2.3-Histograms
Histogram:
We use a visual tool called a histogram to analyze the shape of the distribution of the data. A histogram is a
graph consisting of bars of equal width drawn adjacent to each other (without gaps). The horizontal scale
represents the classes of quantitative data values and the vertical scale represents the frequencies. The heights
of the bars correspond to the frequency values.
A histogram is basically a graphic version of a frequency distribution. The bars on the horizontal scale are
labeled with one of the following:
(1) Class boundaries
(2) Class midpoints
(3) Lower class limits (introduces a small error)
Interpreting Histograms
Our objective is not simply to construct a histogram, but rather to understand something about the data.
When graphed, a normal distribution has a “bell” shape. Characteristic of the bell shape are that the
frequencies increase to a maximum, and then decrease, and symmetry, with the left half of the graph roughly
a mirror image of the right half. The histogram on the next slide illustrates this.
Example: Given the data below, create a histogram. Is the data normally distributed?
Grades in MAT 250
55.0
77.0
77.0
88.0
99.0
55.0
77.0
77.0
88.0
62.0
66.0
77.0
77.0
99.0
66.0
77.0
88.0
99.0
66.0
77.0
88.0
66.0
Solution: The data appears to be normally distributed because it starts low, increases to a high, and then
decreases again.
Histogram of Exam / Data Set #1
9
8
Frequency
7
6
5
4
3
2
1
0
50.00
55.00
60.00
65.00
70.00
75.00
80.00
85.00
90.00
95.00 100.00
Example: Given the data below, create a histogram. Is the data normally distributed?
Grades in MAT 250
95.0
99.0
88.0
77.0
88.0
55.0
99.0
88.0
77.0
88.0
65.0
99.0
99.0
77.0
99.0
99.0
99.0
77.0
99.0
62.0
99.0
88.0
Solution: The data is not normally distributed.
Histogram of Exam / Data Set #1
10
9
8
Frequency
7
6
5
4
3
2
1
0
50.00
55.00
60.00
65.00
70.00
75.00
80.00
85.00
90.00
95.00 100.00
Relative Frequency Histogram
A relative frequency histogram has
as the same shape and horizontal scale as a histogram, but the vertical scale
is marked with relative frequencies instead of actual frequencies.
frequencies
Example: A nurse measured the blood pressure of each person who visited her clinic. Following is a
relative-frequency
frequency histogram for the systolic blood pressure readings for those people aged between 25 and
40. The blood pressure readings were given to the nearest whole number.
Approximately what percentage of the people aged 25-40
25 40 had a systolic blood pressure reading between 110
and 119 inclusive?
Solution: By looking at the bar for the class 110
110-119,
119, the height can be determined to be 0.35 which is 35%.
Example: Consider the frequency distribution below, which has single values as classes:
Construct a new frequency distribution for this data with 4 classes. Now, construct another frequency
distribution for this data with 6 classes. Suppose that you construct a histogram corresponding to each of the
frequency distributions. Describe the shapes of each histogram. Does the histogram with six classes capture
the distribution of the data? Does the histogram with four classes capture the distribution of the data?
Solution: The data is bimodal because it has two peaks, one near 13 and one near 18. This bimodal
characteristic of the data is lost when only four classes are used but is clearly evident when six classes are
used.
Class
10-12.
13-15
16-18
19-21
Frequency
11
32
25
18
Class
10-11.
12-13.
14-15
16-17
18-19
20-21
Frequency
4
25
14
9
26
8