Question

For four observations X,X2,X3,X4, it is given that ,\(∑x_i^{2}=656\) and  \(∑x_i=32\). Then, the variance of these four observations is ?

• A. \(144\)
• B. \(730\)
• C. \(120\)
• D. \(248\)
• E. \(182.5\)
• F. \(100\)

Answer 1

Answer: (F)

Given:

• Four observations: \( x_1, x_2, x_3, x_4 \)
• Sum of squares: \( \sum_{i=1}^4 x_i^2 = 656 \)
• Sum of observations: \( \sum_{i=1}^4 x_i = 32 \)

 

We need to find the variance of these observations.

Step 1: Recall the variance formula for a population: \[ \text{Variance} = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \]

Step 2: Plug in the given values (n=4): \[ \text{Variance} = \frac{656}{4} - \left(\frac{32}{4}\right)^2 = 164 - 64 = 100 \]

Answer 2

1. Calculate the mean:

The mean (\( \mu \)) is the sum of the observations divided by the number of observations:

\[ \mu = \frac{\Sigma x_i}{n} = \frac{32}{4} = 8 \]

2. Calculate the population variance:

The population variance (\( \sigma^2 \)) is the sum of the squared differences from the mean, divided by the number of observations:

\[ \sigma^2 = \frac{\Sigma (x_i - \mu)^2}{n} \]

We can use the computational formula:

\[ \sigma^2 = \frac{\Sigma x_i^2 - \frac{(\Sigma x_i)^2}{n}}{n} \]

Substituting the given values:

\[ \sigma^2 = \frac{656 - \frac{(32)^2}{4}}{4} = \frac{656 - 256}{4} = \frac{400}{4} = 100 \]

Therefore, the population variance is 100.