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Question: The following observation \(40,42,45,x-1,x+1,51,54,52\) are arranged in conceding order. If the me...

The following observation 40,42,45,x−1,x+1,51,54,5240,42,45,x-1,x+1,51,54,52 are arranged in conceding order.
If the median is 49, find the value of ‘x’.
Hence, find the mean of the above observation.

Explanation

Solution

In order to find the value of 'x' in the given question, at first, we find the median of the given data from the series. As here, the even number of terms are present, so the median can be calculated by dividing the sum of two middle terms by 2 and equating it with the value of the median.
Median =sum of middle two terms2\text{=}\dfrac{\text{sum}\ \text{of}\ \text{middle}\ \text{two}\ \text{terms}}{\text{2}} for even number of observation.
Mean can be calculated using the formula:-
Mean =sum of all observationNo. of observation=\dfrac{\text{sum}\,\text{of}\,\text{all}\,\text{observation}}{\text{No}\text{.}\,\text{of}\,\text{observation}}

Complete step by step solution: Given observation:
40,42,45,x−1,x+1,51,54,62.40, 42, 45, x-1, x+1, 51, 54, 62.
Given: -Median given for the observation is 49.
We need to find: - mean and value of x.
At first arrange the observation in ascending order:
40,42,45,x−1,x+1,51,54,62.40, 42, 45, x-1, x+1, 51, 54, 62.
Here number of terms are 8 (even number)
So, the medium of these terms will be
⇒sum of middle two terms2=49\Rightarrow \dfrac{\text{sum of middle two terms}}{2}=49
⇒x−1+x+12=2x2==49\Rightarrow \dfrac{x-1+x+1}{2}=\dfrac{2x}{2}==49
⇒\Rightarrow
Now, mean of these terms =sum of all termnumber of term=\dfrac{\text{sum}\ \text{of}\ \text{all}\ \text{term}}{\text{number}\ \text{of}\ \text{term}}
⇒So, x−1 =49−1=48 and, x+1=49+1=50\Rightarrow \text{So,}\ x-1\ =49-1=48\ \text{and,}\ x+1=49+1=50
∴ \therefore \ observation:
40,42,45,48,50,51,54,62.40, 42, 45, 48, 50, 51, 54, 62.
∴ mean =40+42+45+48+50+51+54+628\therefore \ \text{mean}\ \text{=}\dfrac{40+42+45+48+50+51+54+62}{8}
=3928  =\dfrac{392}{\begin{aligned} & 8 \\\ & \\\ \end{aligned}}
=49=49
Therefore, the mean of given numbers is 49.and value of x=49.x=49.

Note: Median is the middle observation of a set of observations that are arranged in increasing (or decreasing) order. To locate the median, we must arrange the data in either increasing or decreasing order. If the sample size ’n’ is an odd number, the median is the middle observation. and in case of an even number, the median is the average of the two middle observations.
An important relation between mean and median can be distinguished from the skewness of data. Comparison between mean and median generally reveals information about the shape of the distribution.
For a set of data, if mean = median, then it is symmetric distribution.
If mean > median, it is a positively skewed distribution.
And, if mean < median, it is negatively skewed.