Histograms, Cumulative Frequency & Box Plots

Histograms

Frequency density = frequency ÷ class width

Cumulative Frequency

Running total, S-curve, reading quartiles

Quartiles & IQR

LQ at n/4, Median at n/2, UQ at 3n/4

Box Plots

5-number summary: min, LQ, med, UQ, max

Reading Graphs

Extract statistics from cumulative frequency curves

Comparing Data

Use median and IQR to compare distributions

1. Histograms

A histogram is used for continuous grouped data. Unlike a bar chart, the y-axis shows frequency density, not frequency.

Frequency Density = Frequency ÷ Class Width

The area of each bar (not the height) represents the frequency.
Area = Frequency Density × Class Width = Frequency

Example:
Class 10–20 (width=10), frequency=40 → fd = 40÷10 = 4
Class 20–25 (width=5), frequency=25 → fd = 25÷5 = 5

Bars in a histogram have no gaps between them. Always label the y-axis "Frequency Density" and the x-axis with the variable.

2. Cumulative Frequency

A cumulative frequency table shows the running total — the total number of data values up to and including each class boundary.

Example: Frequencies: 5, 10, 20, 15, 8 (classes 0-10, 10-20, ...)
Cumulative freq: 5, 15, 35, 50, 58

Plotting cf against the upper class boundary and joining with a smooth curve gives the cumulative frequency curve (S-shaped/ogive).

3. Reading Quartiles from a CF Curve

For n data values in total:

Median: read across at n/2
Lower Quartile (LQ): read across at n/4
Upper Quartile (UQ): read across at 3n/4
IQR = UQ − LQ

Draw a horizontal line from the cumulative frequency value to the curve, then drop vertically to the x-axis to read off the value.

4. Box Plots (Box and Whisker)

A box plot displays the 5-number summary:

Minimum | LQ | Median | UQ | Maximum

Drawing a box plot:
1. Draw a number line
2. Mark the five values
3. Draw a box from LQ to UQ
4. Mark the median inside the box
5. Draw whiskers from the box to min and max

IQR = UQ − LQ measures the spread of the middle 50% of data. A small IQR means the data is clustered. A large IQR means more spread.

5. Comparing Distributions

When comparing two datasets using box plots or cf curves:
Compare the median (central tendency — which is higher/lower?)
Compare the IQR (spread/consistency — which is more spread out?)
Always write a conclusion in context.

Example 1 — Calculating Frequency Density

Class 20–30 has frequency 16. Find the frequency density.

Class width = 30 − 20 = 10

fd = 16 ÷ 10 = 1.60

Example 2 — Finding Frequency from fd

A bar has fd = 3 and class width = 5. Find the frequency.

Frequency = fd × class width = 3 × 5 = 15

Example 3 — Cumulative Frequency Table

Frequencies for classes 0–10, 10–20, 20–30, 30–40, 40–50 are: 8, 12, 20, 10, 5. Find the cumulative frequency at 30.

Up to 30: 8 + 12 + 20 = 40

Example 4 — Finding the Median

60 values in total. Find the median from the cf curve.

n/2 = 60/2 = 30

Read off the value at cf = 30 on the curve. Suppose this gives x = 25.

Example 5 — IQR

LQ = 18, UQ = 34. Find the IQR.

IQR = UQ − LQ = 34 − 18 = 16

Example 6 — Box Plot 5-Number Summary

Data: min=5, LQ=12, median=18, UQ=26, max=40. Describe the spread.

IQR = 26 − 12 = 14. The middle 50% of data lies between 12 and 26.

Range = 40 − 5 = 35. The overall spread is 35 units.

The distribution is roughly symmetric as the median is roughly central in the box.

📊 Histogram Builder

Enter up to 5 classes (boundary + frequency). The histogram will be drawn with frequency density on the y-axis.

Start End₁ f₁ End₂ f₂ End₃ f₃ End₄ f₄ End₅ f₅

Enter data and click Draw Histogram.

Exercise 1 — Calculate Frequency Density (to 2 d.p.)

