The variance of a set of observations measures the spread or dispersion of the data points around the mean. It is calculated by taking the average of the squared differences between each observation and the mean.

Given that the variance of 25 observations is 4, we can use the formula for variance to find the mean of the observations.

Let's assume the mean of the original observations is μ. Then, the variance is given by:

Variance = (Sum of squared differences from the mean) / (Number of observations)

4 = (Sum of (xᵢ - μ)²) / 25

Multiplying both sides of the equation by 25, we get:

100 = Sum of (xᵢ - μ)²

This equation tells us that the sum of the squared differences from the mean is 100.

Now, let's consider what happens when we add 2 to each observation. Let's call the new observations xᵢ'.

The new observations are given by:

xᵢ' = xᵢ + 2

To find the new variance, we need to calculate the sum of the squared differences from the new mean, which we'll call μ'.

Using the formula for variance with the new observations:

Variance' = (Sum of squared differences from the new mean) / (Number of observations)

We want to find the new variance, so let's substitute the new observations into the formula:

Variance' = (Sum of (xᵢ' - μ')²) / 25

Expanding the squared differences:

Variance' = (Sum of (xᵢ + 2 - μ')²) / 25

Now, let's consider the relationship between the new observations and the original observations:

xᵢ' = xᵢ + 2

This means that the new mean is the original mean plus 2:

μ' = μ + 2

Substituting this into the formula for the new variance:

Variance' = (Sum of (xᵢ + 2 - (μ + 2))²) / 25

Simplifying the expression inside the sum:

Variance' = (Sum of (xᵢ - μ)²) / 25

But we know that the sum of the squared differences from the mean is 100, which remains the same regardless of the constant added. So:

Variance' = 100 / 25 = 4

Therefore, adding 2 to each observation does not change the variance. The new variance remains 4, which corresponds to option B.