Formulas Used
Mean: sum / N. Population Variance (σ²): sum of (xi - mean)² / N. Sample Variance (s²): sum of (xi - mean)² / (N-1). Standard Deviation: square root of variance. Use population SD when data represents the full population; use sample SD when data is a sample drawn from a larger population.
Frequently Asked Questions
Standard deviation measures how spread out values are around the mean. A low standard deviation means values are clustered close to the mean; a high standard deviation means values are spread far from the mean. It is the square root of the variance.
Population standard deviation (σ) is used when you have data for every member of a group and divides by N. Sample standard deviation (s) is used when your data is a subset (sample) of a larger population and divides by N-1 (Bessel's correction). If you're analyzing a dataset representing a full population, use σ. If it's a sample, use s.
The value of a standard deviation is only meaningful relative to the mean and the units of the data. A standard deviation of 1 in a dataset with a mean of 100 is very small (1% spread). A standard deviation of 1 in a dataset with a mean of 2 is relatively large (50% spread). The coefficient of variation (SD/mean × 100%) helps compare spread across different scales.
For data that follows a normal (bell curve) distribution, approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule is used to identify outliers and understand the distribution of data.
Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance, which brings the units back to the original scale. Variance is useful mathematically (it has additive properties) while standard deviation is more interpretable because it's in the same units as the data.
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