Variance

Definition

• The variance is a measure of variability. It is calculated by taking the average of squared deviations from the mean.
• Variance tells you the degree of spread in your data set. The more spread the data, the larger the variance is in relation to the mean.

Variance vs. standard deviation

The standard deviation is derived from variance and tells you, on average, how far each value lies from the mean. It’s the square root of variance.

Both measures reflect variability in a distribution, but their units differ:

• Standard deviation is expressed in the same units as the original values (e.g., meters).
• Variance is expressed in much larger units (e.g., meters squared)

Since the units of variance are much larger than those of a typical value of a data set, it’s harder to interpret the variance number intuitively. That’s why standard deviation is often preferred as a main measure of variability.

However, the variance is more informative about variability than the standard deviation, and it’s used in making statistical inferences.

Population vs. sample variance

Different formulas are used for calculating variance depending on whether you have data from a whole population or a sample.

Population variance

When you have collected data from every member of the population that you’re interested in, you can get an exact value for population variance.

The population variance formula looks like this:

\[\sigma^2 = \frac{\Sigma(X - \mu)^2}{N}$$ $$\mathcal{Where,} \ \sigma^2 = population\ variance $$ $$\mu = population\ mean $$ $$N = number \ of \ records \ in\ population\]

Sample variance

When you collect data from a sample, the sample variance is used to make estimates or inferences about the population variance.

The sample variance formula looks like this:

\[s^2 = \frac{\Sigma(X - \hat{x})^2}{n - 1}$$ $$\mathcal{Where,} \ s^2 = sample\ variance $$ $$\hat{x} = sample\ mean $$ $$n = number \ of \ records \ in\ sample\]

With samples, we use n – 1 in the formula because using n would give us a biased estimate that consistently underestimates variability. The sample variance would tend to be lower than the real variance of the population.

Reducing the sample n to n – 1 makes the variance artificially large, giving you an unbiased estimate of variability: it is better to overestimate rather than underestimate variability in samples.

It’s important to note that doing the same thing with the standard deviation formulas doesn’t lead to completely unbiased estimates. Since a square root isn’t a linear operation, like addition or subtraction, the unbiasedness of the sample variance formula doesn’t carry over the sample standard deviation formula.

Steps

1. Find the mean: To find the mean, add up all the scores, then divide them by the number of scores.
2. Find each score’s deviation from the mean: Subtract the mean from each score to get the deviations from the mean. Since x̅ = 50, take away 50 from each score.
3. Square each deviation from the mean: Multiply each deviation from the mean by itself. This will result in positive numbers.
4. Find the sum of squares: Add up all of the squared deviations. This is called the sum of squares.
5. Divide the sum of squares by n – 1 or N: Divide the sum of the squares by n – 1 (for a sample) or N (for a population). Since we’re working with a sample, we’ll use  n – 1, where n = 6.

Variance matters for two main reasons:

• Parametric statistical tests are sensitive to variance.
• Comparing the variance of samples helps you assess group differences.