Representing Data: Histograms, Cumulative Frequency & Box Plots
Representing Data
Cumulative Frequency
Cumulative frequency is a running total of frequencies. Plot it to find the median and quartiles for grouped data.
How to draw:
1. Add a cumulative frequency column (running total)
2. Plot the upper class boundary against the cumulative frequency
3. Join points with a smooth curve (an S-shape or ogive)
Worked Example:
| Time (mins) | Frequency | Cumulative Frequency |
|---|---|---|
| 0 < t ≤ 10 | 5 | 5 |
| 10 < t ≤ 20 | 12 | 17 |
| 20 < t ≤ 30 | 18 | 35 |
| 30 < t ≤ 40 | 10 | 45 |
| 40 < t ≤ 50 | 5 | 50 |
Plot: (10, 5), (20, 17), (30, 35), (40, 45), (50, 50).
Reading values: For 50 pieces of data:
- Median at 50/2 = 25th value → read across from 25 on the y-axis to the curve, then down to the x-axis
- Q1 at 50/4 = 12.5th value
- Q3 at 3 × 50/4 = 37.5th value
- IQR = Q3 − Q1
Box Plots (Box and Whisker Diagrams)
A box plot displays the five-figure summary:
1. Minimum value
2. Lower quartile (Q1)
3. Median (Q2)
4. Upper quartile (Q3)
5. Maximum value
The box spans from Q1 to Q3, with a line at the median. Whiskers extend to the minimum and maximum.
Reading a box plot: The width of the box = IQR. The total span = range.
Comparing box plots: Compare medians (centre) and IQR (consistency). Also comment on skewness — if the median is closer to Q1, the data is positively skewed; if closer to Q3, negatively skewed.
Outliers are sometimes shown as separate points beyond the whiskers. A common rule is that an outlier is any value more than 1.5 × IQR beyond Q1 or Q3.
Histograms (Higher)
A histogram looks like a bar chart but uses frequency density on the y-axis, not frequency. This allows it to represent grouped data with unequal class widths.
Frequency density = frequency ÷ class width
Key rule: The area of each bar represents the frequency, not the height.
Worked Example:
| Height h (cm) | Frequency | Class Width | Frequency Density |
|---|---|---|---|
| 140 < h ≤ 150 | 8 | 10 | 0.8 |
| 150 < h ≤ 155 | 15 | 5 | 3.0 |
| 155 < h ≤ 160 | 20 | 5 | 4.0 |
| 160 < h ≤ 170 | 12 | 10 | 1.2 |
| 170 < h ≤ 190 | 6 | 20 | 0.3 |
Draw bars with no gaps, using frequency density for the height.
Reading from a histogram: frequency = frequency density × class width (area of bar).
Worked Example: A histogram bar has frequency density 2.4 and class width 5. What is the frequency?
- Frequency = 2.4 × 5 = 12
Frequency Polygons
A frequency polygon joins the midpoints of the tops of bars in a frequency diagram with straight lines. It can be drawn directly from a frequency table by plotting (midpoint, frequency) and connecting.
Scatter Graphs and Correlation
- Positive correlation: as one variable increases, the other increases
- Negative correlation: as one increases, the other decreases
- No correlation: no clear relationship
A line of best fit should pass through the mean point (x̄, ȳ) and have roughly equal numbers of points above and below. Use it to interpolate (estimate within the data range — reliable) but be cautious about extrapolation (estimating outside the data range — unreliable).
Pie Charts
The angle for each category = (frequency ÷ total) × 360°.
To find a frequency from a pie chart: frequency = (angle ÷ 360) × total.
Exam Tips
- In histograms, the y-axis is FREQUENCY DENSITY, not frequency — this is the most common mistake
- Cumulative frequency is always plotted at the UPPER boundary
- When comparing distributions, always reference both an average AND spread
- Box plots drawn on the same scale allow direct visual comparison — use them