Cumulative frequency graphs

Cumulative frequency graphs are used to display the accumulation of data as values increase. They are especially useful for visualising percentiles, medians, and quartiles, and for comparing distributions of data sets.

1. Drawing a Cumulative Frequency Graph:

To construct a cumulative frequency graph, follow these steps:

1. Create a cumulative frequency table:
   • Organise the data into class intervals (if not already grouped).
   • For each class, add the frequency of that class to the total of all preceding frequencies to calculate the cumulative frequency.
2. Plot the cumulative frequencies:
   • On the horizontal axis, mark the upper boundary of each class interval.
   • On the vertical axis, mark the cumulative frequency.
   • Plot a point for each class interval at the upper boundary and its corresponding cumulative frequency.
3. Draw a smooth curve: Connect the points with a smooth curve or straight lines to form the cumulative frequency graph.

Example: Consider the following data:

Step 1: Create a cumulative frequency table:

Step 2: Plot the cumulative frequencies against the upper boundaries:

(10, 5), (20, 13), (30, 25), (40, 35), (50, 40).

Step 3: Connect the points with a smooth curve.

2. Interpreting a Cumulative Frequency Graph:

Cumulative frequency graphs provide insights into the data’s distribution:

• Median: To find the median, locate the middle cumulative frequency (total frequency ÷ 2) on the vertical axis. Draw a horizontal line to the curve, then drop a vertical line to the horizontal axis to read the median value.
• Quartiles: Quartiles divide the data into four equal parts:
  • Lower quartile (\(Q_1\)): Find \(\frac{n}{4}\) on the vertical axis and follow the same method as for the median.
  • Upper quartile (\(Q_3\)): Find \(\frac{3n}{4}\) on the vertical axis.
• Interquartile Range (IQR): Subtract \(Q_1\) from \(Q_3\):\[
  \text{IQR} = Q_3 - Q_1
  \]
• Percentiles: To find the \(p\%\) value, locate \(\frac{p}{100} \times n\) on the vertical axis and follow the same method as for the median.

3. Comparing Cumulative Frequency Graphs:

When comparing cumulative frequency graphs for two datasets, focus on:

• Medians: A higher or lower median indicates differences in central tendency.
• Spread: Compare the steepness of the curves. A steeper curve indicates a more concentrated distribution, while a flatter curve shows greater variability.
• IQR: Differences in the interquartile range reflect differences in data spread.

4. Summary:

• Cumulative frequency graphs visually represent the accumulation of data and help identify medians, quartiles, and percentiles.
• They are useful for comparing distributions and assessing data spread and concentration.
• Ensure you calculate cumulative frequencies correctly and use the upper boundaries when plotting.