Histogram
A histogram is a graphical representation of data where data is grouped into class intervals (also called bins). It uses rectangular bars to show the frequency of data within each interval.
Unlike a bar graph, the bars in a histogram are drawn without gaps between them. This is because a histogram represents continuous data, where one class interval ends exactly where the next one begins.
Histograms are used when dealing with large amounts of numerical data that need to be grouped. They help us see the distribution of data — where most values lie, whether the data is spread out or clustered, and whether it is symmetric or skewed.
In NCERT Class 8 Mathematics, histograms are an important part of the Data Handling chapter. You will learn how to draw histograms, read information from them, and understand how they differ from bar graphs.
What is Histogram?
Definition: A histogram is a graphical representation of a frequency distribution where continuous data is grouped into non-overlapping class intervals, and each class interval is represented by a rectangular bar whose height (or area) corresponds to the frequency of that interval.
Key Terms:
- Class Interval (Bin): A range of values into which data is grouped. For example, 0-10, 10-20, 20-30.
- Class Width: The difference between the upper and lower boundaries of a class interval. For 10-20, the class width is 10.
- Frequency: The number of data values that fall within a class interval.
- Frequency Distribution Table: A table showing all class intervals and their corresponding frequencies.
- Upper Boundary: The higher value of a class interval (e.g., 20 in the interval 10-20).
- Lower Boundary: The lower value of a class interval (e.g., 10 in the interval 10-20).
- Continuous Data: Data that can take any value within a range (e.g., height, weight, marks, temperature).
Difference from Bar Graph:
- Bar graphs have gaps between bars; histograms have no gaps.
- Bar graphs represent discrete/categorical data (colours, cities, subjects); histograms represent continuous/grouped data.
- In bar graphs, the width of bars can vary; in histograms, all bars usually have equal width.
- In bar graphs, bars can be reordered; in histograms, the order is fixed (numerical order of intervals).
Histogram Formula
Constructing a Histogram — Steps:
- Organise the data into a frequency distribution table with class intervals and frequencies.
- If class intervals are not continuous (e.g., 1-10, 11-20), convert them to continuous form (e.g., 0.5-10.5, 10.5-20.5) by adjusting boundaries.
- Draw the x-axis (horizontal) and mark the class intervals.
- Draw the y-axis (vertical) and mark the frequency scale.
- For each class interval, draw a rectangular bar with height equal to the frequency.
- Bars must touch each other — no gaps.
- Add a title and label both axes.
Adjustment for Non-Continuous Intervals:
Adjustment = (Lower limit of 2nd class - Upper limit of 1st class) / 2
Subtract this value from all lower limits and add it to all upper limits to make the intervals continuous.
Derivation and Proof
How to Read a Histogram:
Reading and interpreting histograms is as important as drawing them.
- Identify the class intervals from the x-axis.
- Read the frequency of each interval from the height of the bar (y-axis).
- Find the class with highest frequency — this is the modal class.
- Find the total frequency by adding the heights of all bars.
- Observe the shape:
- If bars increase then decrease: bell-shaped (symmetric).
- If bars are taller on the left: right-skewed (most data on the left).
- If bars are taller on the right: left-skewed (most data on the right).
- If all bars are similar height: uniform distribution.
Converting Non-Continuous to Continuous Intervals:
If the data is given as 1-10, 11-20, 21-30, there is a gap between 10 and 11. To make it continuous:
- Adjustment = (11 - 10) / 2 = 0.5
- New intervals: 0.5-10.5, 10.5-20.5, 20.5-30.5
This ensures the bars touch each other without gaps.
Types and Properties
Histograms can be classified based on the shape of the distribution:
1. Uniform Histogram:
- All bars have approximately the same height.
- Data is evenly distributed across all intervals.
2. Symmetric (Bell-Shaped) Histogram:
- The bars rise gradually, peak in the middle, and decrease symmetrically.
- Most data is clustered around the central value.
