Standard Deviation

Standard deviation is a measure of the dispersion or spread of values in a dataset around the mean. A low standard deviation indicates that values tend to cluster closely around the mean, while a high standard deviation indicates that values are widely scattered. It is the most commonly used measure of variability in descriptive and inferential statistics and is calculated as the square root of the variance.

Standard deviation provides essential context for interpreting the mean. A mean blood pressure of 130 mmHg in a sample has very different implications if the standard deviation is 5 mmHg (about 95% of values between 120 and 140, if the data are roughly normal) versus 25 mmHg (about 95% between 80 and 180). Standard deviation is a key input for sample size calculations, confidence interval construction, effect size computation, and meta analysis. It is also central to understanding normal distributions, where approximately 68% of values fall within one standard deviation of the mean and 95% within two standard deviations.

The standard deviation is calculated in several steps. First, the mean of the data is computed. Second, each value's deviation from the mean is calculated (value minus mean). Third, each deviation is squared to eliminate negative values. Fourth, the squared deviations are averaged (dividing by n for the population standard deviation or by n minus 1 for the sample standard deviation, where the n minus 1 correction, known as Bessel's correction, accounts for the loss of one degree of freedom when estimating the population parameter from a sample). Fifth, the square root of the average squared deviation is taken, returning the measure to the original units of the data.

Dividing by n minus 1 makes the sample variance an unbiased estimator of the population variance; its square root, the sample standard deviation, still slightly underestimates the population standard deviation on average, but it is the version usually reported in research. It is denoted as s or SD, while the population standard deviation is denoted as sigma.

Standard deviation is in the same units as the original data, making it directly interpretable. If a test score has a mean of 100 and a standard deviation of 15, values between 85 and 115 encompass about 68% of the distribution (assuming approximate normality).

Report the standard deviation alongside the mean whenever summarizing continuous data. Use it for computing confidence intervals, effect sizes (such as Cohen's d, which divides the mean difference by the pooled standard deviation), and sample size calculations. Standard deviation is appropriate for interval and ratio scale data. For heavily skewed distributions, the median and interquartile range may be more informative summary statistics than the mean and standard deviation.

A common error is confusing the standard deviation with the standard error. The standard deviation describes the variability of individual observations, while the standard error describes the variability of the sample mean as an estimate of the population mean. Another mistake is reporting the standard deviation for data that are not approximately normally distributed, where it may not adequately characterize the spread. Researchers sometimes report the standard deviation without the mean, which makes the SD uninterpretable. Finally, using the population formula (dividing by n) instead of the sample formula (dividing by n minus 1) produces a slight underestimate of the true variability.

A study of resting heart rate in 500 healthy adults reports a mean of 72 beats per minute with a standard deviation of 10 beats per minute. Assuming approximate normality, approximately 68% of participants have resting heart rates between 62 and 82 bpm, and approximately 95% between 52 and 92 bpm. The standard error of the mean is 10 divided by the square root of 500, or approximately 0.45, yielding a 95% confidence interval for the population mean of approximately 71.1 to 72.9 bpm.