Histograms
I. Summary
A histogram represents percents by area. It consists of a set of blocks, and the area of
each block represents the percent of cases in the corresponding class interval. With a
density scale, the height of each block equals the percent of cases in the corresponding
class interval divided by the length of that interval. With a density scale, the total area
of the histogram is 100%.
II. Examples
1. The following are 25 final averages in a math class:
46 64 72 79 89
49 66 74 79 91
53 66 75 80 94
60 67 76 83 95
61 71 79 88 98
Draw the histogram using the following class intervals:
A : 92 x 100
B : 80 x 92
C : 64 x 80
D : 52 x 64
F : 44 x 52
Class Number in % in Width of Height of
interval interval interval interval block
A 3 3 25 12% 100-92=8 12% 8 1.5% / Pt
B 5 5 25 20% 92-80=12 20% 12 1.7% / Pt
C 12 12 25 48% 80-64=16 48% 16 3% / Pt
D 3 3 25 12% 64-52=12 12% 12 1% / Pt
F 2 2 25 8% 52-44=8 8% 8 1% / Pt
25 100%
1
%/Pt
(3.0)
3.0
2.5
2.0 (1.7)
(1.5)
1.5
(1.0) (1.0)
1.0
.5
0
44 52 64 80 92 100
Final average (pts)
2. The following histogram was plotted showing the number of cigarettes per
day smoked by each subject. The class intervals include the right endpoint,
not the left. The block over the class interval 20 to 40 is missing.
%/cig
4
(3.0)
3
2
(1.0)
1 (.5)
0
0 10 20 40 80
Number of cigarettes
What is the height of the missing block?
2
Class Width of Height of % in
interval interval block interval
0 x 10 10 0 10cig 1% / cig (10cig )(1% / cig ) 10%
10 x 20 20 10 10cig 3% / cig (10cig)(3% / cig) 30%
20 x 40 40 20 20cig ? ??
40 x 80 80 40 40cig 0.5% / cig (40cig )(0.5% / cig ) 20%
Since the total percent in all the blocks is 100%, the percent in the class interval
20 x 40 must be 40% [40% 100% (10% 30% 20%)]. Thus,
(20cig )(?) 40% height of the missing block 40 % 20 cig 2% / cig.
3. In a small town, the number of children in each family was tabulated with the
following results:
Number of children Percent of families having
in family indicated number of children
0 20%
1 25%
2,3 40%
4,5,6 15%
Draw the histogram.
Class interval % Width of interval Height of block
.5 x .5 20% .5 (.5) 1 20% 1child 20% / child
.5 x 1.5 25% 1.5 .5 1 25% 1child 25% / child
1.5 x 3.5 40% 3.5 1.5 2 40% 2children 20% / child
3.5 x 6.5 15% 6.5 3.5 3 15% 3children 5% / child
%/child (25)
25 (20) (20)
20
15
10 (5)
5
0
0 1 2 3 4 5 6
Number of children
3
Practice Sheet – Histograms
I. The following table gives the distribution of final averages in a math class:
Class interval Number in interval
A : 93 x 100 5
B : 85 x 93 12
C : 78 x 85 31
D : 70 x 78 18
F : 50 x 70 4
Draw the histogram for this data.
II. The following histogram was plotted, using the results from a final exam. The
class interval include the left endpoint. The block over the class interval 65 to 80
is missing.
%/Pt
4
3
2 (1.5)
(1.0) (1.0)
1 (.5)
0
20 50 65 80 90 100
Points on final exam
What is the height of the missing block?
III. Four fair coins are tossed and the number of heads is recorded. This process is done
50 times with the following results:
Number of heads Frequency
0 2
1 13
2 22
3 10
4 3
Draw the histogram for this data.
4
Solution Key for Histograms
I. %/Pt (6.3)
6
5
4
(3.3)
3
(2.1)
2
1 (1)
(.3)
0
50 70 78 85 93 100
Final average (pts)
II. Height = 3.0%/Pt
III. %/Head
50 (44)
40
30 (26)
(20)
20
10 (6)
(4)
0
0 1 2 3 4
Number of heads
5