medium difficulty6 min

Mode of grouped data

The mode of grouped data is the value that occurs most frequently, estimated using the modal class and a standard formula.

In one line: Mode lives near the tallest class, but the formula finds the exact estimate.

Practice This Concept

Understand it

For grouped data, the exact most frequent value is not directly visible, so we estimate mode from the modal class. The formula uses the lower boundary of the modal class, the frequency of the modal class, and the frequencies of the class just before and just after it. This gives a practical estimate of the most common value.

Example

If the modal class is 20-30, the mode lies near that class and is calculated using the grouped-data mode formula.

Formula and rule

Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] x h, where l is lower boundary of modal class, f1 is modal class frequency, f0 is preceding frequency, f2 is succeeding frequency, and h is class width.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

If modal class is 20-30 with l = 20, h = 10, f1 = 12, f0 = 8, and f2 = 5, then mode = 20 + [(12 - 8)/(24 - 8 - 5)] x 10 = 20 + (4/11)x10 = 23.64 approximately.

Method you can reuse

1Identify the modal class.
2Note l, h, f1, f0, and f2.
3Substitute into the formula.
4Simplify the fraction carefully.
5Write the final value with units if needed.

Sketch it in your answer

Use a frequency table and mark the modal class along with the preceding and succeeding frequencies before applying the formula.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is the mode of grouped data only an estimate?

Think first, then show answer

It is only an estimate because grouped data does not show every original value separately. We know the interval with the highest frequency, so we calculate a value that best represents the most common observation.

Common wrong answer

A common wrong answer is to use the class with the highest frequency but stop there, instead of applying the formula.

Don't mix this up

Students often use the frequency of the modal class as the mode itself, which is not correct for grouped data.

Exam tip

Always identify the modal class first, then use the three frequencies around it carefully in the formula.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Board questions often ask for the mode of grouped data after giving a frequency table and may require identifying the modal class first.

1
Enough for 1 mark
The mode of grouped data is the value that occurs most frequently, estimated using the modal class and a standard formula.
2
Add for 2 marks
Example: If the modal class is 20-30, the mode lies near that class and is calculated using the grouped-data mode formula.
3
Add for 3 marks
The formula uses the lower boundary of the modal class, the frequency of the modal class, and the frequencies of the class just before and just after it.

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