Sample Standard Deviation
s = โ[ฮฃ(xแตข - xฬ)ยฒ / (n - 1)]
The sample standard deviation measures the average amount of variability or spread in a data set. It quantifies how far individual observations tend to fall from the sample mean. The denominator uses n - 1 (Bessel's correction) to provide an unbiased estimate of the population variance.
Variables
A measure of the spread of the data around the sample mean
Each data value in the sample
The arithmetic average of all observations
The total number of observations in the sample
Example Calculation
Scenario
Five measurements of a chemical solution's pH are: 7.2, 7.5, 7.1, 7.4, and 7.3. Calculate the sample standard deviation.
Given Data
Calculation
s = โ(0.10 / 4) = โ0.025
Result
s = 0.158
Interpretation
The pH measurements vary by about 0.158 units from the mean of 7.3. This relatively small standard deviation indicates the measurements are consistent and tightly clustered around the average.
When to Use This Formula
- โMeasuring the variability or dispersion in a sample
- โConstructing confidence intervals for the population mean
- โPerforming t-tests and other inferential procedures
- โComparing the spread of different data sets
Common Mistakes
- โDividing by n instead of n - 1 when computing the sample standard deviation
- โForgetting to take the square root after computing the variance
- โConfusing standard deviation with standard error (s/โn)
- โInterpreting standard deviation as a percentage without computing the coefficient of variation
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Common questions about this formula
Dividing by n - 1 is called Bessel's correction. When we use the sample mean in place of the unknown population mean, we lose one degree of freedom. Dividing by n would systematically underestimate the population variance. Using n - 1 produces an unbiased estimate of the population variance.
Variance (sยฒ) is the average of the squared deviations from the mean. Standard deviation (s) is the square root of the variance. Standard deviation is in the same units as the original data, making it easier to interpret, while variance is in squared units.