1. Class 0–10, frequency=8. fd?

2. Class 10–20, frequency=12. fd?

3. Class 20–24, frequency=10. fd?

4. Class 5–10, frequency=3. fd?

5. Class 10–15, frequency=9. fd?

6. Class 0–5, frequency=15. fd?

7. Class 20–30, frequency=4. fd?

8. Class 40–45, frequency=10. fd?

9. Class 0–20, frequency=30. fd?

10. Class 10–50, frequency=10. fd?

Exercise 2 — Find Frequency from fd and Class Width

1. fd=0.8, class width=10. Frequency?

2. fd=1.5, class width=10. Frequency?

3. fd=2.5, class width=10. Frequency?

4. fd=0.6, class width=10. Frequency?

5. fd=1.8, class width=10. Frequency?

6. fd=3.0, class width=10. Frequency?

7. fd=0.8, class width=5. Frequency?

8. fd=4.0, class width=5. Frequency?

9. fd=1.5, class width=10. Frequency?

10. fd=0.25, class width=20. Frequency?

Exercise 3 — Cumulative Frequency at a Given Point

Frequencies: 5, 10, 15, 20, 10, 15, 10, 20, 5, 15 for classes of width 10 starting at 0.

1. Cumulative frequency at end of class 1 (after 0–10, freq=5). CF?

2. CF after classes 1+2 (f=5+10+20). Hmm — use the given pattern: freq=5,10,20... CF at end of class 3?

3. For a dataset with total 100, CF at 60th percentile?

4. Frequencies: 10,15,15,20,20. CF at end of class 4?

5. Frequencies: 5,10,15,20,25,20. CF at end of class 5?

6. Freq: 8,12,20,10,5. CF at end of class 1?

7. Freq: 8,12,20,10,5. CF at end of class 3?

8. Freq: 5,10,20,15,12,8. CF at end of class 4?

9. Freq: 3,5,10,20,25,20,10,5. CF at end of class 5?

10. If n=100 and CF at 88, what is CF at end (all classes done)?

Exercise 4 — Interquartile Range (IQR = UQ − LQ)

1. LQ=10, UQ=20. IQR?

2. LQ=15, UQ=30. IQR?

3. LQ=22, UQ=30. IQR?

4. LQ=5, UQ=25. IQR?

5. LQ=18, UQ=30. IQR?

6. LQ=12, UQ=30. IQR?

7. LQ=4, UQ=10. IQR?

8. LQ=10, UQ=35. IQR?

9. LQ=16, UQ=30. IQR?

10. LQ=11, UQ=20. IQR?

Exercise 5 — Median from Cumulative Frequency Graph

For each dataset of given total, the median is the value at n/2 on the cf curve. The answers given are the median values read from typical exam cf graphs.

1. n=50. Median read from cf curve ≈ ?

2. n=80. Median ≈ ?

3. n=100. Median ≈ ?

4. n=60. Median ≈ ?

5. n=90. Median ≈ ?

6. n=120. Median ≈ ?

7. n=60. Median ≈ ? (different dataset)

8. n=100. Median ≈ ? (n/2=50)

9. n=80. Median ≈ ?

10. n=100. Median ≈ ?

🏋️ Practice — 20 Questions

1. Class 0–10, freq=8. fd?

2. Class 10–20, freq=12. fd?

3. Class 20–24, freq=10. fd?

4. Class 5–10, freq=3. fd?

5. Class 10–15, freq=9. fd?

6. Class 0–5, freq=15. fd?

7. Class 20–30, freq=4. fd?

8. Class 40–45, freq=10. fd?

9. Class 0–20, freq=30. fd?

10. Class 10–50, freq=10. fd?

11. fd=0.8, width=10. freq?

12. fd=1.5, width=10. freq?

13. fd=2.5, width=10. freq?

14. fd=0.6, width=10. freq?

15. fd=1.8, width=10. freq?

16. fd=3.0, width=10. freq?

17. fd=0.8, width=5. freq?

18. fd=4.0, width=5. freq?

19. fd=1.5, width=10. freq?

20. fd=0.25, width=20. freq?

🏆 Challenge — 8 Questions

1. n=50, cf at n/2=25. What is the median position to read from the curve?

2. n=100. Cumulative freq up to value 35 is 35. What % of data is below 35?

3. Freq: 8,12,20,10,5. CF at end of class 3?

4. Freq: 5,10,20,15,12,8. CF at end of class 4?

5. Freq: 3,5,10,20,25,20,10,5. CF at end of class 5?