Variance and Standard Deviation

Statistics · Axiom Academy

LESSON Variance and Standard Deviation Understanding how to measure data spread in populations and samples Variance measures the average squared distance of each data point from the mean. By squaring the differences, we ensure all values are positive and give more weight to outliers. Where: N = total number of values in the population, x i = each individual value, = population mean 2. Population vs Sample Variance The key difference between population and sample variance lies in the denominator. Sample variance uses (n - 1) instead of n to correct for bias when estimating population parameters from sample data. The (n - 1) denominator is called Bessel's correction and provides an unbiased estimate of population variance. 3. Standard Deviation: Bringing It Back to Scale Standard deviation is simply the square root of variance. This operation returns our measure of spread to the same units as the original data, making it more interpretable. Population Standard Deviation: 4. Computing Variance and Standard Deviation Let's compute both measures for a sample dataset: 4, 8, 6, 5, 3, 7 Step 1: Calculate the mean: x̄ = (4 + 8 + 6 + 5 + 3 + 7) / 6 = 33 / 6 = 5.5 Step 2: Find squared deviations: Step 3: Sum of squared deviations = 17.5 Step 4: Sample variance: s 2 = 17.5 / (6 - 1) = 17.5 / 5 = 3.5 Step 5: Sample standard deviation: s = 3.5 = 1.87

This is the written version of the interactive lesson above. See the full Statistics course.