Frequency Diagrams
Frequency diagrams display how often values occur. This topic covers bar charts, frequency polygons, histograms, and cumulative frequency graphs — each suited to different data types.
📖 Learn
Bar Charts and Vertical Line Charts
Used for discrete or categorical data. The height (or length) of each bar represents the frequency.
- Bars do NOT touch (gaps between them for categorical/discrete)
- y-axis labelled "Frequency"
- All bars the same width
💡 A vertical line chart (frequency diagram) is like a bar chart with lines instead of bars — used for discrete data.
Frequency Polygons
Used for grouped continuous data. Plot the frequency against the midpoint of each class, then join with straight lines.
Midpoint of class a ≤ x < b = (a + b) / 2
Plot (midpoint, frequency) and join with lines.
Frequency polygons allow easy comparison of two data sets on the same axes.
💡 Start and end the polygon at the midpoints of imaginary empty classes at each end (zero frequency).
Histograms
Used for continuous grouped data, especially with unequal class widths. The y-axis shows frequency density, not frequency.
Frequency Density = Frequency ÷ Class Width
Frequency = Frequency Density × Class Width
Area of bar = Frequency
If class widths are equal, histograms look like bar charts but bars must touch.
💡 In a histogram, bars touch each other. The area of each bar represents the frequency, not the height!
Cumulative Frequency Graphs
Plot cumulative frequency (running total) against the upper boundary of each class. Join with a smooth S-curve.
Cumulative frequency after each class = sum of all frequencies up to and including that class.
Median ≈ value at n/2
Lower quartile (Q1) ≈ value at n/4
Upper quartile (Q3) ≈ value at 3n/4
IQR = Q3 − Q1
💡 Always start the cumulative frequency graph at (lower boundary of first class, 0).
✏️ Worked Examples
Example 1: Frequency polygon midpoints
Classes: 0≤x<10, 10≤x<20, 20≤x<30. Find midpoints.
Midpoint 1 = (0+10)/2 = 5
Midpoint 2 = (10+20)/2 = 15
Midpoint 3 = (20+30)/2 = 25
Example 2: Frequency density
Class 20≤x<30, frequency = 15. Class 30≤x<50, frequency = 20. Find frequency density for each.
Class 20–30: width = 10. FD = 15/10 = 1.5
Class 30–50: width = 20. FD = 20/20 = 1.0
Example 3: Reading a histogram
A histogram bar has height (FD) = 2.5 and class width = 4. What is the frequency?
Frequency = FD × class width = 2.5 × 4 = 10
Example 4: Cumulative frequency — finding median
Cumulative frequencies at x = 10, 20, 30, 40, 50 are: 5, 18, 32, 44, 50. Find the median (n=50).
Median at n/2 = 25th value
25 falls between cumulative 18 (at x=20) and 32 (at x=30)
Interpolate: median ≈ 20 + (25−18)/(32−18) × 10 = 20 + 7/14 × 10 = 20 + 5 = 25