A histogram is a graphical representation used in statistics to show the distribution of continuous numerical data. The data is grouped into class intervals (bins), and the height of each bar shows the frequency of values.
Example: Suppose you measure the height of every student in a school class in centimeters (cm). The heights range from 120 cm to 180 cm.
You decide to group the results into intervals of 10 cm:
- The 120 cm to just below 130 cm range
- The 130 cm to just below 140 cm range
- The 140 cm to just below 150 cm range
Height (cm) | Number of Students |
|---|---|
120-130 | 5 |
130-140 | 12 |
140-150 | 20 |
150-160 | 30 |
160-170 | 18 |
170-180 | 10 |
Types
There are various variations of the histograms based on their shapes:
Uniform Histogram
A uniform histogram is a histogram in which all the bars are nearly equal in height, showing that the data is evenly distributed among all class intervals.
Bimodal Histogram
A histogram is called bimodal if it has two distinct peaks. This shows that the data consists of observations from two different groups or categories, and each group has its own centre or pattern of values.
Symmetric Histogram
A symmetric histogram is also called a bell-shaped histogram. It is said to be symmetric when a vertical line drawn through the centre divides the histogram into two identical halves. The left and right sides are equal in size and shape, showing a balanced distribution of data.
Right-Skewed Histogram
A right-skewed histogram is a histogram in which the bars extend more towards the right side. This means that most of the data values are on the left side, while a few very large values lie on the right.
Example: In a histogram showing family incomes, most families may have lower incomes, but a few very rich families stretch the data towards the right.
Left-Skewed Histogram
A left-skewed histogram shows bars that extend towards the left side. This means that most of the data values are on the right side, while a few very low values lie on the left.
Example: In a histogram showing test scores, most students may score high marks, but a few students with very low scores stretch the distribution towards the left.
Frequency Histogram
A frequency histogram is a graph that shows how often values occur in a data set. Each bar shows a group of values, and its height shows how many times the values appear.
Example: Let’s say we have the ages of 12 people: Ages: [12, 15, 17, 18, 18, 19, 21, 22, 24, 25, 25, 26]
Frequency Table
Range(Bins) | Frequency |
|---|---|
10-14 | 1 |
15-19 | 4 |
20-24 | 3 |
25-29 | 4 |
Relative Frequency
A relative frequency histogram is a graph that shows the proportion or percentage of data values in each class interval instead of the exact number. Each bar represents the relative frequency of the data, which helps us understand the distribution pattern easily.
Example: Given the test scores of 10 students:
Scores: 55, 60, 62, 70, 75, 78, 80, 82, 85, 90
The frequency table for the scores is as follows:
| Interval (Bins) | Frequency | Relative Frequency |
|---|---|---|
50–59 | 1 | 1/10 = 0.10 |
60–69 | 2 | 2/10 = 0.20 |
70-79 | 3 | 3/10 = 0.30 |
80-89 | 3 | 3/10 = 0.30 |
90-99 | 1 | 1/10 = 0.10 |
Cumulative Frequency
A cumulative frequency histogram is a graph that shows the total number of observations up to a certain value, with the cumulative frequency increasing as we move along the graph.
Cumulative Frequency Table
The cumulative frequency table below shows the distribution of test scores for 10 students:
| Interval | Frequency | Cumulative Frequency |
|---|---|---|
50-59 | 1 | 1 |
60-69 | 2 | 1 + 2 = 3 |
70-79 | 3 | 3 + 3 = 6 |
80-89 | 3 | 6 + 3 = 9 |
90-99 | 1 | 9 + 1 = 10 |
Cumulative Relative Frequency
A cumulative relative frequency histogram is a graph that shows the percentage of data values that fall below a given value, with each bar representing the total of relative frequencies up to that point.
Example: Suppose you have exam scores from 10 students: Scores: 55, 60, 62, 70, 75, 78, 80, 82, 85, 90
| Interval | Frequency | Cumulative Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|---|
50-59 | 1 | 1 | 0.10 | 0.10 |
60-69 | 2 | 3 | 0.20 | 0.30 |
70-79 | 3 | 6 | 0.30 | 0.60 |
80-89 | 3 | 9 | 0.30 | 0.90 |
90-99 | 1 | 10 | 0.10 | 1.00 |
Cumulative Histogram
A cumulative histogram is a graphical representation of cumulative frequencies that shows the running total of frequencies across class intervals. It helps determine the number of observations below or above a particular class boundary.
Formula:
CF_i = \sum_{j=1}^{i} f_j
Number of Bins and Width
1.Bins
The number of bins in a histogram refers to the number of intervals used to divide a dataset into groups. Each bin represents a range of data values, and its height indicates the frequency of observations within that range. The appropriate number of bins helps present the distribution clearly without hiding important patterns or showing unnecessary detail.
2.Bin Width
The bin width in a histogram is the size of each interval used to group data values. It is calculated by dividing the difference between the maximum and minimum data values by the number of bins, assuming all bins have equal widths. A smaller bin width provides more detail, whereas a larger bin width gives a broader overview of the data distribution.
Rice Rule
The Rice Rule is a statistical method used to determine the appropriate number of bins for a histogram based on the number of observations in a dataset. It helps choose a suitable bin count to represent the distribution of data clearly.
Formula:
k = 2n1/3
Where:
- k = Number of bins in the histogram
- n = Total number of observations in the dataset
Doane's Formula
Doane's Formula is a statistical method used to determine the appropriate number of bins in a histogram. It improves upon Sturges' Rule by accounting for the skewness of the data, making it more suitable for datasets that are not normally distributed.
Formula:
k = 1 + \log_2(n) + \log_2\left(1 + \frac{|g_1|}{\sigma_{g_1}}\right)
Where:
- k = Number of bins in the histogram
- n = Total number of observations
- g1 = Sample skewness
\sigma_{g_1} = Standard error of skewness, given by
\sigma_{g_1} = \sqrt{\frac{6(n-2)}{(n+1)(n+3)}}
Terrell–Scott Rule
The Terrell–Scott Rule is a statistical method used to determine the optimal bin width for a histogram, particularly for displaying the distribution of continuous data.
Formula:
h = \frac{3.5s}{n^{1/3}}
Where:
- h = Bin width
- s = Sample standard deviation
- n = Total number of observations
Applications
A histogram is used in the following situations:
- When the data is numerical.
- To understand the shape and distribution of the data.
- To see whether the data is evenly spread or grouped.
- To compare changes in a process over time.
- To compare the results of two or more processes.
- To check whether a process meets customer requirements.
- To present data clearly and quickly for easy understanding.