Q.The following information relates to a sample of size 60: $\sum x^2 = 18000$, $\sum x = 960$ The variance is
(A) 6.63
(B) 16
(C) 22
(D) 44

The variance measures the average squared deviation from the mean. We need the mean first, then apply the computational formula.

Step 1. Find the mean:

Step 2. Use the computational formula for variance:

Step 3. Substitute the values: …

The variance quantifies how spread out the data points are from their mean. While the definitional formula $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$ captures the concept directly—average of squared deviations—it's computationally tedious when you have raw data summaries. That's where the computational formula shines: it lets you calculate variance directly from $\sum x$ and $\sum x^2$ without finding every individual deviation.

The computational formula is:

This expands to "mean of squares minus square of mean," a pattern worth memorizing for quick calculations.

Let me work through this step by step with the given information.

1. Identify what we know

We have:

• Sample size: $n = 60$
• Sum of observations: $\sum x = 960$
• Sum of squared observations: $\sum x^2 = 18000$

2. Calculate the mean

The mean is simply the sum divided by the count:

3. Calculate the mean of squares

This is the average of all the squared values:

4. Apply the computational formula

Now substitute into the variance formula: …