Q.Find the variance of the following data: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24

The key idea is that variance is the average of the squared deviations from the mean.

First, find the mean of the data. The numbers are evenly spaced, so the mean is the average of the first and last terms:

Now, find the deviations from the mean and square them. Since the data is symmetric, we can compute efficiently:

Why Mean Deviation About Mean?

Variance measures how spread out numbers are. The most natural way to think about spread is: "how far is each number from the center?" The center we use is the mean (average). If we just averaged the distances (deviations), positive and negative ones would cancel out — so we square them first. That gives us the mean squared deviation, which is exactly the variance.

For a population (which this ungrouped data represents), the formula is:

Where $\bar{x}$ is the mean and $N$ is the number of observations.

Step-by-step solution

1. Find the mean ($\bar{x}$)

The data: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24.

There are $N = 10$ numbers.

Notice they form an arithmetic progression with common difference 2. For an AP, the mean equals the average of the first and last terms:

This shortcut works because the data is symmetric around the middle. Let's verify by direct sum anyway:

Sum = $6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 = 150$

$\bar{x} = 150 / 10 = 15$. Confirmed.

2. Compute deviations from the mean

For each $x_i$, find $x_i - \bar{x}$:

Notice the symmetry: deviations are $-9, -7, -5, -3, -1, 1, 3, 5, 7, 9$. They sum to zero (always true for deviations from the mean), which is why we need to square them.

3. Square each deviation