Question

The mean of the data : 4, 7, $x$, 13, 16 is 10. Then the mean deviation of the data is

• A. 3
• B. 3.3
• C. 3.6
• D. 3.8
• E. 4

Hint:

Mean deviation always uses absolute values. Ensure you don't subtract without the modulus, or the sum of deviations will always be zero!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given an incomplete dataset and its mean. First, we must find the missing value \(x\).
Then, we calculate the mean deviation, which is the average of the absolute differences between each data point and the mean.
Step 2: Key Formula or Approach:
Mean: \(\mu = \frac{\sum x_i}{n}\).
Mean Deviation (M.D.) = \(\frac{\sum |x_i - \mu|}{n}\).
Step 3: Detailed Explanation:
Find the value of \(x\):
Given \(\mu = 10\) and \(n = 5\).
\[ \frac{4 + 7 + x + 13 + 16}{5} = 10 \] \[ \frac{40 + x}{5} = 10 \] \[ 40 + x = 50 \implies x = 10 \] The complete data set is \(4, 7, 10, 13, 16\).
Calculate the absolute deviations from the mean (\(\mu = 10\)):
\(|4 - 10| = 6\)
\(|7 - 10| = 3\)
\(|10 - 10| = 0\)
\(|13 - 10| = 3\)
\(|16 - 10| = 6\)
Sum of absolute deviations = \(6 + 3 + 0 + 3 + 6 = 18\).
Calculate the Mean Deviation:
\[ \text{M.D.} = \frac{18}{5} = 3.6 \] Step 4: Final Answer:
The mean deviation is 3.6.