Histograms
A histogram can be defined as a set of rectangles with bases along with the intervals between class boundaries. Each rectangle depicts some sort of data and all the rectangles are adjacent. The heights of rectangles are proportional to corresponding frequencies of similar as well as for different classes. Let's learn about histograms more in detail.
| 1. | What is Histogram? |
| 2. | Histogram Calculator |
| 3. | Frequency Histogram |
| 4. | Histogram Shapes |
| 5. | Difference Between a Bar Chart and a Histogram |
| 6. | How to Make a Histogram? |
What is Histogram?
A histogram is the graphical representation of data where data is grouped into continuous number ranges and each range corresponds to a vertical bar.
- The horizontal axis displays the number range.
- The vertical axis (frequency) represents the amount of data that is present in each range.
The number ranges depend upon the data that is being used.
Histogram Example
Let's explore more with an example. Uncle Bruno owns a garden with 30 black cherry trees. Each tree is of a different height. The height of the trees (in inches): 61, 63, 64, 66, 68, 69, 71, 71.5, 72, 72.5, 73, 73.5, 74, 74.5, 76, 76.2, 76.5, 77, 77.5, 78, 78.5, 79, 79.2, 80, 81, 82, 83, 84, 85, 87. We can group the data as follows in a frequency distribution table by setting a range:
| Height Range (ft) | Number of Trees (Frequency) |
|---|---|
| 60 - 75 | 3 |
| 66 - 70 | 3 |
| 71 - 75 | 8 |
| 76 - 80 | 10 |
| 81 - 85 | 5 |
| 86 - 90 | 1 |
This data can be now shown using a histogram. In a histogram, there shouldn’t be any gaps between the bars.
Histogram Calculator
Here is the Histogram Calculator. Here, we can enter a list of values of data and it gives us the corresponding histogram. This allows you to change the class interval also by sliding the slider.
Frequency Histogram
A frequency histogram is a histogram that shows the frequencies (the number of occurrences) of the given data items. For example, in a hospital, there are 20 newborn babies whose ages in increasing order are as follows: 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5. This information can be shown in a frequency distribution table as follows:
| Age (in days) | Frequency |
|---|---|
| 1 | 4 |
| 2 | 5 |
| 3 | 8 |
| 4 | 2 |
| 5 | 1 |
This data can be now shown using a frequency histogram.
Histogram Shapes
The histogram can be classified into different types based on the frequency distribution of the data. There are different types of distributions, such as normal distribution, skewed distribution, bimodal distribution, multimodal distribution, comb distribution, edge peak distribution, dog food distribution, heart cut distribution, and so on. The histogram can be used to represent these different types of distributions. We have mainly 6 types of histogram shapes. They are:
- Bell Shaped Histogram
- Bimodal Histogram
- Skewed Right Histogram
- Skewed Left Histogram
- Uniform Histogram
- Random Histogram
Bell-Shaped Histogram
A bell-shaped histogram has a single peak. The histogram has just one peak at this time interval and hence it is a bell-shaped histogram. For example, the following histogram shows the number of children visiting a park at different time intervals. This histogram has only one peak. The maximum number of children who visit the park between 5.30 PM to 6 PM.
Bimodal Histogram
A bimodal histogram has two peaks and it looks like the graph given below. For example, the following histogram shows the marks obtained by the 48 students of Class 8 of St.Mary’s School. The maximum number of students have scored either between 40 to 50 marks OR between 60 to 70 marks. This histogram has two peaks (between 40 to 50 and between 60 to 70) and hence it is a bimodal histogram.
Skewed Right Histogram
A skewed right histogram is a histogram that is skewed to the right. In this histogram, the bars of the histogram are skewed to the right, hence called a skewed right histogram. For example, the following histogram shows the number of people corresponding to different wage ranges. The histogram is skewed to the right. For the maximum number of people, wages ranged from 10-20(thousands)
Skewed Left Histogram
A skewed left histogram is a histogram that is skewed to the left. In this histogram, the bars of the histogram are skewed to the left, hence, called a skewed left histogram. For example, the following histogram shows the number of students of Class 10 of Greenwood High School according to the amount of time they spent on their studies on a daily basis. The maximum number of students study 4.5-5(hours) on daily basis.
Uniform Histogram
A uniform histogram is a histogram where all the bars are more or less of the same height. In this histogram, the lengths of all the bars are more or less the same. Hence, it is a uniform histogram. For example, Ma’am Lucy, the Principal of Little Lilly Playschool, wanted to record the heights of her students. The following histogram shows the number of students and their varying heights. The height of the students ranges between 30 inches to 50 inches.
Random Histogram
A random histogram is a histogram where the heights of bars have no pattern. In this histogram, all bars are of different heights and there is no pattern. Hence, it is a random histogram. For example, the following histogram shows the different age groups of the people residing in Carroll Avenue, Los Angeles. The residents belong to varying age groups.
Difference Between a Bar Chart and a Histogram
The fundamental difference between histograms and bar graphs from a visual aspect is that bars in a bar graph are not adjacent to each other.
- A bar graph is the graphical representation of categorical data using rectangular bars where the length of each bar is proportional to the value they represent.
- A histogram is the graphical representation of data where data is grouped into continuous number ranges and each range corresponds to a vertical bar.
The main differences between a bar chart and a histogram are as follows:
| Bar Graph | Histogram |
|---|---|
| Equal space between every two consecutive bars. | No space between two consecutive bars. They should be attached to each other. |
| X-axis can represent anything. | X-axis should represent only continuous data that is in terms of numbers. |
But in both graphs, Y-axis represents numbers only. We can understand these differences from the following figure:
How to Make a Histogram?
