What is a histogram?
Definition: A histogram is a bar graph of a frequency distribution.
Each bar represents a class, with height = frequency (or relative frequency).
The cumulative-frequency curve (ogive): read the median and quartiles by going up to the curve, then across.
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Key difference: Bars in histograms TOUCH (continuous data).
Bars in bar charts have gaps (categorical data).
| Feature | Histogram | Bar chart |
|---|---|---|
| Data type | Continuous/grouped | Categorical |
| Bars touch? | Yes | No |
| Shows | Frequency distribution | Counts of categories |
Reading histograms: Area of each bar represents frequency (for unequal class widths, use frequency density on y-axis).
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Drawing histograms
Steps: 1.
Find frequency for each class 2.
Choose y-axis scale (frequency or relative frequency) 3.
Draw bars touching each other, height = frequency 4.
Label axes, title, include units
Worked example: draw histogram
The frequency table below shows a set of grouped data.
| Class | [10-20) | [20-30) | [30-40) |
|---|---|---|---|
| Frequency | 4 | 7 | 5 |
Draw a histogram to represent this data.
Solution
- Set up axes: x-axis shows class intervals, y-axis shows frequency
- Bar 1: [10-20) height=4
- Bar 2: [20-30) height=7 (tallest)
- Bar 3: [30-40) height=5
- Bars touch to show continuous data
Final answer
Histogram complete with all bars touching and correct heights.
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Cumulative frequency and ogives
Ogive (cumulative frequency curve): A smooth curve connecting points of cumulative frequency.
Used to find medians, quartiles, and percentiles.
Worked example: draw cumulative frequency curve
Classes [10-20), [20-30), [30-40) with cumulative frequencies 4, 11, 16.
Plot ogive.
Solution
- Plot cumulative frequency at upper class boundary
- Point 1: x=20, y=4
- Point 2: x=30, y=11
- Point 3: x=40, y=16
- Connect with smooth curve (not straight lines)
Final answer
Ogive shows cumulative trend. Rising from left to right, gradually flattening.
Reading from ogive: Find median: locate n/2 on y-axis, read x-value.
Similarly for quartiles.
Frequency density (unequal class widths)
Problem: When class widths differ, frequency alone misleads.
Must use frequency density.
Worked example
Class [0-10) has f=8, class [10-30) has f=12.
Heights are different widths.
Find frequency densities.
Solution
- Class [0-10): width=10, frequency density=8/10=0.8
- Class [10-30): width=20, frequency density=12/20=0.6
- Plot histogram with y-axis as frequency density
Final answer
Frequency density corrects for width differences. Bar area still represents frequency.