Variance and Standard Deviation

Ages 16–18

Difficulty adapts to you

Variance and Standard Deviation

For a population of n ungrouped values, variance σ² = Σ(x − x̄)² ÷ n and standard deviation σ = √σ². An equivalent calculation formula is σ² = (Σx² ÷ n) − x̄². For a sample, use s² = Σ(x − x̄)² ÷ (n − 1) when the question specifies sample statistics. For grouped data, use midpoints: σ² ≈ (Σfm² ÷ Σf) − x̄². A larger standard deviation means more spread in the same units as the data.

Your goals

✓You can measure how far values spread from their mean for ungrouped and grouped data.
✓You can use Variance and Standard Deviation, model correctly.
✓You can use the definitions, units, assumptions, and examples to check your answer.

Words to know

Variance and Standard Deviation:: Measure how far values spread from their mean for ungrouped and grouped data.
model:: A rule or representation used to describe a situation and make a prediction.

Examples

1For 2, 4, 6, x̄ = 4; squared deviations are 4, 0, 4, so variance = 8 ÷ 3 and standard deviation = √(8 ÷ 3).
2When every value is increased by 5, the mean increases by 5 but the standard deviation stays the same.

Check these

• Do not skip the meaning of a term or give a result without explaining what it means.
• Keep every number, symbol, label, and unit in its correct place.
• Show enough working to check how you reached your answer.

Helpful habits

• Read the whole question before choosing what to do.
• Show one clear step at a time and check each step.
• If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own variance and standard deviation example, solve it, and explain how you checked it.

Professor Wise Owl

Wisdom & Understanding

“Mistakes are proof that you are trying. Keep going!”