Step 1: Understanding the Question:
We are given a frequency distribution table and asked to calculate the mean deviation about the mean. Step 2: Key Formula or Approach:
The process involves three main steps:
1. Calculate the mean ($\bar{x}$) of the distribution: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
2. Calculate the absolute deviations from the mean: $|x_i - \bar{x}|$.
3. Calculate the mean deviation (M.D.) about the mean: M.D. = $\frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$. Step 3: Detailed Explanation:
Let's organize the calculations in a table. Step 3.1: Calculate the Mean ($\bar{x}$)
First, find the total number of observations, $N = \sum f_i$.
$N = 8 + 6 + 2 + 2 + 2 + 6 = 26$.
Next, find $\sum f_i x_i$.
$\sum f_i x_i = (5 \times 8) + (7 \times 6) + (9 \times 2) + (10 \times 2) + (12 \times 2) + (15 \times 6)$
$\sum f_i x_i = 40 + 42 + 18 + 20 + 24 + 90 = 234$.
Now, calculate the mean:
$\bar{x} = \frac{\sum f_i x_i}{N} = \frac{234}{26} = 9$. Step 3.2: Calculate the Mean Deviation
Now we build a table to calculate $\sum f_i |x_i - \bar{x}|$, with $\bar{x}=9$.
From the table, $\sum f_i |x_i - \bar{x}| = 32 + 12 + 0 + 2 + 6 + 36 = 88$.
Finally, calculate the mean deviation:
M.D. = $\frac{\sum f_i |x_i - \bar{x}|}{N} = \frac{88}{26}$.
Simplify the fraction:
M.D. = $\frac{44}{13}$. Step 4: Final Answer:
The mean deviation about the mean is $\frac{44}{13}$.
