What is median of following observations?

Median of a group of observation is the value which lies in the middle of the data (when arranged in an ascending or descending order) with half of the observations above it and the other half below it.

● When the number of observations (n) is odd.

Then, median is (n + 1)/2 th observation.

● When the number of observations (n) is even.

Then median is the mean of (n/2)th and (n + 1/2)th observation.

i.e., Median =

Let us observe the following solved problems using step-by-step explanation.

Worked-out examples on median:

1. Find the median of the data 25, 37, 47, 18, 19, 26, 36.

Solution:

Arranging the data in ascending order, we get 18, 19, 25, 26, 36, 37, 47

Here, the number of observations is odd, i.e., 7.

Therefore, median = (n + 1/2)th observation.

= (7 + 1/2)th observation.

= (8/2)th observation

= 4th observation.

4th observation is 26.

Therefore, median of the data is 26.

2. Find the median of the data 24, 33, 30, 22, 21, 25, 34, 27.

Solution:

Here, the number of observations is even, i.e., 8.

Arranging the data in ascending order, we get 21, 22, 24, 25, 27, 30, 33, 34

Therefore, median = {(n/2)th observation + (n + 1/2)th observation}/2

= (8/2)th observation + (8/2 + 1)th observation

= 4th observation + (4 + 1)th observation

= {25 + 27}/2 = 52/2 = 26

Therefore, the median of the given data is 26.

● Statistics

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11, 12, 14, 18, x + 2, x + 4, 30, 32, 35n = 10 (even), Median = 24

∴ Median = `(("n"/2)^"th" "term" + ("n"/2 + 1)^"th" "term")/(2)`

= `((10/2)^"th" "term" + (10/2 + 1)^"th" "term")/(2)`

Median = `(5^"th" "term" + 6^"th" "term")/(2)`

24 = `(x + 2 + x + 4)/(2)`2x + 6 = 24 x 2⇒ 2x = 48 - 6⇒ 2x = 42

x = 21.

Solution:

The median is the value of the given number of observations, which divides it into exactly two parts. So, when the data is arranged in ascending (or descending) order the median of ungrouped data can be calculated based on the number of observations being even or odd.

It can be observed that the total number of observations in the given data 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95 is 10 (even number). Therefore, the median of this data will be the average of 10/2 i.e., 5th and 10/2 + 1 i.e., 6th observation.

Therefore, median of the data = (5th observation + 6th observation) / 2

⇒ 63 = (x + x + 2) / 2

⇒ 63 = (2x + 2) / 2

⇒ 63 = x + 1

∴ x = 62

☛ Check: Class 9 Maths NCERT Solutions Chapter 14

Video Solution:

NCERT Solutions for Class 9 Maths Chapter 14 Exercise 14.4 Question 3

Summary:

If the median of the data 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95 is 63, the value of x = 62.

☛ Related Questions:

Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.
Find the mean salary of 60 workers of a factory from the following table:
Give one example of a situation in which(i) The mean is an appropriate measure of central tendency(ii) The mean is not an appropriate measure of central tendency, but the median is an appropriate measure of central tendency.

Answer

Verified

Hint: Median of a distribution is the value of the observation which divides the distribution into two equal parts. If the values in the ungrouped data are arranged in order of increasing or decreasing magnitude, then the median is the value of the middlemost observation. Also we should look whether the number of observations are even or odd because it creates a difference in the answer. In this given data, the number of terms are even, therefore,Median $ = $Value of ${(\dfrac{n}{2})^{th}}$observation + Value of ${(\dfrac{n}{2} + 1)^{th}}$observation $2$

Complete step-by-step answer:

Given observations are $5,8,6,9,11,4$Arranging the given data in ascending order, we get $4,5,6,8,9,11$.Here, the number of observations $ = 6$, which is even ($n = 6$).Therefore, we apply the formula of median, where the number of terms are even, Median $ = $Value of ${(\dfrac{n}{2})^{th}}$observation + Value of ${(\dfrac{n}{2} + 1)^{th}}$observation $2$$\Rightarrow $Median $ = $Value of ${(\dfrac{6}{2})^{th}}$observation + Value of ${(\dfrac{6}{2} + 1)^{th}}$observation $2$$\Rightarrow $Median $ = $Value of ${3^{^{rd}}}$observation + Value of ${4^{th}}$observation $2$On substituting the values in the above expression, we get$\Rightarrow $Median $ = \dfrac{{6 + 8}}{2} = 7$Therefore, we get the median of the given observations as $7$.

Hence, Option B is the correct choice.

Note: Median is referred to as the middle most value of a series.

If the number of terms are even, then,Median $ = $Value of ${(\dfrac{n}{2})^{th}}$observation + Value of ${(\dfrac{n}{2} + 1)^{th}}$observation $2$If the number of observations are odd, then,Median $ = $ Value of ${(\dfrac{{n + 1}}{2})^{th}}$observation $2$

The median is the value that is in the middle of the data, with half of the observations above it and half below it.

Arranging the given data in ascending order, we have:

33, 35, 41, 46, 55, 58, 64, 77, 87, 90, 92

Here, the number of observations n is 11 (odd).

Since the number of observations is odd, therefore,

Therefore median = ((n+1)/2)th term

Median = value of 5th term

= 58.

Hence, median = 58.

If 92 is replaced by 99 and 41 by 43, then the new observations arranged in ascending

order are:

33, 35, 43, 46, 55, 58, 64, 77, 87, 90, 99