Understanding Mean Deviation
Mean deviation is a measure of dispersion that tells us how much, on average, each value in a dataset deviates from the central tendency (the mean).
Step-by-Step Solution
Given data set: $6, 8, 10, 13, 5$
Step 1: Calculate the Mean ($\bar{x}$) To find the mean, sum the observations and divide by the count ($n = 5$): $$\bar{x} = \frac{6 + 8 + 10 + 13 + 5}{5} = \frac{42}{5} = 8.4$$
Step 2: Calculate Absolute Deviations from the Mean Subtract the mean ($8.4$) from each observation ($x_i$) and take the absolute value ($|x_i - \bar{x}|$):
- $|6 - 8.4| = |-2.4| = 2.4$
- $|8 - 8.4| = |-0.4| = 0.4$
- $|10 - 8.4| = |1.6| = 1.6$
- $|13 - 8.4| = |4.6| = 4.6$
- $|5 - 8.4| = |-3.4| = 3.4$
Step 3: Calculate the Mean of these Deviations Sum the absolute deviations and divide by $n$: $$\text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} = \frac{2.4 + 0.4 + 1.6 + 4.6 + 3.4}{5}$$ $$\text{Mean Deviation} = \frac{12.4}{5} = 2.48$$
Intuition
The mean deviation provides a singular number summarizing how spread out the data points are. A higher value indicates the data points are further from the average, while a value of zero indicates that all points are identical to the mean.