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Mean deviation of $39,40,40,41,41,42,42,43,43,44,44,45$ from their median is?
A. 15
B. 1.5
C. 42
D. 35

seo-qna
Last updated date: 20th Jun 2024
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Answer
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Hint: To solve this problem we have to know about the mean, median and deviation from mean. Mean and median are statistical values in statistics and probability theory. Mean is the average of the given set of data which can be expressed as the ratio of the sum of all the observations to the total number of observations. Median is the middle value of the observation in a given set of data. Deviation from mean is how each observation is deviated or varied from their mean.

Complete step-by-step solution:
Here given the set of data, we have to find the mean deviation from mean.
We know that the mean is the average of the given set of data, which is given the ratio of the sum of observations to the total number of observations, which is mathematically expressed below :
Let ${x_1},{x_2},{x_3},.....{x_n}$ be the observations of a set of data, then the mean is given by:
$ \Rightarrow \overline x = \dfrac{{\sum\limits_{i = 1}^n {{x_i}} }}{n}$
Here $\overline x $ is the arithmetic mean of the given set of data.
$\sum\limits_{i = 1}^n {{x_i}} $ is the sum of the observations in the given set of data.
$n$ is the total number of observations.
Now the median of the observations is given by the middle most observation out of the given set of data.
Here the total number of observations are 12.
$\therefore n = 2$
For an even no. of total observations the median is the average of the two middle most observations of the given data.
Here the median of the given data is the average of the 6th and 7th observation, as given below:
$ \Rightarrow M = \dfrac{{42 + 42}}{2}$
$\therefore M = 42$
Thus the median of the given data is 42.
Now we have to calculate the mean deviation of the data from the median, which is given by:
$ \Rightarrow \dfrac{{\sum\limits_{i = 1}^{12} {({x_i} - M)} }}{{12}} = \dfrac{{(39 - 42) + (40 - 42) + .......... + (45 - 42)}}{{12}}$
$ \Rightarrow \dfrac{{\sum\limits_{i = 1}^{12} {({x_i} - M)} }}{{12}} = \dfrac{{3 + 2 + 2 + 1 + 1 + 0 + 0 + 1 + 1 + 2 + 2 + 3}}{{12}}$
$ \Rightarrow \dfrac{{\sum\limits_{i = 1}^{12} {({x_i} - M)} }}{{12}} = 1.5$
$\therefore $ Mean deviation from the median is 1.5

Option B is the correct answer.

Note: While finding the median of any given data it is always important to observe the total number of observations correctly, else everything could go wrong. As already discussed, if the total number of observations are odd, then the mean is the middle value of the observation else the median would be the average of the two middle values in the data.