Find the middle class
Walk down the cumulative frequencies until you pass n/2.
The middle student
53 students stand in a line, ordered by their marks. The middle student has 26 students on one side and 26 on the other, so he or she is the 27th. We write n2 = 26.5.
The table of cumulative frequencies tells us how many students are below each upper limit.
Which class do you think holds the middle student?
What this lesson covers
The idea
The median class is the class whose cumulative frequency is greater than and nearest to n/2, where n is the number of observations.
The middle student
53 students stand in a line, ordered by their marks. The middle student has 26 students on one side and 26 on the other, so he or she is the 27th. We write n/2 = 26.5.
The table of cumulative frequencies tells us how many students are below each upper limit.
Which class do you think holds the middle student?
- 40–50
- 60–70
- 80–90
Find the class that passes n/2
Each bar is the cumulative frequency of a class. The dashed line is n/2. Tap the class whose bar is the first to go past the dashed line. Do it for all three tables.
The median class
The median class is the class whose cumulative frequency is greater than (and nearest to) n/2, where n is the number of observations.
Going down the cf column, the totals grow. The first total that passes n/2 belongs to the class that holds the middle observation. For the 53 students n/2 = 26.5, and the first cf above 26.5 is 29, so the median class is 60–70.
Notes
The median class is the class whose cumulative frequency is greater than (and nearest to) n/2.
Check yourself
For the heights of the 51 girls (NCERT Example 7), what is n/2?
Answer: 25.5
51/2 = 25.5.
A table of 30 observations has the cumulative frequencies 3, 9, 20, 26, 30 in the classes 0–5, 5–10, 10–15, 15–20, 20–25. Which is the median class?
In the table of 53 students the median class is 60–70. What is the cumulative frequency of the class just before it, 50–60?
Answer: 22
5 + 3 + 4 + 3 + 3 + 4 = 22. This number is called cf in the median formula you meet next.
For the 53 students the cumulative frequencies are 22, 29 and 38 for the classes 50–60, 60–70 and 70–80, and n/2 = 26.5. Which cf do we use to pick the median class?
- 10–15 — correct. Yes! n/2 = 15. The cf 9 is less than 15, and 20 is the first cf greater than 15.
- 5–10. Its cf is 9, which is less than n/2 = 15. Only 9 observations are below 10.
- 15–20. Its cf, 26, is greater than 15, but the class before it already passed 15.
- 29 — correct. Yes! It is the first cf greater than 26.5, and it is the nearest one above 26.5.
- 22. 22 is less than 26.5. Fewer than half of the students are below 60.
- 38. 38 is also greater than 26.5, but 29 is nearer to it, and it comes first.