How to Calculate Standard Deviation: Step-by-Step

Standard deviation is one of the most important measures in statistics. It tells you how spread out the values in a data set are relative to the mean. Whether you are analyzing test scores, stock returns, manufacturing tolerances, or scientific measurements, understanding standard deviation gives you a clear picture of data variability that the mean alone cannot provide.

This guide walks you through both population and sample standard deviation formulas with step-by-step calculations, shows you how to interpret results using the 68-95-99.7 rule, and provides practical real-world examples that make the concept concrete.

What Is Standard Deviation?

Standard deviation measures the average distance between each data point and the mean of the data set. A small standard deviation means the values cluster tightly around the mean, while a large standard deviation means the values are spread over a wider range.

Consider two classes that both averaged 80 on an exam. In Class A, scores ranged from 75 to 85 with a standard deviation of 3.2. In Class B, scores ranged from 50 to 100 with a standard deviation of 15.8. The same mean tells a very different story when paired with these different standard deviations. Class A performed uniformly, while Class B had dramatic variation between high and low performers.

Formally, standard deviation is the square root of variance. Variance calculates the average of the squared differences from the mean. Taking the square root converts the result back into the original units of measurement, making standard deviation far more intuitive to interpret than variance.

Population vs Sample Standard Deviation

There are two versions of the standard deviation formula, and choosing the right one depends on whether your data represents an entire population or a sample drawn from a larger population.

The key difference is the denominator: population standard deviation divides by N, while sample standard deviation divides by N minus 1. This adjustment, known as Bessel's correction, compensates for the fact that a sample tends to underestimate the variability of the population it was drawn from. When you calculate the mean from a sample, you have already used one piece of information from the data, leaving only N minus 1 independent data points (degrees of freedom) for estimating variability.

Use population standard deviation when you have data for every member of the group: all employees in a company, all students in a classroom, all products in a batch. Use sample standard deviation when your data is a subset: a survey of 500 out of 10,000 customers, 30 randomly selected widgets from a production run of 10,000, or blood pressure readings from 200 patients representing a larger population.

Step-by-Step Calculation

Both formulas follow the same five-step process. The only difference is step 4, where you divide by N for population or N minus 1 for sample data.

1. Calculate the mean — Add all values and divide by the number of values.
2. Find each deviation from the mean — Subtract the mean from each data point.
3. Square each deviation — This eliminates negative signs and gives more weight to larger deviations.
4. Calculate the average of the squared deviations — Divide the sum of squared deviations by N (population) or N minus 1 (sample). This result is the variance.
5. Take the square root of the variance — This converts the result back to the original units, giving you the standard deviation.

Population Standard Deviation Example

Suppose a teacher has a class of 6 students who scored the following on a quiz (out of 20): 12, 15, 18, 14, 16, 13. Since this represents the entire class, we use population standard deviation.

Sample Standard Deviation Example

Rachel is a quality control analyst at a factory that produces thousands of bolts daily. She randomly selects 8 bolts and measures their length in millimeters: 50.2, 49.8, 50.1, 50.3, 49.9, 50.0, 50.4, 49.7. Since these 8 bolts are a sample from the full production, she uses sample standard deviation.

Interpreting Standard Deviation

Calculating standard deviation is only useful if you know how to interpret the result. The number itself has no universal threshold for "high" or "low" — it depends entirely on the context of your data and what you are measuring.

A standard deviation of 5 means very different things depending on the scale. For a data set of exam scores out of 100, a standard deviation of 5 indicates tightly grouped scores. For a data set of ratings on a 1-to-10 scale, a standard deviation of 5 would indicate extreme variability across nearly the entire range.

The coefficient of variation (CV) helps compare variability across different scales. It is calculated as the standard deviation divided by the mean, multiplied by 100 to express as a percentage. A CV of 10% means the standard deviation is 10% of the mean, providing a scale-free comparison of relative variability.

The 68-95-99.7 Rule

For data that follows a normal (bell-shaped) distribution, the standard deviation has a precise interpretation known as the empirical rule or the 68-95-99.7 rule:

For example, if the mean height of adult men in a country is 70 inches with a standard deviation of 3 inches, approximately 68% of men are between 67 and 73 inches tall, about 95% are between 64 and 76 inches, and roughly 99.7% are between 61 and 79 inches. Any man taller than 79 inches or shorter than 61 inches would be an extremely rare outlier — falling beyond three standard deviations from the mean.

This rule only applies to normally distributed data. For skewed or multi-modal distributions, the percentages will differ. However, Chebyshev's theorem guarantees that regardless of the distribution shape, at least 75% of data falls within 2 standard deviations and at least 89% falls within 3 standard deviations.

Practical Examples with Real Data

The following examples show how standard deviation applies across different fields with fictional but realistic scenarios.

Standard Deviation Reference Table

The following table provides typical standard deviation ranges across common applications. These values offer a frame of reference for interpreting your own calculations.

Tips for Working with Standard Deviation

Understanding the formula is just the beginning. Applying standard deviation effectively requires knowing its assumptions, limitations, and relationship to other statistical measures.

• Always check for outliers first. A single extreme value can dramatically inflate standard deviation. Before calculating, plot your data or check for values that seem unreasonably far from the rest. Consider whether outliers represent genuine variation or data entry errors.
• Use standard deviation alongside the mean. Neither measure is complete without the other. Reporting "average salary: $60,000" without the standard deviation hides whether most employees earn between $55,000 and $65,000 or between $30,000 and $90,000.
• Verify normality before applying the 68-95-99.7 rule. The empirical rule only holds for approximately normal distributions. Create a histogram of your data to check for symmetry, or use a formal normality test like the Shapiro-Wilk test for smaller data sets.
• Compare using the coefficient of variation when scales differ. You cannot directly compare the standard deviation of heights (measured in inches) with the standard deviation of weights (measured in pounds). The CV normalizes both to percentages of their respective means.
• Use standard error when estimating population parameters. Standard error (SE = s / √n) measures the precision of your sample mean as an estimator of the population mean. It decreases as sample size increases, unlike standard deviation itself.
• Know when to use alternatives. For highly skewed data, the interquartile range (IQR) may be a better measure of spread because it is not affected by extreme values. The mean absolute deviation (MAD) is another robust alternative.

Common Mistakes to Avoid

• Using population formula for sample data. This is the most common error. If your data is a subset of a larger group, always divide by n minus 1, not n. Dividing by n underestimates the true population standard deviation.
• Forgetting to square root the variance. Variance is in squared units, which are not directly interpretable. If your data is in dollars, variance is in dollars-squared. Always take the square root to get back to meaningful units.
• Comparing standard deviations across different scales. A standard deviation of 10 on a 1,000-point scale is proportionally tiny, while 10 on a 20-point scale is enormous. Use the coefficient of variation for cross-scale comparisons.
• Ignoring the shape of the distribution. Standard deviation assumes nothing about distribution shape, but its interpretation changes dramatically. For skewed data, the mean and standard deviation may not effectively describe the center and spread.
• Calculating standard deviation from summary statistics. You cannot correctly compute the standard deviation of combined groups by simply averaging their individual standard deviations. The combined standard deviation requires knowledge of group means, group sizes, and group variances.

Last updated: February 23, 2026