Question

The mean deviation about the mean from the data 400, 410, 420, 430, 440 is:

• A. 14
• B. 10
• C. 20
• D. 12
• E. 16

Hint:

The mean of an arithmetic progression is always the middle term if the number of terms is odd.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Mean deviation about the mean is a measure of dispersion. It is the average of the absolute differences between each data point and the mean of the data set.
Step 2: Key Formula or Approach:
1. Calculate the mean (\( \bar{x} \)) of the data set.
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \] 2. Calculate the mean deviation (M.D.) using the formula:
\[ M.D.(\bar{x}) = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n} \] where \( x_i \) are the data points and n is the number of data points.
Step 3: Detailed Explanation:
The given data set is \( \{400, 410, 420, 430, 440\} \). The number of data points is \( n=5 \).
Step 3a: Calculate the mean (\( \bar{x} \))
The data is an arithmetic progression, so the mean is the middle value.
\[ \bar{x} = 420 \] Alternatively, by calculation: \[ \bar{x} = \frac{400 + 410 + 420 + 430 + 440}{5} = \frac{2100}{5} = 420 \] Step 3b: Calculate the absolute deviations from the mean
We need to find \( |x_i - \bar{x}| \) for each data point.
- \( |400 - 420| = |-20| = 20 \)
- \( |410 - 420| = |-10| = 10 \)
- \( |420 - 420| = |0| = 0 \)
- \( |430 - 420| = |10| = 10 \)
- \( |440 - 420| = |20| = 20 \)
Step 3c: Calculate the sum of the absolute deviations
\[ \sum |x_i - \bar{x}| = 20 + 10 + 0 + 10 + 20 = 60 \] Step 3d: Calculate the mean deviation
\[ M.D.(\bar{x}) = \frac{\sum |x_i - \bar{x}|}{n} = \frac{60}{5} = 12 \] Step 4: Final Answer:
The mean deviation about the mean is 12.