The process of making a histogram using the given data is described below:
- Step 1: Choose a suitable scale to represent weights on the horizontal axis.
- Step 2: Choose a suitable scale to represent the frequencies on the vertical axis.
- Step 3: Then draw the bars corresponding to each of the given weights using their frequencies.
Example: Construct a histogram for the following frequency distribution table that describes the frequencies of weights of 25 students in a class.
| Weights (in lbs) | Frequency (Number of students) |
|---|---|
| 65 - 70 | 4 |
| 70 - 75 | 10 |
| 75 - 80 | 8 |
| 80 - 85 | 4 |
Steps to draw a histogram:
- Step 1: On the horizontal axis, we can choose the scale to be 1 unit = 11 lb. Since the weights in the table start from 65, not from 0, we give a break/kink on the X-axis.
- Step 2: On the vertical axis, the frequencies are varying from 4 to 10. Thus, we choose the scale to be 1 unit = 2.
- Step 3: Then draw the bars corresponding to each of the given weights using their frequencies.
Tips and Tricks
Choose the scale on the vertical axis while drawing a histogram, check for the highest number that divides all the frequencies. If there is no such number exists, then check for the highest number that divides most of the frequencies.
Important Points
- A histogram is a graph that is used to summarise continuous data.
- A histogram gives the visual interpretation of continuous data.
- The scales of both horizontal and vertical axes don’t need to start from 0.
- There should be no gaps between the bars of a histogram.
Solved Examples on Histograms
-
Example 1: Consider the following histogram that represents the weights of 34 newborn babies in a hospital. If the children weighing between 6.5 lb to 8.5 lb are considered healthy, then find the percentage of the children of this hospital that are healthy.
Solution:
We have to first find the number of children weighing between 4.4 lb to 6.6 lb. From the given histogram, the number of children weighing between:
6.5 lb - 7.5 lb = 10
7.5 lb - 8.5 lb = 18
Therefore, the number of children weighing between 6.5 lb to 8.5 lb = (10+18=28). The total number of children in the hospital = 34. Hence, the required percentage is: 28/34 × 100 = approx 83%. ∴ Required percentage = 83%. -
Example 2: A random survey is done on the number of children belonging to different age groups who play in government parks and the information is tabulated in the table given below.
(i) Draw a histogram representing the data.
(ii) Identify the number of children belonging to the age groups 2, 3, 4, 5, 6, and 7 who play in government parks.Age (in years) Frequency 1 - 2 8 2 - 4 10 4 - 7 18 7 - 9 10 9 - 11 12 11 - 15 6 Solution:
(i) We take the age (in years) on the horizontal axis of the graph and by observing the first column of the table, we choose the scale to be: 1 unit = 1 year. We take frequency on the vertical axis of the graph and by observing the second column of the table, we choose the scale to be: 1 unit = 2. Now, we will draw the corresponding histogram.
(ii) From the table/graph, the number of children belonging to the age groups:
2 to 4 years = 10
4 to 7 years = 18So, the number of children belonging to the age groups 2, 3, 4,5,6, and 7 who play in government parks is 10+18= 28. ∴ Required number of children = 28
FAQs on Histogram
What is a Histogram Used For?
A histogram is used for showing the frequencies of different data. It is the graphical representation of data where the data is grouped into continuous number ranges and each range corresponds to a vertical bar.
What is a Histogram Graph?
A histogram is a type of graph for the graphical representation of data. This data is grouped into number ranges and each range corresponds to a vertical bar.
How do you Construct a Histogram?
The steps to construct a histogram are as follows:
- Step 1: We place the intervals on the horizontal axis by choosing a suitable scale.
- Step 2: We place frequencies on the vertical axis by choosing a suitable scale.
- Step 3: We construct vertical bars according to the given frequencies.
What is the Difference Between a Bar Graph and a Histogram?
The fundamental difference between histograms and bar graphs from a visual aspect is that bars in a bar graph are not adjacent to each other. A bar graph has equal space between every two consecutive bars and X-axis can represent anything. On the other hand, a histogram has no space between two consecutive bars. They should be attached to each other and the X-axis should represent only continuous data that is in terms of numbers.
What is a Relative Frequency Histogram?
A relative frequency histogram is a kind of graphical representation only, that uses the same information as a frequency histogram. These compare each class interval to the total number of items.
How are Histograms Used Around?
A histogram is a type of bar chart only that is used to display the variation in continuous data, such as time, weight, size, or temperature. A histogram helps to recognize and analyze patterns in data that are not apparent simply by looking at a table of data, or by finding the average or median.
How does a Histogram Represent Data?
A histogram is a graphical display of data with bars of different heights, where each bar groups numbers into ranges. The taller the bars, the more the data falls in that range. It displays the shape as well as the spread of continuous sample data.
Why is a Histogram Two-Dimensional?
A histogram is a visual representation of data, a two-dimension graph that uses a set of vertical rectangles(emphasizing both the lengths and widths of the rectangles) to represent class frequencies of the given distribution.
How do you Interpret the Skewness of a Histogram?
We can interpret the skewness of a histogram by looking into the following aspects.
- Normal distribution will have a skewness of 0.
- If the tail on the right side of the distribution will be longer, the skewness will be positive.
- If the tail on the left side of the distribution will be longer, the skewness will be negative.