Understanding Mean Deviation from Median
Mean deviation is a measure of dispersion that describes the average spread of values in a dataset. Unlike variance or standard deviation, which use squared differences, mean deviation uses the absolute values of the differences from a central tendency, such as the mean or the median.
The Formula
The mean deviation from the median is calculated as:
$$MD_{median} = \frac{\sum |x_i - \text{Median}|}{n}$$
Where:
- $x_i$ represents each value in the dataset.
- $n$ is the total number of observations.
Step-by-Step Solution
Problem: Find the mean deviation from the median of the numbers 5, 7, 10, 12, and 6.
Step 1: Arrange the data in ascending order
To find the median, we first sort the data: Original set: 5, 7, 10, 12, 6 Sorted set: 5, 6, 7, 10, 12
Step 2: Find the Median
Since there are $n = 5$ terms (an odd number), the median is the middle value at position $\frac{n+1}{2} = 3$. Median = 7.
Step 3: Calculate the absolute deviations from the median
Calculate $|x_i - 7|$ for each value:
- $|5 - 7| = |-2| = 2$
- $|6 - 7| = |-1| = 1$
- $|7 - 7| = |0| = 0$
- $|10 - 7| = |3| = 3$
- $|12 - 7| = |5| = 5$
Step 4: Find the sum of these deviations
Sum $= 2 + 1 + 0 + 3 + 5 = 11$
Step 5: Calculate the Mean Deviation
Divide the sum by $n = 5$: $$MD_{median} = \frac{11}{5} = 2.2$$
Final Answer: The mean deviation from the median is 2.2.