards
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Relative Frequency (p. 49)
Smooth Curves (p. 49)
Symmetrical Distribution (p. 50)
Skewed Distribution (p. 50)
Tail (p. 51)
Positively Skewed (p. 51)
Negatively Skewed (p. 51)
Percentile Rank (p. 53)
Percentile (p. 53)
More Key Terms: Think Notecards
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Cumulative Frequency (p. 54)
Cumulative Percentage (p. 54)
Interpolation (p. 55)
Stem and Leaf Displays (p. 60)
Frequency Distributions
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Question 1: Construct a frequency distribution for the
following set of scores. (see next slide for answer)
Scores: 3, 2, 3, 2, 4, 1, 3, 3, 5

Question 2: Find each of the following values for the sample
in the following frequency distribution table. (see p. 42, #2 for
answers)
1)
2)
3)
n
ΣX
ΣX2
X
f
5
1
4
2
3
2
2
4
1
1
Frequency Distributions
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Question 1 Answers:
X
f
Proportion
Percentage
5
1
.11
11%
4
1
.11
11%
3
4
.44
44%
2
2
.22
22%
1
1
.11
11%
REMEMBER: The values in the X column
should always be listed from the
highest to the lowest!
Frequency Distributions

Question 2: Find each of the following values for the
sample in the following frequency distribution table.
1)
2)
3)
n
ΣX
ΣX2
X
f
5
1
4
2
3
2
2
4
1
1
Frequency Distributions
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Question 2 Answers:
1)
2)
3)
n=10
ΣX=28
ΣX2 =92 (square then add all 10 scores)
Grouped Frequency
Distributions
Question 3: An instructor has obtained the set of N=25
exam scores. To help organize these scores, place them in
a frequency distribution table. The scores are:
82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84,
70, 76, 89, 75, 80, 73, 78, 60
See next slide for answer.
Grouped Frequency
Distributions
Question 3 Answer:
X
f
Proportion
Percentage
90-94
3
.12
12%
85-89
4
.16
16%
80-84
5
.20
20%
75-79
4
.16
16%
70-74
3
.12
12%
65-69
1
.04
4%
60-64
3
.12
12%
55-59
1
.04
4%
50-54
1
.04
4%
Frequency Distribution Graphs
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Question 4: A researcher records the gender and
academic major for each student at a college basketball
game. If the distribution of majors is shown in a
frequency distribution graph, what type of graph should
be used? (see next slide for answer)
Frequency Distribution Graphs
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Question 4 Answer:

In this instance, a bar graph would be used. Remember, bar
graphs are used to represent nominal data. Gender and
academic major are nominal categories.
Frequency Distribution Graphs
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Question 5: If the results from a research study are
presented in a frequency distribution histogram, would it
also be appropriate to show the same results in a
polygon? Explain your answer. (see next slide for
answer)
Frequency Distribution Graphs
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Question 5 Answer:

Yes. Histograms and polygons are both used for data from
interval or ratio scales.
Grouped Frequency
Distributions
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Question 6: Create a histogram using the following
grouped frequency distribution table: (see next slide for
answers)
Question 7: What is the shape of the distribution?
X
f
25-29
1
20-24
1
15-19
5
10-14
9
5-9
4
Grouped Frequency
Distributions
Question 6 Answer:
Question 7 Answer:
The distribution is positively skewed. Please note that when a
distribution is positively skewed, the tail gets thinner at the higher
scores.
Grouped Frequency
Distributions

Question 7: What is the shape of the distribution?
X
f
25-29
1
20-24
1
15-19
5
10-14
9
5-9
4
Grouped Frequency
Distributions
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Question 7 Answer:

The distribution is positively skewed. Please note that when a
distribution is positively skewed, the tail gets thinner at the higher
scores.
WARNING!!!!

WATCH OUT!!
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APLIA WILL TAKE POINTS OFF WHEN YOU CREATE
HISTOGRAMS AND BAR CHARTS THAT DO NOT CONNECT
THE BOTTOM OF THE BAR TO THE X-AXIS.
IF YOU LEAVE ANY GAP BETWEEN THE BOTTOM OF A
BAR AND THE X-AXIS, YOU WILL NOT GET THE PROBLEM
CORRECT.
WHEN DRAWING A BAR, ALWAYS MAKE SURE THAT YOU
RELEASE THE MOUSE BUTTON WHEN THE CURSOR
ACTUALLY CONTACTS THE X-AXIS WITHOUT GAPS.
SEE ILLUSTRATIONS ON NEXT SLIDE
WARNING!!!!

HISTOGRAM EXAMPLES

HOW SHOULD THEY LOOK?
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NOTE: SAME RULES APPLY TO POLYGONS
WRONG WAY
CORRECT WAY
NOTICE THAT THERE IS
A GAP BETWEEN THE
BAR AND THE X-AXIS
NOTICE THAT THERE IS
NO GAP BETWEEN THE
BAR AND THE X-AXIS
WARNING!!!!

