Question

For a data containing 100 observations, the mean and variance are 8 and 10.5 respectively. For 50 observations selected from these 100 observations, the mean and variance are 10 and 4 respectively. Find the mean and standard deviation of the other 50 observations.

Answer 1

Mean for other 50 observations is 6 and standard deviation  is 3

Given:

• A data containing 100 observations
• Mean = 8 and Variance = 10.5
• 50 Observations selected
• Mean = 10 and Variance is 4

To Find:

• Mean and standard deviation of the other 50 observations.

Solution:

• Variance = (Standard deviation)²
• Variance = ∑x² / N  - (Mean)²
• Mean = (sum of observations)/(number of observations)

Step 1:

Calculate ∑x²  for 100 observations

10.5  = ∑x² / 100   - (8)²

=> 10.5 = ∑x² / 100 - 64

=> 74.5 =  ∑x² / 100

=>  ∑x² = 7450

Step 2:

Calculate ∑x²  for 50 observations

4  = ∑x² /50   - (10)²

=> 4= ∑x² / 50 - 100

=> 104 =  ∑x² / 50

=>  ∑x² = 5200

Step 3:

Calculate ∑x²  for Remaining 50 observations

∑x²  = 7450 - 5200  = 2250

Step 4:

Calculate Mean for remaining 50 observations

( 8 x 100  - 10 x 50)/50  = 6

Step 5:

Calculate Variance for remaining observation

Variance = 2250/50  - 6²

Variance = 45  - 36

Variance = 9

Step 6:

Calculate Standard deviation

Standard deviation² = 9

Standard deviation = 3

Mean for other 50 observations is 6 and standard deviation  is 3