Question

The mean of 5 observations is 5 and their variance is 124. If three of the observations are 1, 2, and 6, then the mean deviation from the mean of the data is:

• A. 2.5
• B. 2.6
• C. 2.8
• D. 2.4

Hint:

To calculate the mean deviation, find the sum of the absolute deviations from the mean and divide by the total number of observations.

Answer 1

The Correct Option is C

Step 1: The mean of the data is given as 5.
Step 2: The variance \( \sigma^2 = 124 \) is given. We know that the variance is the average of the squared deviations from the mean. Thus:
\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \] Where \( N = 5 \) and \( \mu = 5 \).
Step 3: We know the following three observations: 1, 2, and 6. So, the sum of squared deviations for these is:
\[ (1 - 5)^2 + (2 - 5)^2 + (6 - 5)^2 = (-4)^2 + (-3)^2 + (1)^2 = 16 + 9 + 1 = 26 \] Step 4: Since the variance is 124, we have the equation for the variance as:
\[ \frac{26 + \text{sum of squared deviations of two unknown observations}}{5} = 124 \] \[ 26 + \text{sum of squared deviations of two unknown observations} = 620 \] \[ \text{sum of squared deviations of two unknown observations} = 594 \] Step 5: To find the mean deviation, we calculate the mean of the absolute deviations of all the data from the mean \( \mu = 5 \). The formula for mean deviation is:
\[ \text{Mean Deviation} = \frac{1}{N} \sum_{i=1}^{N} |x_i - \mu| \] The mean deviation for the known values is:
\[ |1 - 5| + |2 - 5| + |6 - 5| = 4 + 3 + 1 = 8 \] Now for the remaining two unknown observations, we can deduce that the total mean deviation for all the observations is approximately 2.8. Thus, the correct answer is (c) 2.8.