Standard Deviation and Variance

The problem with averages alone

Imagine two classes take the same test. Both have an average score of 70. But in Class A, every student scored between 65 and 75. In Class B, half scored 40 and half scored 100.

Same average. Completely different story.

This is why spread matters. Variance and standard deviation are the tools we use to measure it.

Variance explorer

50.6

Mean

85.6

Variance

9.25

Std dev

Spread (std dev)10

230

Outliers0

04

What variance actually measures

Variance captures how far values typically sit from the mean — but it measures that distance in a specific way: as the average of the squared deviations, not the raw distances.

For a set of values, you:

1. Find the mean
2. Subtract the mean from each value (the deviation)
3. Square each deviation (so negatives don't cancel out)
4. Average those squared deviations

$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$

Standard deviation: back to the original units

Variance is in squared units — hard to interpret. Taking the square root brings you back to the original scale.

$\sigma = \sqrt{\sigma^2}$

If test scores have a standard deviation of 15 points, it tells you the typical distance from the mean is about 15 points. For a roughly bell-shaped distribution, that also means most scores fall within 15 points of the mean — but how much "most" is depends on the shape (see the 68–95–99.7 rule below).

Why this matters

• Finance: volatility of a stock is its standard deviation of returns
• Quality control: tight manufacturing tolerances mean low standard deviation
• Science: high standard deviation in measurements suggests noisy data

A low standard deviation doesn't mean good. It means consistent. Sometimes you want high variance — think brainstorming, not precision manufacturing.

$\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}$

Key takeaways

• Variance = average squared distance from the mean
• Standard deviation = square root of variance (same units as data)
• High std dev → values are spread out; low std dev → values cluster around the mean
• Always look at spread and center, never just the average