3. Right-Skewed Histogram:
- Most data is concentrated on the left side.
- The "tail" extends to the right.
- Example: Income distribution (many people earn less, few earn very high).
4. Left-Skewed Histogram:
- Most data is concentrated on the right side.
- The "tail" extends to the left.
- Example: Age at retirement (most retire around 60, few retire very early).
5. Bimodal Histogram:
- Two peaks (modes) are visible.
- The data may come from two different groups combined.
Solved Examples
Example 1: Example 1: Drawing a histogram from a frequency table
Problem: Draw a histogram for the following data showing marks obtained by 40 students:
- 0-10: 5 students
- 10-20: 8 students
- 20-30: 12 students
- 30-40: 10 students
- 40-50: 5 students
Solution:
- The class intervals are already continuous (0-10, 10-20, etc.).
- X-axis: Mark the class intervals (0, 10, 20, 30, 40, 50).
- Y-axis: Mark the frequency scale (0, 2, 4, 6, 8, 10, 12).
- Draw bars: height 5 for 0-10, height 8 for 10-20, height 12 for 20-30, height 10 for 30-40, height 5 for 40-50.
- All bars must touch each other.
Observations:
- Modal class = 20-30 (highest frequency = 12).
- Total students = 5 + 8 + 12 + 10 + 5 = 40.
- The distribution is approximately symmetric (bell-shaped).
Example 2: Example 2: Reading a histogram
Problem: A histogram shows the weights (in kg) of students. The bars have the following heights: 10-20 kg: 3, 20-30 kg: 8, 30-40 kg: 15, 40-50 kg: 10, 50-60 kg: 4. Find the total number of students and the modal class.
Solution:
Total students:
- 3 + 8 + 15 + 10 + 4 = 40 students
Modal class:
- The class with highest frequency is 30-40 kg (frequency = 15).
- Modal class = 30-40 kg
Answer: Total students = 40, Modal class = 30-40 kg.
Example 3: Example 3: Converting non-continuous to continuous intervals
Problem: Convert the following non-continuous class intervals to continuous form and draw a histogram: 1-5: 4, 6-10: 7, 11-15: 10, 16-20: 6, 21-25: 3.
Solution:
Step 1: Find the adjustment
- Gap between classes = 6 - 5 = 1
- Adjustment = 1/2 = 0.5
Step 2: Adjust the boundaries
- 0.5-5.5: 4
- 5.5-10.5: 7
- 10.5-15.5: 10
- 15.5-20.5: 6
- 20.5-25.5: 3
Step 3: Draw the histogram using the adjusted continuous intervals. Bars of heights 4, 7, 10, 6, 3 are drawn touching each other.
Answer: The adjusted intervals are 0.5-5.5, 5.5-10.5, 10.5-15.5, 15.5-20.5, 20.5-25.5.
Example 4: Example 4: Finding frequency from a histogram
Problem: In a histogram, the heights of the bars for class intervals 0-10, 10-20, 20-30, 30-40 are 6, 14, 10, and x. If the total frequency is 40, find x.
Solution:
Given:
- Total frequency = 40
- Sum of known frequencies = 6 + 14 + 10 = 30
Calculation:
- 6 + 14 + 10 + x = 40
- 30 + x = 40
- x = 10
Answer: The frequency of the class 30-40 is 10.
Example 5: Example 5: Histogram with unequal class widths
Problem: The following data shows the time (in minutes) taken by students to complete a task: 0-5: 4, 5-10: 8, 10-20: 20, 20-40: 16. The class widths are unequal. How should the histogram be drawn?
Solution:
When class widths are unequal, use frequency density (frequency per unit class width) for the height of bars.
Calculating frequency density:
- 0-5 (width 5): density = 4/5 = 0.8
- 5-10 (width 5): density = 8/5 = 1.6
- 10-20 (width 10): density = 20/10 = 2.0
- 20-40 (width 20): density = 16/20 = 0.8
Draw bars with heights 0.8, 1.6, 2.0, and 0.8 on the y-axis (labelled "Frequency Density").
