Cumulative frequency and the median

What you will be able to do

• build a cumulative frequency table;
• find the median class and use the median formula;
• find the median when the classes are inclusive.

The middle value

Seven students are lined up in order of height. The middle one is the fourth. The median is the value in the middle of the ordered data. Half of the values are below it and half are above. With grouped data the values are not listed, so we use the cumulative frequency to find where the middle falls.

Cumulative frequency

The cumulative frequency of a class is the number of values that are in that class or in the classes before it. It is a running total. Fifty patients waited these times at a clinic.

The last cumulative frequency is always $N$, the total. Here it says that $13$ patients waited less than $20$ minutes and $30$ waited less than $30$ minutes. This is the less than type of cumulative frequency. The next course draws it as a graph.

The median class

Half of the patients is $\dfrac{N}{2} = 25$. Which class holds the $25$th patient? Only $13$ patients waited less than $20$ minutes, and $30$ waited less than $30$ minutes. So the $25$th patient is in the class $20$ to $30$. This is the median class. It is the first class whose cumulative frequency is at least $\dfrac{N}{2}$.

The median formula

We know that $13$ patients are below $20$ minutes. We need $25 - 13 = 12$ more to reach the middle. The median class holds $17$ patients, spread evenly over $10$ minutes. So we go $\dfrac{12}{17}$ of the way through the class:

• $l$ is the lower limit of the median class.
• $cf$ is the cumulative frequency of the class before the median class.
• $f$ is the frequency of the median class.
• $h$ is the class width.

For the clinic, $l = 20$, $cf = 13$, $f = 17$ and $h = 10$:

The formula assumes that the patients are spread evenly through each class. A-Level books call this idea linear interpolation, and the answer the interpolated median.

Inclusive classes

Classes such as $1$ to $10$ and $11$ to $20$ must be turned into continuous classes first. Move each lower limit down by $0.5$ and each upper limit up by $0.5$. Here are the marks of $50$ students.

$\dfrac{N}{2} = 25$, so the median class is $21$ to $30$. Its boundaries are $20.5$ and $30.5$, so $l = 20.5$. Then $cf = 14$, $f = 16$ and $h = 10$:

If we had used $l = 21$, the answer would be $0.5$ too large.

When $N$ is odd. Still use $\dfrac{N}{2}$ as it stands, for example $\dfrac{45}{2} = 22.5$. For grouped data this is the rule in the Indian syllabus.

Practise with new numbers

Try it in Python

The program builds the cumulative frequencies with a running total. Then it finds the median class and the median of the clinic table.

Common mistakes

• Using the cumulative frequency of the median class in place of the class before it. The $cf$ in the formula is the one just before the median class.
• Comparing $\dfrac{N}{2}$ with the frequencies rather than with the cumulative frequencies.
• Forgetting to subtract $cf$. The top of the fraction is $\dfrac{N}{2} - cf$.
• Keeping inclusive limits. Use $20.5$, not $21$, as the lower limit of $21$ to $30$.

Remember these

• 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 before the median class.
• For inclusive classes, move the limits by $0.5$ first.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. The frequencies of a table are $5, 8, 12, 9, 6$. Write the cumulative frequencies.

2. In the table of question 1, which class is the median class?

3. A median class is $30$ to $40$ with $f = 10$, $cf = 17$ and $\dfrac{N}{2} = 21$. Find the median.

4. What is the lower limit $l$ of the median class $41$ to $50$ when the classes are inclusive?

5. Why does the median formula subtract $cf$?

References

• NCERT, Mathematics for Class 10, the chapter Statistics: the median of grouped data.
• OpenStax, Introductory Statistics, Chapter 2 Descriptive Statistics: the median.