Histogram of Grouped Data
A histogram is a graphical representation of grouped frequency distribution data. It uses adjacent rectangular bars with no gaps between them to show the distribution of continuous data.
Unlike a bar graph (which represents categorical data with gaps), a histogram represents continuous numerical data where each bar covers a class interval. The width of each bar represents the class width, and the height (or area) represents the frequency.
This topic is part of NCERT Class 9 Mathematics, Chapter: Statistics. Histograms are essential for visualising how data is spread across intervals.
What is Histogram of Grouped Data?
Definition: A histogram is a graphical representation of a frequency distribution in which:
- Each class interval is represented by a rectangular bar.
- Bars are placed adjacent to each other (no gaps).
- The width of each bar = class width (upper boundary − lower boundary).
- The height of each bar = frequency of that class (for equal class widths).
For unequal class widths:
Height = Frequency / Class width (frequency density)
Important:
- Histograms are used for continuous data only.
- The x-axis represents class intervals and the y-axis represents frequency (or frequency density).
- If the given data has gaps between classes (e.g., 10–19, 20–29), convert to continuous classes (e.g., 9.5–19.5, 19.5–29.5) before drawing.
- The area of each bar is proportional to the frequency of that class.
Histogram of Grouped Data Formula
Key Formulas and Rules:
1. For equal class widths:
- Height of bar = Frequency of the class
- All bars have the same width.
2. For unequal class widths:
Frequency density = Frequency / Class width
- The height of each bar = frequency density (not frequency).
- This ensures that the area of the bar equals the frequency.
3. Adjusting class intervals:
- If classes are inclusive (10–19, 20–29), adjust to exclusive (9.5–19.5, 19.5–29.5).
- Adjustment = (lower limit of next class − upper limit of current class) / 2
- Subtract from lower limits, add to upper limits.
4. Class mark (mid-value):
- Class mark = (Lower boundary + Upper boundary) / 2
- Used for frequency polygons drawn from histograms.
Derivation and Proof
Steps to Draw a Histogram:
Step 1: Check class intervals
- Verify the data is in continuous (exclusive) form.
- If inclusive (e.g., 10–19), convert to exclusive (9.5–19.5).
Step 2: Choose the axes
- X-axis: Class intervals
- Y-axis: Frequency (or frequency density for unequal widths)
Step 3: Choose a scale
- Select appropriate scales for both axes.
- If the data starts from a large value, use a break (kink) on the x-axis.
Step 4: Draw the bars
- For each class interval, draw a rectangle with width = class width.
- Height = frequency (equal class widths) or frequency density (unequal widths).
- Bars must be adjacent — no gaps between consecutive bars.
Step 5: Label the diagram
- Label both axes with variable names and units.
- Write the scale used.
- Add a title to the histogram.
Step 6: Reading a histogram
- The tallest bar represents the modal class (class with highest frequency).
- The total frequency = sum of all bar heights (equal widths) or sum of all bar areas (unequal widths).
Types and Properties
Types of histogram problems:
1. Equal class width histogram
- All classes have the same width (e.g., 0–10, 10–20, 20–30).
- Height of each bar = frequency directly.
- Most common type in Class 9.
2. Unequal class width histogram
- Classes have different widths (e.g., 0–10, 10–30, 30–40).
- Height = frequency density = frequency / class width.
- Area of bar = frequency.
3. Histogram with inclusive classes
- Classes given as 1–10, 11–20, etc. (gaps between them).
- Convert to 0.5–10.5, 10.5–20.5, etc. before drawing.
4. Histogram with a break (kink)
- When data starts from a large value (e.g., 200–210).
- A zigzag line (kink) is drawn on the x-axis to indicate the scale break.
5. Reading and interpreting histograms
- Given a histogram, determine the frequency table.
- Identify the modal class, total frequency, and distribution pattern.
Solved Examples
Example 1: Example 1: Drawing a histogram (equal class widths)
Problem: Draw a histogram for the following data showing marks of students.
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Students | 5 | 12 | 18 | 10 | 5 |
Solution:
Steps:
- Classes are already continuous and equal-width (width = 10).
- X-axis: Marks (0 to 50); Y-axis: Number of students
- Draw bars: 0–10 (height 5), 10–20 (height 12), 20–30 (height 18), 30–40 (height 10), 40–50 (height 5)
- No gaps between bars.
Modal class: 20–30 (highest frequency = 18).
Total students: 5 + 12 + 18 + 10 + 5 = 50.