Answer: Use frequency density on the y-axis when class widths are unequal.
Example 6: Example 6: Comparing two histograms
Problem: A histogram of daily temperatures in January shows modal class 5-10°C with frequency 12. A histogram for July shows modal class 30-35°C with frequency 15. What can you conclude?
Solution:
- January: Most days had temperatures in the 5-10°C range (cool month).
- July: Most days had temperatures in the 30-35°C range (hot month).
- The distribution shifted from lower values (winter) to higher values (summer).
- July has a higher peak frequency (15 vs 12), meaning temperatures were more concentrated in one range during July.
Answer: January temperatures cluster around 5-10°C, while July temperatures cluster around 30-35°C, showing a clear seasonal shift.
Example 7: Example 7: Finding percentage from a histogram
Problem: A histogram shows test scores of 50 students: 0-20: 5, 20-40: 10, 40-60: 18, 60-80: 12, 80-100: 5. What percentage of students scored 60 or above?
Solution:
Students scoring 60 or above:
- 60-80: 12 students
- 80-100: 5 students
- Total = 12 + 5 = 17 students
Percentage:
- = (17/50) x 100
- = 34%
Answer: 34% of students scored 60 or above.
Example 8: Example 8: Histogram from raw data
Problem: The heights (in cm) of 20 students are: 140, 142, 145, 148, 150, 151, 153, 155, 156, 158, 160, 161, 162, 163, 165, 167, 168, 170, 172, 175. Group into class intervals of width 5 starting from 140 and draw a histogram.
Solution:
Step 1: Create frequency table
- 140-145: 3 (140, 142, 145 — note: 145 goes to next class in exclusive method. Using inclusive: 140, 142)
Using exclusive method (upper boundary excluded):
- 140-145: 2 (140, 142)
- 145-150: 2 (145, 148)
- 150-155: 3 (150, 151, 153)
- 155-160: 3 (155, 156, 158)
- 160-165: 4 (160, 161, 162, 163)
- 165-170: 3 (165, 167, 168)
- 170-175: 2 (170, 172)
- 175-180: 1 (175)
Step 2: Draw histogram with these 8 bars of heights 2, 2, 3, 3, 4, 3, 2, 1.
Modal class: 160-165 (frequency 4).
Example 9: Example 9: Histogram vs Bar Graph identification
Problem: State whether a histogram or bar graph should be used for each: (a) Number of students in each house (Red, Blue, Green, Yellow), (b) Weight distribution of 50 apples, (c) Favourite sport of students.
Solution:
- (a) Bar graph — Houses are discrete categories, not numerical ranges.
- (b) Histogram — Weight is continuous numerical data that can be grouped into intervals.
- (c) Bar graph — Sports are discrete categories.
Rule: Use histograms for continuous/grouped numerical data. Use bar graphs for categorical/discrete data.
Example 10: Example 10: Finding the class with most and least frequency
Problem: A histogram represents the ages of 60 people in a locality: 0-10: 8, 10-20: 12, 20-30: 18, 30-40: 10, 40-50: 7, 50-60: 5. Find: (a) the modal class, (b) the class with least frequency, (c) how many people are below 30 years.
Solution:
(a) Modal class:
- Highest frequency = 18 (class 20-30)
- Modal class = 20-30
(b) Class with least frequency:
- Least frequency = 5 (class 50-60)
- Answer: 50-60
(c) People below 30 years:
- 0-10: 8, 10-20: 12, 20-30: 18
- Total = 8 + 12 + 18 = 38 people
Real-World Applications
Education: Teachers use histograms to see how marks are distributed in a class. It shows whether most students scored well, poorly, or average.
Weather Data: Meteorologists use histograms to display temperature, rainfall, or wind speed distribution over a period.
Healthcare: Hospitals use histograms to study age distribution of patients, weight distribution of newborns, or duration of hospital stays.
