Making Histograms
Histograms are like bar graphs, but the bars are drawn so they touch each other. Histograms are used only
with numerical data.
Example. These are prices of hair dryers in three stores (in dollars). Make a histogram.
14 15 19 20 20 20 21 24 25 34 34 35 35 37 42 45 55
We need to decide how many bins to make and how “wide” they are. For that, we first calculate the range, or
the difference between the greatest and smallest data item. It is 55 − 14 = 41. Then we divide the range into
equal parts (bins) to get the approximate bin width.
If we make five bins, we get 41 ÷ 5 = 8.2 for the bin width. The bins would be 8.2 units apart. However, in
this case it is nice to have the bins go by whole numbers, so we round 8.2 up to 9 and use 9 for the bin width.
The important part is that each data item needs to be in one of the bins. You may have to try out slightly
different bin widths and starting points to see how it works. This time, starting the first bin at 13 makes the
last bin to end at 57, which works, because the data will “fit” into the bins. (Starting at 14 would work, too.)
The frequency describes how many data items fall into that bin. Lastly, all we need is to draw the histogram,
remembering that the bars touch each other.
Price ($) Frequency
13..21 7
22..30 2
31..39 5
40..48 2
49..57 1
This is a double-peaked distribution and skewed to the right (the direction of skewness is where the “long
tail” of the distribution is, in this case to the right). Since it definitely is not bell-shaped, the mean is not a
good measure of center. Therefore, the median is the better choice for measure of center.
Consequently, we need to use interquartile range, and not mean absolute deviation, as a measure of variation.
The median is underlined below:
14 15 19 20 20 20 21 24 25 34 34 35 35 37 42 45 55
The 1st quartile is $20 and the 3rd quartile is $36 (verify those). So the interquartile range is $16.
This means that half of the data is found between 20 and 36 dollars. This price range is quite large for devices
with a median price of only $25. A large range compared to the median describes data that is widely
scattered. (We can also see that from the dot plot.)
Since the median is $25, which is nearer the
low end of the interval from $20 to $36, the
prices are somewhat more concentrated in
the lower end of that interval.
For a comparison, look at the dot plot as well.
It has a similar shape to the histogram.
Sample worksheet from
www.MathMammoth.com
1. This data lists the heights of 24 swimmers in centimeters. Make a histogram with five bins.
155 155 156 157 158 159 159 160 162 162 163 163
164 165 166 167 167 168 168 170 172 174 175 177
Height (cm) Frequency
2. Make a histogram from this data, which lists all the scores a basketball team had in the games in
one season. Make six bins.
60 62 68 71 72 72 73 74 74 74 75 75 76 77 77 77
78 78 78 79 79 81 81 82 83 83 85 86 88 90 92 95
Score Frequency
3. a. As this distribution in problem 2 has its peak near the center, you could use either mean or median as a
measure of center. This time, find the median and the interquartile range.
median ___________ interquartile range ___________
b. Describe the variation in the data. Is it very scattered (varied), somewhat so, or not very much so?
Sample worksheet from
www.MathMammoth.com
4. This data is the finishing times of a group of athletes for the 100-meter dash. Make two histograms, one wit
h
four bins and the other with five, and compare them.
Notice the differences in how they look. Both show a peak, but the one with more bins shows more detail.
The look of a histogram changes depending on the number of bins and the starting point of the first bin.
For this reason, if the data set is small, a dot plot or a stem-and-leaf plot may work better than a histogram.
a. Make a histogram with 4 bins.
11.8 12.0 12.1 12.1 12.3 12.4 12.5 12.5 12.6 12.6 12.7
12.7 12.7 12.7 12.7 12.8 12.8 12.8 12.8 12.9 12.9 13.1
Finishing times
(seconds)
Frequency
b. Make a histogram with 5 bins (from the same data).
11.8 12.0 12.1 12.1 12.3 12.4 12.5 12.5 12.6 12.6 12.7
12.7 12.7 12.7 12.7 12.8 12.8 12.8 12.8 12.9 12.9 13.1
Finishing times
(seconds)
Frequency
Find a quick, mental math method for calculating the mean
for this data set. 102, 94, 99, 105, 96, 107, 101, 104
(the weights of a litter of kittens at birth, in grams)
Sample worksheet from
www.MathMammoth.com
