Question

Mean & Variance of 7 observations are 8 & 16 respectively, if number 14 is omitted then a & b are new mean & variance. The value of a + b is

Answer 1

Correct Answer: 19

Given Information:

• The mean of 7 observations is 8.
• The variance of 7 observations is 16.
• One observation, 14, is omitted.

We need to find the new mean a and variance b of the remaining 6 observations, and then calculate a + b.

 

Step 1: Calculate the Sum of the 7 Observations

The mean of 7 observations is given by: 
μ = sum of observations / 7 
Given that the mean is 8: 
8 = sum of observations / 7 
Thus, the sum of the observations is: 
sum of observations = 8 × 7 = 56

Step 2: Calculate the Sum of Squared Deviations for the 7 Observations

The formula for variance is: 
σ² = Σ(xᵢ - μ)² / 7 
We are told that the variance is 16, so: 
16 = Σ(xᵢ - 8)² / 7 
Multiplying both sides by 7: 
Σ(xᵢ - 8)² = 16 × 7 = 112 
This represents the sum of squared deviations of the 7 observations from the mean.

Step 3: Omit the Observation 14

When the number 14 is omitted, we are left with 6 observations. We need to find the new sum of squared deviations and the new mean for these 6 observations.

New Mean (a):

After removing the observation 14, the new sum of the remaining 6 observations is: 
new sum of observations = 56 - 14 = 42 
The new mean of the remaining 6 observations is: 
a = new sum of observations / 6 = 42 / 6 = 7

New Variance (b):

To calculate the new variance, we first subtract the squared deviation of 14 from the total sum of squared deviations. The squared deviation of 14 from the mean is: 
(14 - 8)² = 6² = 36 
So, the new sum of squared deviations for the remaining 6 observations is: 
Σ(xᵢ - 8)² = 112 - 36 = 76

Now, the new variance b is: 
b = 76 / 6 = 38 / 3 ≈ 12.67

Step 4: Calculate a + b

Now, we calculate a + b, where a = 7 and b = 38 / 3. 
a + b = 7 + 38 / 3 = 21 / 3 + 38 / 3 = 59 / 3 
This simplifies to approximately: 
a + b ≈ 19.67