Business: Companies analyse sales data using histograms — for example, the number of products sold in different price ranges.
Quality Control: Factories use histograms to check whether products meet specifications. If most measurements cluster around the target value, the process is under control.
Census Data: Government agencies use histograms to represent population distribution by age groups, income levels, or household sizes.
Key Points to Remember
- A histogram represents continuous grouped data using rectangular bars with no gaps.
- The x-axis shows class intervals; the y-axis shows frequency.
- Bars in a histogram must touch each other because the data is continuous.
- A bar graph is for discrete/categorical data (with gaps); a histogram is for continuous/grouped data (no gaps).
- If class intervals are not continuous, adjust the boundaries to make them continuous before drawing.
- The modal class is the class interval with the highest frequency (tallest bar).
- When class widths are unequal, use frequency density (frequency / class width) for bar heights.
- The area of each bar (not just height) represents the frequency when class widths are unequal.
- Histograms can be symmetric, right-skewed, left-skewed, or bimodal depending on the data distribution.
- The total frequency equals the sum of frequencies of all class intervals.
Practice Problems
- Draw a histogram for the data: Marks 0-10: 6, 10-20: 10, 20-30: 14, 30-40: 8, 40-50: 2. Identify the modal class.
- Convert the intervals 11-20, 21-30, 31-40, 41-50 to continuous form suitable for a histogram.
- A histogram has bars of heights 5, 12, 18, 10, and 5 for class intervals 10-20, 20-30, 30-40, 40-50, 50-60. Find the total frequency and the percentage of values in the modal class.
- The daily wages (in Rs) of 30 workers are: 100-150: 4, 150-200: 8, 200-250: 12, 250-300: 4, 300-350: 2. Draw a histogram and state two conclusions.
- State whether a histogram or bar graph is appropriate: (a) Number of books read per month, (b) Time taken by students to run 100 m, (c) Favourite colour of students.
- A histogram shows that 45% of students scored in the interval 40-60. If the total number of students is 80, how many students scored in this interval?
Frequently Asked Questions
Q1. What is a histogram?
A histogram is a graph that displays the frequency of continuous grouped data using rectangular bars with no gaps between them. The x-axis shows class intervals and the y-axis shows frequency.
Q2. How is a histogram different from a bar graph?
A histogram represents continuous numerical data (no gaps between bars), while a bar graph represents discrete or categorical data (with gaps between bars). Histograms must maintain numerical order; bar graphs can be reordered.
Q3. Why are there no gaps in a histogram?
There are no gaps because histograms represent continuous data where one class interval ends exactly where the next begins (e.g., 10-20, 20-30). There is no break in the data range.
Q4. What is a modal class?
The modal class is the class interval with the highest frequency. It is represented by the tallest bar in the histogram.
Q5. How do you convert non-continuous intervals to continuous?
Find the gap between the upper limit of one class and the lower limit of the next (e.g., 10 and 11 gives gap = 1). Adjustment = gap/2 = 0.5. Subtract 0.5 from all lower limits and add 0.5 to all upper limits.
Q6. What should the y-axis represent in a histogram?
The y-axis represents frequency (number of observations in each class). If class widths are unequal, the y-axis should show frequency density (frequency divided by class width).
Q7. Can a histogram have bars of different widths?
Yes, if the class intervals have different widths. In such cases, use frequency density (frequency / class width) for the bar heights to ensure accurate representation.
Q8. How do you find the total number of observations from a histogram?
Add the frequencies (heights of all bars) together. If unequal widths are used with frequency density, multiply each bar's height by its width and add.
Q9. What types of data are shown using histograms?
Continuous numerical data that can be grouped into intervals: height, weight, marks, temperature, time, distance, age, salary, etc. Categorical data (colours, names, cities) uses bar graphs instead.
Q10. What does the shape of a histogram tell us?
A symmetric histogram shows data clustered around the centre. A right-skewed histogram has most data on the left. A left-skewed histogram has most data on the right. The shape reveals how data is distributed.