Example 2: Example 2: Histogram with inclusive classes
Problem: Draw a histogram for the following data.
| Weight (kg) | 30–39 | 40–49 | 50–59 | 60–69 | 70–79 |
|---|---|---|---|---|---|
| Students | 8 | 15 | 20 | 12 | 5 |
Solution:
Step 1: Convert to exclusive classes
- Adjustment = (40 − 39) / 2 = 0.5
- New classes: 29.5–39.5, 39.5–49.5, 49.5–59.5, 59.5–69.5, 69.5–79.5
Step 2: Draw histogram
- X-axis: Weight; Y-axis: Number of students
- Draw 5 adjacent bars with heights 8, 15, 20, 12, 5.
Modal class: 49.5–59.5 (frequency = 20).
Example 3: Example 3: Histogram with unequal class widths
Problem: Draw a histogram for the following data.
| Age | 0–10 | 10–20 | 20–40 | 40–60 | 60–80 |
|---|---|---|---|---|---|
| People | 5 | 10 | 30 | 16 | 8 |
Solution:
Step 1: Find frequency density
- Minimum class width = 10
- 0–10 (width 10): density = 5/10 = 0.5, adjusted frequency = 5
- 10–20 (width 10): density = 10/10 = 1, adjusted frequency = 10
- 20–40 (width 20): density = 30/20 = 1.5, adjusted frequency = 15
- 40–60 (width 20): density = 16/20 = 0.8, adjusted frequency = 8
- 60–80 (width 20): density = 8/20 = 0.4, adjusted frequency = 4
Step 2: Draw bars using adjusted frequencies as heights
Note: The height represents frequency density, not frequency. The area of each bar represents the actual frequency.
Example 4: Example 4: Reading a histogram
Problem: A histogram shows the following bar heights for class intervals 0–5, 5–10, 10–15, 15–20, 20–25: heights are 4, 8, 12, 6, 2. Find the total frequency and the modal class.
Solution:
Total frequency:
- 4 + 8 + 12 + 6 + 2 = 32
Modal class:
- The tallest bar has height 12, corresponding to class 10–15.
Answer: Total frequency = 32; Modal class = 10–15.
Example 5: Example 5: Histogram with kink
Problem: The daily earnings of 30 workers are given below. Draw a histogram.
| Earnings (Rs) | 200–250 | 250–300 | 300–350 | 350–400 | 400–450 |
|---|---|---|---|---|---|
| Workers | 3 | 7 | 12 | 5 | 3 |
Solution:
Steps:
- Classes are continuous and equal-width (width = 50).
- X-axis starts at 200 (not 0), so draw a kink (zigzag) near the origin.
- Draw bars: 200–250 (height 3), 250–300 (height 7), 300–350 (height 12), 350–400 (height 5), 400–450 (height 3).
Modal class: 300–350 (frequency = 12).
Example 6: Example 6: Constructing frequency table from histogram
Problem: A histogram has bars over intervals 10–20, 20–30, 30–40, 40–50 with heights 6, 14, 10, 4. Construct the frequency distribution table and find total frequency.
Solution:
| Class Interval | Frequency |
|---|---|
| 10–20 | 6 |
| 20–30 | 14 |
| 30–40 | 10 |
| 40–50 | 4 |
| Total | 34 |
Answer: Total frequency = 34.
Example 7: Example 7: Histogram for test scores
Problem: Test scores of 40 students: 0–20 (3), 20–40 (8), 40–60 (15), 60–80 (10), 80–100 (4). (a) Draw the histogram. (b) Which class has the maximum students? (c) How many students scored below 40?
Solution:
(a) Draw 5 bars with heights 3, 8, 15, 10, 4 over intervals 0–20, 20–40, 40–60, 60–80, 80–100.
(b) Maximum students in class 40–60 (frequency 15).
(c) Students below 40 = 3 + 8 = 11.
Example 8: Example 8: Histogram vs bar graph
Problem: State whether each of the following should be represented by a histogram or a bar graph: (a) Marks of students in ranges, (b) Favourite fruit of students, (c) Heights of plants in intervals.
Solution:
- (a) Marks in ranges — Histogram (continuous numerical data in class intervals)
- (b) Favourite fruit — Bar graph (categorical data, not continuous)
- (c) Heights in intervals — Histogram (continuous numerical data in class intervals)
Example 9: Example 9: Finding class width
Problem: In a histogram, the class intervals are 5–15, 15–25, 25–35. What is the class width? If the frequencies are 6, 10, 8, verify the total frequency.
