Mean and Median
Mean: The sum of all the observation is divided by the number of observations is called mean and it is denoted by x̅.
(i) Mean of ungrouped (or) individual data. If x₁, x₂, …, xn are n observations, then mean x̅
\(\left( \overline{x} \right)=\frac{{{x}_{1}}+{{x}_{2}}+…+{{x}_{n}}}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{x}_{i}}}\)(ii) Mean of grouped (or) continuous data. Whose corresponding frequencies are f₁, f₂, …, fn
\(\left( \overline{x} \right)=\frac{{{x}_{1}}{{f}_{1}}+{{x}_{2}}{{f}_{2}}+…..+{{x}_{n}}{{f}_{n}}}{{{f}_{1}}+{{f}_{2}}+……+{{f}_{n}}}=\frac{\sum\limits_{i=1}^{n}{{{f}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{n}{{{f}_{i}}}}\).
Mean deviation about the Mean \(=\frac{\sum\limits_{i=1}^{10}{\left| {{x}_{i}}-\overline{x} \right|}}{n}\).
Median: Median for ungrouped (or) individual data
Suppose n observations are arranged in ascending (OR) descending order
(i) If n is an odd number, then Median = Value of (n+1)/2th term
(ii) If n is an even number, then Median = [Value of (n/ 2)th term + Value of (n + 1)/ 2th term]/ 2.
Mean deviation about Median \(=\frac{\sum\limits_{i=1}^{11}{\left| {{x}_{i}}-M \right|}}{n}\).
Example 1: Find the mean deviation about the mean for the following data 38, 70, 48, 40, 42, 55, 63, 46, 54, 44.
Solution: Mean \(\left( \overline{x} \right)=\frac{\sum\limits_{i=1}^{10}{{{x}_{i}}}}{n}\) = (38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44)/ 10
= 500/ 10 = 50
The absolute values of mean deviations are |xi – x̅| = 12, 20, 2, 10, 8, 5, 13, 4, 4, 6.
Mean deviation about the Mean \(=\frac{\sum\limits_{i=1}^{10}{\left| {{x}_{i}}-\overline{x} \right|}}{10}\) = (12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6)/ 10
= 84/ 10 = 8.4
Example 2: Find the mean deviation about the median for 13, 17, 16, 11, 13, 10, 16, 11, 18, 12, 17.
Solution: Given data in the ascending order: 10, 11, 11, 12, 13, 13, 16, 16, 17, 17, 18.
Mean (M) of these 11 observations is 13.
The absolute values of deviations are |xi – M| = 3, 2, 2, 1, 0, 0, 3, 3, 4, 4, 5.
Mean deviation about Median \(=\frac{\sum\limits_{i=1}^{11}{\left| {{x}_{i}}-M \right|}}{n}\) = (3 + 2 + 2 + 1 + 0 + 0 + 3 + 3 + 4 + 4 + 5)/ 11
= 27/ 11 = 2.45.