Find the mean deviation about the median for the data 36, 72, 46, 42, 60, 45, 53, 46, 51, 49.
Solution and Explanation
The given data is
36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Here, the number of observations is 10, which is even.
Arranging the data in ascending order, we obtain
36, 42, 45, 46, 46, 49, 51, 53, 60, 72
Median M= \(\frac{(\frac{10}{2})^{th}observation+(\frac{10}{2}+1)^{th} \text{observation}}{2}\)
\(\frac{5^{th}observation +6^{th} observation}{2}\)
\(\frac{46+49}{4}=\frac{95}{2}=47.5\)
The deviations of the respective observations from the median, i.e. \(x_i-M,\) are
11.5, 5.5, 2.5, 1.5, 1.5, 1.5, 3.5, 5.5, 12.5, 24.5
Thus, the required mean deviation about the median is
M.D.(M)=\(\frac{\sum_{I=1}^{10}|x_i-M|}{10}\)
= \(\frac{11.5+5.5+2.5+1.5+1.5+1.5+3.5+5.5+12.5+24.5}{10}\)
=\(\frac{70}{10}=7\)
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Concepts Used:
Mean Deviation
A statistical measure that is used to calculate the average deviation from the mean value of the given data set is called the mean deviation.
The Formula for Mean Deviation:
The mean deviation for the given data set is calculated as:
Mean Deviation = [Σ |X – µ|]/N
Where,
- Σ represents the addition of values
- X represents each value in the data set
- µ represents the mean of the data set
- N represents the number of data values
Grouping of data is very much possible in two ways:
- Discrete Frequency Distribution
- Continuous Frequency Distribution