Find the median from the following data: (i) 8 3 11 7 12 6 9 (ii) X : 10 5 7 11 8 f : 15 20 15 18 12

Steps to Solve

1. Finding the Median of a Small List

For the first dataset: (8, 3, 11, 7, 12, 6, 9)

First, we need to arrange the numbers in ascending order: $$3, 6, 7, 8, 9, 11, 12$$

Since there are 7 numbers (an odd count), the median will be the middle number: $$\text{Median} = 8$$

2. Finding the Median of a Frequency Distribution

For the second dataset, let’s calculate the cumulative frequency first.

Data given:

• (X: 10, 5, 7, 11, 8)
• (f: 15, 20, 15, 18, 12)

First, calculate the cumulative frequency (CF):

• For (10): (15)
• For (5): (15 + 20 = 35)
• For (7): (35 + 15 = 50)
• For (11): (50 + 18 = 68)
• For (8): (68 + 12 = 80)

Total (N = 80) (which is even).

Thus, the median is found by: $$\text{Median position} = \frac{N}{2} = \frac{80}{2} = 40$$

Now, we look for the median class, where cumulative frequency just exceeds 40, which is (7) (with (50) as its cumulative frequency).

We can calculate the median using the formula: $$ \text{Median} = L + \left( \frac{ \frac{N}{2} - CF }{f} \right) \times h $$

Where:

• (L) is the lower boundary of the median class (which is (7))
• (CF) is the cumulative frequency before the median class ((35))
• (f) is the frequency of the median class ((15))
• (h) is the class width (assumed to be (2) as (5) to (7))

Substituting the values: $$ \text{Median} = 7 + \left( \frac{40 - 35}{15} \right) \times 2 $$ $$ = 7 + \left( \frac{5}{15} \right) \times 2 $$ $$ = 7 + \frac{10}{15} $$ $$ = 7 + \frac{2}{3} $$ $$ \approx 7.67 $$

3. Combining Results

The medians from both datasets are:

• From part (i): Median = 8
• From part (ii): Median ≈ 7.67