Solution:
Class width:
- Width = 15 − 5 = 10 (same for all classes)
Total frequency:
- 6 + 10 + 8 = 24
Answer: Class width = 10; Total frequency = 24.
Example 10: Example 10: Percentage from histogram
Problem: A histogram of test scores of 200 students shows: 0–25 (20), 25–50 (55), 50–75 (80), 75–100 (45). What percentage of students scored 50 or above?
Solution:
Students scoring 50 or above:
- 80 + 45 = 125
Percentage:
- (125/200) × 100 = 62.5%
Answer: 62.5% of students scored 50 or above.
Real-World Applications
Applications of Histograms:
- Education: Visualising the distribution of test scores across a class or school.
- Census data: Representing population distribution by age groups, income levels, or household sizes.
- Medical research: Displaying frequency of blood pressure readings, weights, or recovery times.
- Quality control: Showing the distribution of product measurements (weight, length) in manufacturing.
- Economics: Representing income distribution, price ranges, or expenditure patterns.
- Weather analysis: Displaying temperature distribution, rainfall patterns across months.
Key Points to Remember
- A histogram represents grouped continuous data using adjacent rectangular bars with no gaps.
- For equal class widths: height of bar = frequency.
- For unequal class widths: height of bar = frequency density = frequency / class width.
- The area of each bar is proportional to the frequency.
- If classes are inclusive (10–19, 20–29), convert to exclusive (9.5–19.5, 19.5–29.5) before drawing.
- The modal class is the class with the tallest bar (highest frequency).
- A kink (zigzag) on the x-axis indicates that the scale does not start from zero.
- Histograms are for continuous data; bar graphs are for categorical data.
- Total frequency = sum of all frequencies (for equal widths) or sum of all areas (for unequal widths).
- The x-axis shows class intervals; the y-axis shows frequency or frequency density.
Practice Problems
- Draw a histogram for: 0–10 (4), 10–20 (9), 20–30 (15), 30–40 (8), 40–50 (4). Identify the modal class.
- Convert the following inclusive classes to exclusive and draw a histogram: 1–5 (3), 6–10 (7), 11–15 (12), 16–20 (5).
- A histogram has bars of heights 5, 10, 15, 12, 8 over equal-width classes. Find the total frequency.
- Draw a histogram for unequal class widths: 0–10 (8), 10–30 (20), 30–40 (12), 40–80 (16).
- From a histogram, the frequencies for classes 20–30, 30–40, 40–50, 50–60 are 10, 18, 14, 8. What percentage of data lies in the modal class?
- State whether a histogram or bar graph is more appropriate for: (a) heights of students, (b) favourite colours, (c) daily temperatures.
Frequently Asked Questions
Q1. What is a histogram?
A histogram is a graphical representation of grouped frequency distribution using adjacent rectangular bars. Each bar covers a class interval, and the height represents the frequency (for equal widths) or frequency density (for unequal widths).
Q2. What is the difference between a histogram and a bar graph?
A histogram represents continuous numerical data with no gaps between bars. A bar graph represents categorical (discrete) data with gaps between bars. In a histogram, the width of bars matters; in a bar graph, only the height matters.
Q3. What is frequency density?
Frequency density = frequency / class width. It is used when class widths are unequal, so that the area of each bar (not its height) represents the frequency.
Q4. What is the modal class?
The modal class is the class interval with the highest frequency. In a histogram, it corresponds to the tallest bar.
Q5. How do you convert inclusive classes to exclusive classes?
Find the adjustment factor = (lower limit of next class − upper limit of current class) / 2. Subtract this from all lower limits and add to all upper limits. For example, 10–19, 20–29 becomes 9.5–19.5, 19.5–29.5.
Q6. Why are there no gaps in a histogram?
Because the data is continuous. Each class interval starts where the previous one ends (e.g., 0–10, 10–20). The absence of gaps shows that the data covers a continuous range.
Q7. What is a kink on the x-axis?
A kink (zigzag line) on the x-axis indicates that the scale is broken — there is a gap between the origin and the first class interval. It is used when data values are far from zero.
Q8. Is histogram part of CBSE Class 9?
Yes. Histograms are covered in CBSE Class 9 Mathematics, Chapter: Statistics, under graphical representation of grouped data.
Related Topics
- Frequency Distribution Table
- Frequency Polygon
- Histogram
- Statistics - Collection and Presentation
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Mean of Grouped Data
- Median of Grouped Data
- Mode of Grouped Data
- Cumulative Frequency Distribution
- Ogive (Cumulative Frequency Curve)
- Empirical Relationship Between Mean, Median, Mode
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
