Variance and standard deviation are two ways of measuring the same thing: how spread out the values in a dataset are around the mean. They're calculated from the same underlying formula, but variance squares the deviations while standard deviation takes the square root of that result to bring the units back to the original scale.
For a dataset of heights measured in centimeters, variance would be expressed in cm², which has no intuitive meaning. Standard deviation would be expressed in centimeters — the same unit as the data itself — making it far easier to communicate. A standard deviation of 8 cm means most values fall within roughly 8 cm of the mean; a variance of 64 cm² tells you the same thing, but the unit makes it harder to reason about.
The calculation: population vs sample
For a population (every member of the group you care about), variance is calculated as the average of squared deviations from the mean: σ² = Σ(xᵢ − μ)² / N. For a sample (a subset used to estimate the population), you divide by N−1 instead of N, giving s² = Σ(xᵢ − x̄)² / (N−1). This N−1 correction, called Bessel's correction, compensates for the fact that a sample mean is closer to its own data points than the true population mean is, which causes sample variance to underestimate population variance without the adjustment.
Standard deviation is simply the square root of the appropriate variance: σ for populations, s for samples. The calculation steps are identical up to the final square root.
When to use variance vs standard deviation
Use standard deviation when you need to communicate spread in plain language — in reports, summaries, or anywhere non-statisticians will read the number. Its same-unit property makes statements like 'scores were within one standard deviation of the mean' immediately meaningful.
Variance is more useful in statistical theory and when combining independent sources of variability. Because the variance of the sum of independent random variables equals the sum of their variances (a property that does not hold for standard deviations), variance is the natural measure for things like ANOVA, regression error decomposition, and portfolio risk calculations. In finance, for example, portfolio variance is additive when assets are independent, which is why it's used in the math even though standard deviation (called volatility in that context) is what gets reported to investors.