Question

12. Calculate the median value from the following distribution:
Item:
10 11 12 13 14 15 16 17 18
Frequency: 1
1
9 26 59 72 52 29 7 1​

Answer 1

EXPLANATION.

Calculate the median value from the following distributions.

Item : x = 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18

Frequency : f = 1 , 9 , 26 , 59 , 72 , 52 , 29 , 7 , 1.

As we know that,

Formula of Median.

• If n is odd then, $median = \frac{(n + 1)^{th} }{2} \ term$
• If n is even then, $median = \frac{(\frac{n}{2})^{th} + (\frac{n}{2} + 1)^{th} }{2} \ term$

Using this formula in this question, we get.

First we need to find positions.

positions : 1 , (1 + 9 = 10) , (10 + 26 = 36) , (36 + 59 = 95) , (95 + 72 = 167) , (167 + 52 = 219) , (219 + 29 = 248) , (248 + 7 = 255) , (255 + 1 = 256).

Now, we can write expression as,

Item : x = 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18

Frequency : f = 1 , 9 , 26 , 59 , 72 , 52 , 29 , 7 , 1.

Positions : p = 1 , 10 , 36 , 95 , 167 , 219 , 248 , 255 , 256.

Total number of observations = 1 + 9 + 26 + 59 + 72 + 52 + 29 + 7 + 1.

Total number of observations = 256.

Number of observations is even number.

We need to apply median formula when n is even.

$\sf \displaystyle median = \frac{\bigg(\dfrac{256}{2}\bigg)^{th} + \bigg(\dfrac{256}{2} + 1 \bigg)^{th} }{2} \ term$

$\sf \displaystyle median = \frac{(128)^{th} + (129)^{th} }{2} \ term$

$\sf \displaystyle median = \frac{14 + 14}{2}$

$\sf \displaystyle median = \frac{28}{2}$

$\sf \displaystyle median = 14$