Learning Objectives
  • Construct cumulative-frequency tables and diagrams from grouped data.
  • Estimate medians, quartiles, percentiles and interquartile ranges from cumulative-frequency curves.
  • Calculate frequency density and draw or interpret histograms with unequal class widths.
  • Compare distributions using measures of centre and spread.
  • Interpret scatter diagrams and lines of best fit while recognising limitations.
Key Terms
Cumulative frequency
The running total of frequencies up to a class boundary.
Quartile
A value dividing ordered data into four parts.
Interquartile range
Q_3-Q_1, the spread of the middle 50% of the data.
Percentile
A value below which a stated percentage of the data lies.
Frequency density
Frequency divided by class width.
Histogram
A diagram whose bar area represents frequency.
Correlation
The strength and direction of an association between two variables.
Line of best fit
A straight line drawn by eye to represent the trend in a scatter diagram.
What This Chapter Covers
  • Cumulative-frequency tables and smooth curves
  • Median, quartiles, percentiles and interquartile range
  • Histograms and unequal class intervals
  • Frequency-density calculations
  • Scatter diagrams, correlation and comparison of data sets
Cumulative Frequency

Cumulative frequency is a running total. For grouped continuous data, each cumulative total is plotted against the upper class boundary. The first point may be placed at the lowest boundary with cumulative frequency 0. Points should be clearly marked and joined with a smooth increasing curve rather than a jagged frequency polygon.

Worked Example: Build A Cumulative Table

Question: The frequencies for 0\lt x\le10, 10\lt x\le20, 20\lt x\le30, 30\lt x\le40 are 6, 11, 15 and 8. Find the cumulative frequencies.

  1. First total: 6.
  2. Second total: 6+11=17.
  3. Third total: 17+15=32.
  4. Final total: 32+8=40.

Answer: 6, 17, 32, 40.

Estimating Median And Quartiles

If the total frequency is N, the median corresponds to cumulative frequency N/2, the lower quartile to N/4 and the upper quartile to 3N/4. On the graph, move horizontally from the required cumulative frequency to the curve, then vertically down to the horizontal axis. Because the graph is based on grouped data, the values are estimates.

Q_1\text{ at }\frac{N}{4},\qquad \text{median at }\frac{N}{2},\qquad Q_3\text{ at }\frac{3N}{4}
\mathrm{IQR}=Q_3-Q_1
Worked Example: Read A Cumulative Curve

Question: A cumulative-frequency curve represents 80 observations. The graph gives Q_1=24, median =31 and Q_3=39. Find the interquartile range and interpret it.

  1. Calculate 39-24=15.
  2. The middle 50% of the observations extend across an interval of about 15 units.

Answer: IQR =15.

Percentiles

The pth percentile is read at cumulative frequency pN/100. For example, the 90th percentile of 200 observations is found at cumulative frequency 180. Percentiles are useful for identifying cut-off values and comparing relative positions.

Worked Example: Percentile Position

Question: A curve represents 150 test scores. At what cumulative frequency is the 60th percentile read?

  1. Calculate 0.60\times150.

Answer: Cumulative frequency 90.

Comparing Two Cumulative-Frequency Curves

Use a measure of centre and a measure of spread. A larger median indicates a higher typical value. A smaller interquartile range indicates that the middle half of the data is more consistent. Avoid saying one group is simply “better” without using context.

Comparison feature Interpretation
Higher median Higher typical central value
Smaller IQR Middle 50% is less spread out
Same median, different IQR Similar centre but different consistency
Crossing curves The relationship varies at different percentiles; one simple claim may be insufficient
Histograms

In a histogram, the horizontal axis represents a continuous variable and the bars touch. When class widths differ, height cannot represent frequency directly. Instead the vertical axis is frequency density, and the area of each bar represents frequency.

\text{frequency density}=\frac{\text{frequency}}{\text{class width}}
\text{frequency}=\text{frequency density}\times\text{class width}
Worked Example: Frequency Density

Question: A class 20\le x\lt35 contains 24 observations. Find its frequency density.

  1. Class width =35-20=15.
  2. Density =24/15=1.6.

Answer: 1.6.

Worked Example: Recover Frequency

Question: A histogram bar has class width 8 and frequency density 3.5. Find its frequency.

  1. Multiply bar width by bar height.
  2. 8\times3.5=28.

Answer: 28.

Drawing A Histogram
Histogram Method
  1. Write the true class boundaries and calculate each class width.
  2. Calculate frequency density for every class.
  3. Choose suitable scales and label the vertical axis “Frequency density”.
  4. Draw touching bars whose widths match the class intervals and whose heights equal the densities.
  5. Check that the relative bar areas correspond to the frequencies.
Scatter Diagrams And Correlation

A scatter diagram displays paired data. Positive correlation means larger values of one variable tend to be associated with larger values of the other. Negative correlation means one tends to decrease as the other increases. Zero correlation means no clear linear association. Correlation does not prove causation.

A line of best fit should be a single ruled line drawn by inspection across the full data set, with a roughly even distribution of points on both sides. It is used for interpolation within the observed range. Extrapolation beyond the range is less reliable.

Worked Example: Using A Line Of Best Fit

Question: A line of best fit for revision time x hours and score y is y=6x+42. Estimate the score for 5 hours of revision.

  1. Substitute x=5.
  2. y=6(5)+42=72.

Answer: Estimated score 72.

Restrictions On Conclusions

Statistical evidence may be limited by small samples, biased sampling, unequal group sizes, omitted variables or inappropriate extrapolation. A graph can suggest an association but does not identify the cause. Grouped data also hides exact individual values.

Examination Guidance
  • Plot cumulative frequency against upper class boundaries.
  • Join cumulative points with a smooth increasing curve.
  • Label a histogram axis as frequency density, not frequency, when widths differ.
  • Use both median and IQR when comparing two distributions.
  • Draw a best-fit line across the full scatter and keep roughly equal numbers of points on each side.
Common Mistakes
  • Plotting ordinary frequency instead of cumulative frequency.
  • Using class midpoints rather than upper boundaries on a cumulative-frequency graph.
  • Using frequency as histogram height when class widths differ.
  • Calling an association a cause-and-effect relationship.
  • Comparing only medians and ignoring spread.
Knowledge Check And Practice

1. For 120 observations, at what cumulative frequencies are Q_1, the median and Q_3 read?

Answer: 30, 60 and 90.

2. If Q_1=18 and Q_3=31, find the IQR.

Answer: 13.

3. A class of width 5 has frequency 18. Find its frequency density.

Answer: 3.6.

4. A histogram bar has density 2.4 and width 10. Find the frequency.

Answer: 24.

5. What type of correlation is suggested when points trend downward from left to right?

Answer: Negative correlation.

6. Why is extrapolation less reliable than interpolation?

Answer: It predicts beyond the observed data range, where the established trend may not continue.
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