Step 1: Understanding the Question:
We are given a frequency distribution table for a discrete data set. The objective is to compute the mean deviation about the mean. Step 2: Key Formula or Approach:
The procedure is a three-step process:
1. Calculate the Mean ($\bar{x}$): For a frequency distribution, the mean is calculated as $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
2. Calculate Deviations: Find the absolute difference of each data point from the calculated mean, $|x_i - \bar{x}|$.
3. Calculate Mean Deviation (M.D.): The M.D. is the weighted average of these absolute deviations: M.D. = $\frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$. Step 3: Detailed Explanation:
It is best to organize the calculations in a table. Part 1: Calculate the Mean ($\bar{x}$)
First, we find the total frequency, $N = \sum f_i$:
$N = 8 + 6 + 2 + 2 + 2 + 6 = 26$.
Next, we calculate the sum $\sum f_i x_i$:
$\sum f_i x_i = (8 \times 5) + (6 \times 7) + (2 \times 9) + (2 \times 10) + (2 \times 12) + (6 \times 15)$
$\sum f_i x_i = 40 + 42 + 18 + 20 + 24 + 90 = 234$.
Now, the mean is:
$\bar{x} = \frac{\sum f_i x_i}{N} = \frac{234}{26} = 9$. Part 2: Calculate the Mean Deviation
With the mean $\bar{x}=9$, we can now compute the values for the M.D. formula. Let's extend our table:
\includegraphics[width=0.7\linewidth]{8-sol.png}
Summing the last column gives us $\sum f_i |x_i - \bar{x}|$:
$\sum f_i |x_i - \bar{x}| = 32 + 12 + 0 + 2 + 6 + 36 = 88$.
Finally, we calculate the mean deviation:
M.D. = $\frac{\sum f_i |x_i - \bar{x}|}{N} = \frac{88}{26}$.
Simplifying this fraction by dividing the numerator and denominator by 2, we get:
M.D. = $\frac{44}{13}$. Step 4: Final Answer:
The mean deviation about the mean is $\frac{44}{13}$.
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