Constructing a Cumulative Frequency Table
Start with a frequency distribution table that groups data into class intervals with their corresponding frequencies.
Calculate the cumulative frequency for each class by adding the frequency of the current class to the total frequencies of all previous classes.
The cumulative frequency for the last class equals the total number of data points, which is essential for later calculations.
- Example: For class intervals 0–10, 10–20, 20–30 with frequencies 5, 8, 7, cumulative frequencies are 5, 13 (5+8), and 20 (13+7).
Plotting the Cumulative Frequency Diagram
Use the cumulative frequency table to plot points on a graph with the upper class boundaries on the horizontal axis and cumulative frequencies on the vertical axis.
Mark each point clearly with a small cross (×) to ensure accuracy and visibility.
Connect the points smoothly with a curve rather than straight lines to reflect the continuous nature of the data distribution.
- Example: For the class 10–20, plot a point at x=20 (upper boundary) and y=13 (cumulative frequency).
Estimating the Median from the Diagram
The median corresponds to the value at which half the data lies below it.
Find half the total frequency on the vertical axis and draw a horizontal line from this point to meet the cumulative frequency curve.
From the intersection, draw a vertical line down to the horizontal axis to read the median value approximately.
- If total frequency is 40, half is 20. Locate 20 on the vertical axis, find the curve intersection, then read the median on the horizontal axis.
Determining Quartiles and Percentiles
Quartiles divide data into four equal parts; the first quartile (Q1) is at 25% of total frequency, and the third quartile (Q3) at 75%.
Percentiles divide data into 100 equal parts; for example, the 90th percentile is at 90% of total frequency.
Use the same method as for the median: locate the cumulative frequency corresponding to the desired percentile or quartile, draw horizontal and vertical lines to estimate the value on the horizontal axis.
- For total frequency 40, Q1 is at 10, Q3 at 30 on the vertical axis.
- To find the 90th percentile, locate 36 (0.9 × 40) on the vertical axis.
Calculating and Interpreting the Interquartile Range
The interquartile range (IQR) measures the spread of the middle 50% of data and is calculated as IQR = Q3 − Q1.
Use the cumulative frequency diagram to estimate Q1 and Q3 as described, then subtract to find the IQR.
A smaller IQR indicates data is more concentrated around the median, while a larger IQR shows greater spread.
- If Q1 ≈ 15 and Q3 ≈ 25, then IQR = 25 − 15 = 10.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Construct cumulative frequency table | Add frequencies cumulatively from grouped data | Frequencies 5, 8, 7 become cumulative 5, 13, 20 | Show clear cumulative totals for plotting points |
| Plot points | Use upper class boundaries on x-axis and cumulative frequencies on y-axis | Plot (20,13) for class 10–20 with cumulative frequency 13 | Mark points with crosses (×) for clarity |
| Draw curve | Connect points smoothly, not with straight lines | Smooth curve through plotted points | Demonstrates understanding of continuous data representation |
| Estimate median | Find half total frequency on y-axis, draw horizontal then vertical lines | Half of 40 is 20; locate 20 on y-axis to find median | Accurate reading earns marks |
| Find quartiles/percentiles | Locate 25%, 75%, or other percentiles on y-axis and project to x-axis | Q1 at 10, Q3 at 30 for total 40 | Use correct cumulative frequency values for precision |
| Calculate IQR | Subtract Q1 from Q3 values obtained from diagram | IQR = 25 − 15 = 10 | Show calculation clearly for full marks |
Active recall
Quick check
1. How do you find the cumulative frequency for the third class interval?
Add the frequency of the third class interval to the cumulative frequency of the second class interval.
2. Where do you plot points on a cumulative frequency diagram?
At the upper class boundary on the horizontal axis and the corresponding cumulative frequency on the vertical axis.
3. How is the median estimated from a cumulative frequency diagram?
Locate half the total frequency on the vertical axis, draw a horizontal line to the curve, then drop a vertical line to the horizontal axis to read the median.
4. What does the interquartile range represent?
The spread of the middle 50% of the data, calculated as the difference between the third quartile and the first quartile.