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Less-than cumulative frequency table

A less-than cumulative frequency table lists the number of observations less than each upper class boundary.

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Student-friendly explanation

In this table, we keep adding frequencies and write totals against the upper limits of the classes. It is very useful for finding the median and for drawing ogives. The data becomes easier to interpret because the totals increase smoothly from top to bottom.

How to write this in exams

  1. 1

    Start with the exact idea

    A less-than cumulative frequency table lists the number of observations less than each upper class boundary.

  2. 2

    Then show how to use it

    1. Write the class intervals in order. 2. Add frequencies step by step. 3. Attach each running total to the upper boundary. 4. Check that the last total equals total frequency. 5. Use the table for graphing or median work.

  3. 3

    Add one concrete example

    For classes 0-10, 10-20, 20-30 with frequencies 4, 6, 5, the less-than cumulative frequencies are 4, 10, 15.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to write the totals beside lower boundaries, which breaks the pattern used for an ogive.

Definition

A less-than cumulative frequency table lists the number of observations less than each upper class boundary.

Example

For classes 0-10, 10-20, 20-30 with frequencies 4, 6, 5, the less-than cumulative frequencies are 4, 10, 15.

Rule to remember

Less-than cumulative frequency = running total of frequencies written against upper class boundaries.

Memory hook

Less-than means count up to the upper boundary.

Examples and method

Worked example

If the classes are 0-5, 5-10, 10-15 and frequencies are 2, 4, 3, the less-than cumulative frequencies are 2, 6, 9. These are written against 5, 10, and 15 respectively.

Method to apply

1. Write the class intervals in order. 2. Add frequencies step by step. 3. Attach each running total to the upper boundary. 4. Check that the last total equals total frequency. 5. Use the table for graphing or median work.

Diagram support

Make a table with class interval, frequency, and less-than cumulative frequency against upper boundaries.

How CBSE asks it

Students may be asked to complete a less-than cumulative frequency table before plotting an ogive or finding the median.

Avoid common mistakes

Common confusion

Students sometimes attach the cumulative total to the lower boundary instead of the upper boundary.

Common wrong answer

A common wrong answer is to write the totals beside lower boundaries, which breaks the pattern used for an ogive.

Exam tip

Write the upper class boundary carefully; the graph and median position depend on it.

Quick check

Why is the less-than cumulative frequency table helpful in Statistics?

It is helpful because it turns grouped data into a running total that is easy to read and graph. It is also used directly for finding the median and for drawing an ogive.

Answer writing and exam use

1-mark answer

A less-than cumulative frequency table lists the number of observations less than each upper class boundary.

2-mark answer

A less-than cumulative frequency table lists the number of observations less than each upper class boundary. Less-than cumulative frequency = running total of frequencies written against upper class boundaries. For classes 0-10, 10-20, 20-30 with frequencies 4, 6, 5, the less-than cumulative frequencies are 4, 10, 15.

3-mark answer

In this table, we keep adding frequencies and write totals against the upper limits of the classes. It is very useful for finding the median and for drawing ogives. The data becomes easier to interpret because the totals increase smoothly from top to bottom. Less-than cumulative frequency = running total of frequencies written against upper class boundaries. If the classes are 0-5, 5-10, 10-15 and frequencies are 2, 4, 3, the less-than cumulative frequencies are 2, 6, 9. These are written against 5, 10, and 15 respectively. Students may be asked to complete a less-than cumulative frequency table before plotting an ogive or finding the median. A common wrong answer is to write the totals beside lower boundaries, which breaks the pattern used for an ogive.
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