Mode and Median of Grouped Data: Quick Revision

A brief note on Mode and Median of Grouped Data. Read it before a test. Try each question at the end before you open its answer.

The mode

• The modal class has the greatest frequency.
• $\text{mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$.
• $f_1$ is the frequency of the modal class, $f_0$ of the class before it and $f_2$ of the class after it. A missing neighbour has frequency $0$.
• Two classes with the same largest frequency make the data bimodal. The formula needs equal class widths.

The median

• The cumulative frequency is the running total of the frequencies. The last one is $N$.
• The median class is the first class whose cumulative frequency is at least $\dfrac{N}{2}$.
• $\text{median} = l + \dfrac{\frac{N}{2} - cf}{f} \times h$, with $cf$ the cumulative frequency of the class before the median class.
• For inclusive classes such as $11$ to $20$, use the boundaries $10.5$ and $20.5$.

The three averages

• $3 \times \text{median} = \text{mode} + 2 \times \text{mean}$ is a rule of thumb for moderately skewed data.
• Data with one peak and a long tail on the right usually have mean above median above mode.
• With an open last class the mean cannot be found.

A worked example

Forty students spent these minutes on homework: $0$ to $20$ (frequency $4$), $20$ to $40$ ($10$), $40$ to $60$ ($14$), $60$ to $80$ ($8$) and $80$ to $100$ ($4$). Find the mode and the median.

1. The mode. The modal class is $40$ to $60$, with $f_1 = 14$, $f_0 = 10$ and $f_2 = 8$. So $\text{mode} = 40 + \dfrac{14 - 10}{28 - 10 - 8} \times 20 = 48$ minutes.
2. The cumulative frequencies. $4, 14, 28, 36, 40$.
3. The median. $\dfrac{N}{2} = 20$, so the median class is $40$ to $60$, with $cf = 14$. Then $\text{median} = 40 + \dfrac{20 - 14}{14} \times 20 = 48.57$ minutes.

Practice with new numbers

A program to try

The program finds the median class and the median of the homework table.

Mistakes to avoid

• Swapping $f_0$ and $f_2$ in the mode formula.
• Using the $cf$ of the median class in place of the class before it.
• Using $21$ instead of $20.5$ as the lower limit of an inclusive class.
• Using the rule of thumb as exact. It is only an approximation.

Check yourself

1. The frequencies $6, 11, 16, 9, 4$ are in five classes. Which is the modal class?

2. For the cumulative frequencies $6, 17, 33, 42, 46$, which class holds the median?

3. A moderately skewed data set has mean $30$ and median $28$. Estimate the mode.

4. Can you find the mean of a table whose last class is open?

Sources

• NCERT, Mathematics for Class 10, the chapter Statistics.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics.