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The variance of 6, 7, 5, 9, 12, 15 is calculated as follows. Mean = (6+7+5+9+12+15) / 6 = 9. Squared deviations from the mean: (6−9)²=9, (7−9)²=4, (5−9)²=16, (9−9)²=0, (12−9)²=9, (15−9)²=36. Sum of squares = 74. Population variance (σ²) = 74 / 6 ≈ 12.33. Sample variance (s²) = 74 / 5 = 14.8. Use population variance if these 6 numbers are the whole data set. Use sample variance if they represent sampled data from a larger group.

To calculate variance of 1, 2, 3, 4, 5: Mean = (1+2+3+4+5) / 5 = 3. Deviations from the mean: −2, −1, 0, 1, 2. Squared deviations: 4, 1, 0, 1, 4. Sum of squares = 10. Population variance = 10 / 5 = 2. Sample variance = 10 / 4 = 2.5. Standard deviation is √2 ≈ 1.414 (population) or √2.5 ≈ 1.581 (sample). This data set has low variance because the values are evenly spaced — common in educational assessments and sensitivity analysis.

To calculate variance in Excel, use VAR.S(range) for sample variance and VAR.P(range) for population variance. Example: =VAR.S(A1:A10) returns the sample variance for cells A1 through A10. Google Sheets uses the same function names. In R, var(x) returns sample variance by default. In Python with NumPy, use np.var(data, ddof=1) for sample variance. SPSS, Minitab, JMP, SAS, and StatCrunch also offer built-in variance functions. On a TI-84 calculator, go to STAT → CALC → 1-Var Stats to see both σ² and s².

To calculate sample variance, use the formula s² = Σ(xᵢ − x̄)² / (n − 1). Steps: find the sample mean (x̄), subtract it from each data value, square each result, sum all squared deviations, then divide by n − 1 (not n). The n − 1 denominator is Bessel's correction — it adjusts for the lost degree of freedom when estimating the population mean from sampled data. Sample variance is used in surveys, clinical trials, experimental design, and machine learning feature selection — any situation where your data set is a subset of a larger population.

To calculate variance as a percentage, use the coefficient of variation (CV): CV = (standard deviation / mean) × 100%. Example: if variance = 25, standard deviation = √25 = 5. If the mean is 50, then CV = (5 / 50) × 100% = 10%. The CV is useful for comparing spread across data sets with different scales — common in financial forecasting, portfolio risk analysis, and quality control. A lower CV means less relative variability.

Variance in statistics is a measure of how far numbers in a data set spread from the mean. It is the average of the squared differences from the mean. High variance means data points are far apart; low variance means they cluster near the mean. Variance is the foundation for standard deviation, which uses the same units as the original data. Both population variance and sample variance serve as inputs to hypothesis testing, regression analysis, and analysis of variance (ANOVA).

Sample variance (s²) divides the sum of squared deviations by n − 1 (Bessel's correction). Population variance (σ²) divides by n. Use sample variance when working with sampled data from a larger group — surveys, experiments, clinical trials. Use population variance when the data set covers the whole population — census data, full class test scores, complete batch measurements. The formulas are otherwise identical.

The sample variance formula is s² = Σ(xᵢ − x̄)² / (n − 1). The population variance formula is σ² = Σ(xᵢ − μ)² / n. In both formulas, xᵢ represents each data value, x̄ or μ is the mean, Σ is the summation operator, and n is the count. The variance of a discrete random variable uses a weighted version: Var(X) = Σ[xᵢ² · P(xᵢ)] − μ².

Standard deviation is the square root of variance. Variance gives the spread in squared units. Standard deviation converts it back to the original units. Population: σ = √σ². Sample: s = √s². Both variance and standard deviation measure data spread, but standard deviation is easier to interpret because it shares the same scale as the data. In financial forecasting, standard deviation of returns is called volatility.

No, variance can never be negative. The formula squares every deviation from the mean, and squares are always zero or positive. A variance of 0 means all values in the data set are identical — there is no spread at all. Any non-zero data spread produces a positive variance value.

Dividing by n − 1 is called Bessel's correction. When you estimate variance from a sample, using n as the divisor systematically underestimates the true population variance. The n − 1 divisor corrects this bias and provides an unbiased variance estimate. The number n − 1 represents the degrees of freedom — you lose one degree of freedom because the sample mean constrains the data. This correction matters most for small samples and becomes negligible for large n.

For population variance, you need at least 1 data point (the variance will be 0). For sample variance, you need at least 2 data points because n − 1 = 0 when n = 1, and division by zero is undefined. In practice, more data points produce a more reliable variance estimate — especially for skewed distributions or data with missing values after outlier removal.

The coefficient of variation (CV) equals (standard deviation / mean) × 100%. It measures relative variability — how large the spread is compared to the average. CV is useful when comparing variance across time series, weighted datasets, or data sets measured in different units. In quality control, a low CV signals consistent output. In portfolio risk analysis, CV helps compare investments with different expected returns.