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```							Distributions, Frequencies,
and Graphical Displays
Problem Set & Solutions

Located under “Course Materials”
Looking at my data.
• Let’s say that you have a set of data:
– 5, 6, 4, 7, 3, 3, 7, 2, 1, 5, 3, 6
• How could you rearrange the data to get a
better idea of what the scores are in your
data set?
– 1, 2, 3, 3, 3, 4, 5, 5, 6, 6, 7, 7
• How could you make it even more clear?
Frequency distribution.
X         f     • I could list the raw
score values that I
7.00       2       have from biggest to
6.00       2       smallest, and then
count how many I
5.00       2
have of each, also
4.00       1       known as their
frequency.
3.00       3
• This table is called a
2.00       1       frequency
1.00       1       distribution.
Frequency distribution
X         f   cf    cp      Percentile
7.00       2    12   100.0        P99
0
6.00       2    10   83.33        P83
5.00       2     8   66.67        P66
4.00       1     6   50.00        P50
3.00       3     5   41.67        P41
2.00       1     2   16.67        P16
1.00       1     1    8.33        P8
Grouped Frequency
Distributions
• If you have a large range of scores values,
you may not want to have an interval for
every value.
• To reduce the number of intervals (also
called classes) you could group values
together.
– If the intervals include more than one raw
score value, then it is called a Grouped
Frequency Distribution.
Grouped Frequency Distribution
X       f        X        f
7.00     2
6.00-7.00   4
6.00     2
5.00     2     4.00-5.00   3
4.00     1
3.00     3     2.00-3.00   4
2.00     1
0.00-1.00   1
1.00     1
Calculating Interval width
Interval width: difference between upper real
limit and lower real limit

Interval: 32.00 to 35.00
Lower real limit: 31.50   Upper real limit:
35.50

Difference: 31.50 - 35.50 = 4
Interval Width: 4
Steps to creating a grouped
frequency distribution:
• Find the range of your scores.
• Xmax – Xmin
• Select the number of intervals (classes).
– If you have N < 100, then ten or fewer classes should
be sufficient.
– Always start each interval with a multiple of the class
interval.
– The first class must start below the lowest score, and
the last class must end above the highest score.
• Tally and count the number of observations that
fall into each interval. (frequency)
Grouped Frequency Distribution
X         f        Cf     Cp      Percentile
60.00-69.00    6   67        100.00   P99
50.00-59.00    9   61        92.05    P92
40.00-49.00   11   52        77.62    P77
30.00-39.00   14   41        61.20    P61
20.00-29.00   12   27        40.30    P40
10.00-19.00    8   15        22.39    P22
0.00-9.00     7   7         10.45    P10
The art of creating score
intervals.
• There is not a definite way to select your score
intervals, but there are some suggestions.
– I generally divide my distribution into 10 intervals
(because that looks good on a graph)…
– OR
– I use an interval width of 5 or 10, depending on the
range of the scores.
• If the range is close to 100 then I use a width of 10.
• If the range is less than 50 then I use a width of 5.
– In general we will tell you either the number of
intervals to use, or the interval width.
EDP/COE 502
In-Class Problem Set
Frequency Distributions and Histograms
Ungrouped Frequency Table
– Interval 10.50 – 11.50

cf    5  46100
cf    .108695100
cf    10.8695
cf    10.87
Graphical Displays of Data
• Methods of graphing distributions:
– Histograms
• A frequency distribution where frequencies are represented
by bars.
– Frequency polygons
• A closed figure representing frequency as dots on a
connected line.
– Percentage polygons
• A closed figure representing percents as dots on a connected
line.
– Stem-and-Leaf Displays
• An alternate way to represent a grouped frequency
distribution.
Ungrouped Frequency
Histogram

10

8
Frequency

6

4

2

0
10.0   11.0   12.0   13.0   14.0   15.0   16.0   17.0   18.0   19.0   20.0

Score Interval (w = 1)
Grouped Frequency Histogram
25

20

15
Frequency

10

5
Std. Dev = 3.01
Mean = 15.7
0                                       N = 47.00
10.0   13.0   16.0    19.0   22.0

Score (w = 3)
Tips for histograms:
• Vertical Axis Should be 2/3 to 3/4 as long as the
horizontal axis.
• Scores on X axis increase from left to right.
• Scores on Y axis increase from bottom to top.
• “//” are used to indicate breaks in the sequence
of numbers or frequencies.
• Points on the scale should be compressed or
expanded to fit on the 2/3 – 3/4 guideline.
• Intervals with frequencies of zero should still be
included on the scale
Tips for histograms:
• Vertical Axis Should be 2/3 to 3/4 as long as the
horizontal axis.
• Scores on X axis increase from left to right.
• Scores on Y axis increase from bottom to top.
• “//” are used to indicate breaks in the sequence
of numbers or frequencies.
• Points on the scale should be compressed or
expanded to fit on the 2/3 – 3/4 guideline.
• Intervals with frequencies of zero should still be
included on the scale
Normal distribution
• These distributions are symmetrical and “bell-shaped”.
– Characterized by high frequencies towards the center of the
distribution and low frequencies in the extreme score regions.
– This is a symmetrical distribution.

f

X
Rectangular distribution (Uniform
distribution)
• Every value in the distribution occurs an equal
number of times.
– This is a symmetrical distribution.

f

X
Skewed distribution

• An asymetrical distribution in which the
frequencies of scores are higher on one end of
the distribution than on the other end.

f

X
Bimodal distribution

• A distribution that peaks in two different places. This
happens when two of the scores both occur with equal
frequency, and more frequently than any other score.
– This can be, but does not have to be a symmetrical distribution.

f

X
Datasets & Output

Located under “Course Materials”
SPSS Output
• See Frequencies Output in Course
Materials
Problem Set and Solutions
Located under “Course Materials”

```
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