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Distributions, Frequencies,
and Graphical Displays
Overheads 2
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.
• Define your score limits.
– 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 46100
cf .108695100
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”