WATCH OUT!!
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REMEMBER WHEN USING INTERPOLATION
THAT CHAPTER 2 ONLY USES REAL LIMITS
WITH THE X-VALUE SCALE AND NOT ON THE
PERCENT SCALE.
THE SAME APPLIES TO APLIA FOR THIS WEEK’S
ASSIGNMENT.
Cumulative Frequencies &
Interpolation
Question 8: Fill in the cumulative frequencies and
cumulative percentages for the following table:
a) Find the 70th percentile
b) Find the percentile rank for X=9.5
c) Find the 15th percentile
d) Find the percentile rank for X=13
X
f
20-24
1
15-19
5
10-14
8
5-9
4
0-4
2
cf
c%
Cumulative Frequencies &
Interpolation
Question 8 Answer:
X
f
cf
c%
20-24
1
20
100%
15-19
5
19
95%
10-14
8
14
70%
5-9
4
6
20%
0-4
2
2
10%
Cumulative Frequencies &
Interpolation
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Question 8 Answers:
a) X=14.5 is the 70th percentile
b) X=9.5 has a rank of 20%
c) Because 15% is between the values of 10% and
20% in the table, you must use interpolation. The
score corresponding to a rank of 15% is X=7.
d) Because X=13 is between the real limits of 9.5
and 14.5, you must use interpolation. The
percentile rank for X=13 is 55%.
Cumulative Frequencies &
Interpolation

Question 8C Step-by step: Find the 15th percentile.

Step 1: Find the width of the interval on both scales
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Step 2: Locate position of intermediate value
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15% is located 5 points from top (5/10 = ½ of interval)
Step 3: Use same fraction to determine corresponding position
on other scale. First, determine the distance from the top of the
interval
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
5 and 10 points, respectively
Distance = Fraction x Width = (1/2) * (5 points) = 2.5 Points
Step 4: Use distance from top to determine the position on the
other scale
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
9.5 – 2.5 = 7
Thus, 15th percentile for X is 7.
Cumulative Frequencies &
Interpolation

Question 8D Step-by step: Find the percentile rank for
X=13.
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Step 1: Find the width of the interval on both scales
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Step 2: Locate position of intermediate value

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13 is located 1.5 points from top (1.5/5 = .3)
Step 3: Use same value to determine corresponding position on
other scale. First, determine the distance from the top of the
interval
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5 and 50 points, respectively
Distance = Intermediate Vaule x Width = (.3) * (50 points) = 15 Points
Step 4: Use distance from top to determine the position on the
other scale
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70 – 15 = 55
Thus, X=7 has a percentile rank of 55%.
Stem and Leaf Displays
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Question 10: Use a stem and leaf display to organize the
following set of scores: (Please see next slide for
answers)
74, 103, 95, 98, 81, 117, 105, 99, 63, 86, 94, 107, 96, 100, 98,
118, 107, 82, 84, 71, 91, 107, 84, 77
Stem and Leaf Displays
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Question 10 Answer:
Stem
Leaf
6 3
7 417
8 16244
9 5894681
10 357077
11 78
Stem and Leaf Displays
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Question 11: Explain how a stem and leaf display
contains more information than a grouped frequency
distribution. (Please see next slide for answers)
Stem and Leaf Displays
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Question 11 Answer:
 A grouped frequency distribution table tells only the
number of scores in each interval; it does not identify
the exact value for each score. The stem and leaf
display identifies the individual scores as well as the
number of scores in each interval.
Frequently Asked Questions:
Suggestion from Dr. Kerbs!
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
Question: Is there a way of visually understanding
interpolation?
Answer: Yes. Sometimes it helps to draw a visual depiction
of the x-value intervals and the % intervals. For Question
STEP #2:
8C, consider the following images.
15% is 5 points
down from the
X-Scale
%-Scale
top of a 10-point
STEP 
#3:
We use the fraction
Identified for the % scale (½)
to identify the corresponding
position on the X-scale.
Apply fraction to the X-scale
interval, which spans 5 points
(9.5-4.5=5.0).
½ * (5.0) = 2.5
Thus, the 15th percentile is 2.5
points down from the top of the
X-Scale
15th Percentile = 9.5-2.5 = 7.0
9.5
.
.
. - - > ???
.
.
4.5
STEP #1: DRAW
PICTURES
FOR BOTH SCALES.
THEN START
CALCULATIONS WITH
% SCALE FOR
STEP#2 BECAUSE
THE QUESTION
GIVES PERCENTILE
RANK IN SEARCH OF
X-SCALE VALUE
span that begins
at 20% and ends
at 10%. We turn
this into a fraction
of the total interval
with 5/10 = ½
20%
.
.
. - ->15% about here
.
.
10
REMEMBER: ALWAYS WORK FROM THE TOP DOWN IN THE SCALES