Cut the class in half
Find the point inside the median class with half the observations below it.
4.5 more students
For the 53 students the median class is 60–70. It has 7 students, and 22 students scored less than 60.
The median must have n2 = 26.5 students below it. So we need 26.5 − 22 = 4.5 more students from inside the class 60–70. Imagine the 7 students spread evenly over the marks 60 to 70.
Where in the class 60–70 will the median be?
What this lesson covers
The idea
Median = l + ((n/2 − cf)/f) × h, with l the lower limit and f the frequency of the median class, cf the preceding class's cumulative frequency, h the class size; about half the observations lie below it.
4.5 more students
For the 53 students the median class is 60–70. It has 7 students, and 22 students scored less than 60.
The median must have n/2 = 26.5 students below it. So we need 26.5 − 22 = 4.5 more students from inside the class 60–70. Imagine the 7 students spread evenly over the marks 60 to 70.
Where in the class 60–70 will the median be?
- Nearer to 60
- At 65, the middle
- Nearer to 70
Move the cut
The 7 cells are the students of 60–70. Move the cut with the − and + buttons (half a student at a time). The blue cells are the students below the cut. Stop when exactly half of the 53 students, 26.5, are below it.
The median formula
Median = l + ((n/2 − cf) / f) × h
l = lower limit of the median class, n = number of observations, cf = cumulative frequency of the class before the median class, f = frequency of the median class, h = class size (all sizes equal).
Here l = 60, n/2 = 26.5, cf = 22, f = 7, h = 10, so Median = 60 + (4.5/7) × 10 = 60 + 45/7 = 66.4. About half the students scored less than 66.4, and about half scored more. The formula assumes the students of the class are spread evenly over it.
Notes
Median = l + ((n/2 − cf) / f) × h, with l the lower limit and f the frequency of the median class, cf the cumulative frequency of the class before it, and h the class size. About half the observations lie below it.
Check yourself
NCERT Example 7: for 51 girls the median class is 145–150, with l = 145, cf = 11, f = 18, h = 5. Find the median height correct to 2 decimal places.
Answer: 149.03
n/2 = 25.5, so (25.5 − 11)/18 × 5 = 72.5/18 = 4.027…, and 145 + 4.027… = 149.03.
The 30 students of Table 13.3 have the median class 55–70, with l = 55, cf = 12, f = 6, h = 15. Find the median.
Answer: 62.5
(15 − 12)/6 × 15 = 7.5, and 55 + 7.5 = 62.5.
NCERT Example 8: n = 100 and the median is 525, in the class 500–600 with f = 20, h = 100 and cf = 36 + x. Then 525 = 500 + ((50 − 36 − x)/20) × 100, which gives 25 = (14 − x) × 5. Find x.
Answer: 9
25 = (14 − x) × 5 gives 14 − x = 5, so x = 9.
In Example 8 the total frequency is 100, which gives x + y = 24. With x = 9, find y.
Answer: 15
y = 24 − 9 = 15.