Variance is a number that describes how spread out a dataset is around its mean. A small variance means the values cluster close to the mean, while a large variance means the values are more widely spread. It is useful in science, finance, sports, and many other fields because it turns the idea of variability into a measurable quantity.
Learning the step-by-step method helps students see why each part of the formula matters.
Key Facts
- Mean: x̄ = (sum of values) / n
- Deviation from the mean: deviation = x - x̄
- Squared deviation: (x - x̄)^2
- Population variance: σ^2 = Σ(x - μ)^2 / N
- Sample variance: s^2 = Σ(x - x̄)^2 / (n - 1)
- Variance is measured in squared units, so standard deviation is often used for interpretation: standard deviation = √variance
Vocabulary
- Variance
- Variance is the average squared distance of data values from the mean.
- Mean
- The mean is the sum of all data values divided by the number of values.
- Deviation
- A deviation is the difference between a data value and the mean.
- Population variance
- Population variance measures spread when the dataset includes every member of the group being studied.
- Sample variance
- Sample variance estimates the spread of a larger population using only a sample and divides by n - 1.
Common Mistakes to Avoid
- Forgetting to square the deviations is wrong because positive and negative deviations can cancel out, hiding the true spread.
- Dividing by n for sample variance is wrong because sample variance uses n - 1 to better estimate population spread from limited data.
- Using the original values instead of deviations is wrong because variance measures distance from the mean, not distance from zero.
- Interpreting variance in the original units is wrong because variance is in squared units, so a variance of 16 square centimeters is not the same type of measurement as 16 centimeters.
Practice Questions
- 1 Compute the population variance of the dataset 2, 4, 6, 8. Show the mean, deviations, squared deviations, and final variance.
- 2 Compute the sample variance of the dataset 5, 7, 10, 14. Show the mean, squared deviations, sum of squared deviations, and division by n - 1.
- 3 Two datasets have the same mean: A = 9, 10, 11 and B = 2, 10, 18. Explain which dataset has the larger variance and why.