Question

Let $x_{1},x_{2},\dots,x_{n}$ be the data with respective frequencies $f_{1},f_{2},\dots,f_{n}$. If $\sum_{i=1}^{n}f_{i}(|x_{i}-\overline{x}|)=400$ and the mean deviation from the mean $\overline{x}$ is 10, then $\sum_{i=1}^{n}f_{i}$ is equal to

• A. 55
• B. 50
• C. 36
• D. 40
• E. 25

Hint:

Math Tip: This is a direct formula application question. Always remember that the denominator in any mean deviation or variance formula for grouped data is always the sum of the frequencies ($N = \sum f_i$), not the number of distinct data points ($n$).

Answer 1

The Correct Option is D

Concept:
Statistics - Mean Deviation.
The formula for the Mean Deviation (MD) about the mean for a grouped data distribution is: $$ \text{MD} = \frac{\sum_{i=1}^{n} f_i |x_i - \overline{x}|}{N} $$ where $N = \sum_{i=1}^{n} f_i$ is the total frequency.
Step 1: Identify the given values from the problem.

• The sum of the absolute deviations multiplied by frequencies: $\sum_{i=1}^{n} f_i |x_i - \overline{x}| = 400$
• The mean deviation: $\text{MD} = 10$

Step 2: Substitute the given values into the Mean Deviation formula.
$$ 10 = \frac{400}{\sum_{i=1}^{n} f_i} $$
Step 3: Solve for the total frequency sum.
Rearrange the equation to isolate $\sum_{i=1}^{n} f_i$: $$ \sum_{i=1}^{n} f_i = \frac{400}{10} $$ $$ \sum_{i=1}^{n} f_i = 40 $$