Cumulative frequency analysis is a cornerstone of statistical study, allowing us to gain insights into the distribution of data sets. By constructing cumulative frequency tables and graphs, we can accurately estimate the median, quartiles, and ascertain the percentile values of a data set. This section aims to equip you with the necessary skills to perform these analyses.

Constructing Cumulative Frequency Tables

To create a cumulative frequency table:

1. Sort Data: Arrange your data in ascending order.

2. Create Intervals: Depending on the data range, create suitable intervals.

3. Tally Frequencies: Count the number of data points within each interval.

4. Cumulative Frequency: Add up the frequencies as you move down the table.

Example Problem:

Consider a dataset: 12, 15, 16, 18, 20, 22, 24, 25, 27, 30.

Solution:

1. Sort the data (already sorted in this case).

2. Define intervals: 10-14, 15-19, etc.

3. Tally and calculate cumulative frequencies.

Constructing Cumulative Frequency Graphs

Cumulative frequency graphs, or ogives, visualise the accumulation of frequencies.

1. Plot Points: For each interval, plot the cumulative frequency against the upper boundary.

2. Draw Curve: Connect the points smoothly.

Estimating Median and Quartiles

The median and quartiles divide the data into equal parts and can be estimated from the graph.

• Median is the 50th percentile.
• Lower Quartile (Q1) is the 25th percentile.
• Upper Quartile (Q3) is the 75th percentile.

Practical Application

Consider a class of 20 students with the following test scores:

47, 55, 72, 34, 88, 66, 95, 41, 57, 82, 76, 65, 89, 59, 90, 54, 77, 62, 85, 93.

Tasks:

1. Construct a cumulative frequency table.

2. Plot a cumulative frequency graph.

3. Estimate the median, Q1, Q3, and the 90th percentile.

Cumulative Frequency Table for Student Scores

Here's the cumulative frequency table based on the given scores:

This table helps us visualise how the scores distribute across different ranges.

Statistical Measures from the Data

• Median (Middle Value): 69
• Lower Quartile (Q1 - 25th Percentile): 56.5
• Upper Quartile (Q3 - 75th Percentile): 85.75
• 90th Percentile: Approximately 90.3

This means that half of the scores are above 69, a quarter of the scores are below 56.5, three quarters of the scores are below 85.75, and 90% of the scores are below approximately 90.3.

To visually interpret these results and estimate these values, you would plot the cumulative frequency against the upper boundary of the score ranges on a graph. The cumulative frequency graph (ogive) would show a smooth curve that helps in estimating the median, quartiles, and any specific percentile such as the 90th percentile we calculated.