Variance

Variance measures how spread out a set of numbers / results is - how varied they are.

It’s a better measurement of spread than the range, because it takes into account all the values in the set, not just the absolute extremes.

Variance symbol

Because variance is just the standard deviation squared, we write it as \sigma^2 as \sigma is the symbol for standard deviation.

\text{variance:}\quad\sigma^2

Calculating variance

To calculate variance, we first need to calculate the mean of the set of numbers.

Then, for each number in the set, we calculate the difference between that number and the mean, and square that difference.

Then finally, we average that sum of squared differences by dividing it by the number of values in the set (to find the mean).

Formula

There are three formulas we can use, they’re the same thing, but sometimes one is easier than the others, depending on what we know.

Calculating from mean of squares and square of mean

If we know the mean of squares (the average of the squares of the numbers) - which we write as \overline{x^2} - and the square of the mean (the square of the average of the numbers) - which we write as \overline{x}^2 - then we can calculate the variance using this formula:

\sigma^2 = \overline{x^2} - \overline{x}^2

(Remember that the \overline{\quad} (bar) symbol means “mean of”

Calculating from just the values

If we just have the values, then we can calculate the variance using this formula:

\sigma^2 = \frac{\sum(x-\overline{x})^2}{n}

That does mean we need to calculate the difference between the mean and each value. We can rearrange it to stop us having to do this:

\sigma^2 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2

Which, actually, is just the same as the first formula! We’ve just substituted \overline{x^2} for \frac{\sum x^2}{n} and \overline{x}^2 for \left(\frac{\sum x}{n}\right)^2 into our formula of \sigma^2 = \overline{x^2} - \overline{x}^2.

flashcards

Question	Answer
What symbol represents variance?	\sigma^2
How does variance compare to range as a measure of spread?	Variance is a better measurement because it takes into account all values in the set, not just the absolute extremes.
What is the relationship between variance and standard deviation?	Variance is the standard deviation squared, written as \sigma^2.
What is the formula for variance using the mean of squares and square of the mean?	\sigma^2 = \overline{x^2} - \overline{x}^2, where \overline{x^2} is the mean of squares and \overline{x}^2 is the square of the mean.
What is the formula for variance when calculating directly from values?	\sigma^2 = \frac{\sum(x-\overline{x})^2}{n}
What is the rearranged formula for variance that avoids calculating differences from the mean?	\sigma^2 = \frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2
How do you calculate variance step by step?	1. Calculate the mean of the set. 2. For each number, calculate the difference between it and the mean, then square it. 3. Average the sum of squared differences by dividing by the number of values.