How to Calculate Standard Deviation: Step-by-Step Guide

If you’ve ever stared at a spreadsheet and wondered what those numbers actually tell you, standard deviation is probably the statistic you’re after. It answers the question: “How spread out is this data, really?” Whether you’re grading papers, tracking investment returns, or just trying to understand a dataset your manager dumped on you, knowing how to calculate standard deviation by hand, in Excel, or with Python gives you a real edge. One popular example — the dataset {5, 5, 9, 9, 9, 10, 5, 10, 10} — has a mean of 7.78 and yields a standard deviation of approximately 2.60, a number that immediately tells you how much the values jump around the average.

Population formula: √[Σ(xi – μ)² / N] · Sample formula: √[Σ(xi – x̄)² / (n-1)] · 3 SD rule: 99.7% of data falls within three standard deviations of the mean

The table below consolidates the core definitions and values used throughout this guide.

How do I calculate standard deviation?

Calculating standard deviation follows a clear six-step sequence that works for any dataset. Khan Academy confirms the process works identically whether you’re calculating by hand or using software — the math doesn’t change (Khan Academy).

Step-by-step by hand

The NIH recommends starting by organizing your data and calculating the mean first — this single number becomes your reference point for every subsequent calculation (NIH).

1. Calculate the mean (sum all values, divide by count)
2. Find each deviation (subtract the mean from every data point)
3. Square every deviation (eliminates negative values)
4. Sum the squared deviations
5. Divide by n-1 (for sample) or N (for population)
6. Take the square root

For the dataset {2, 8, 10, 13, 17, 17, 19, 21, 23, 30}, this process yields a sample standard deviation of approximately 8.069 (Outlier.org).

Population vs sample

The critical difference between population and sample standard deviation lies in the divisor. Population standard deviation divides by N (the total number of data points), while sample standard deviation divides by n-1 (degrees of freedom). Outlier.org explains that sample standard deviation uses n-1 to provide an unbiased estimate of the population standard deviation — without this adjustment, you’d consistently underestimate variability (Outlier.org).

As MathsIsFun notes: “But hang on … we are calculating the Sample Standard Deviation, so instead of dividing by how many (N), we’ll divide by N−1” (MathsIsFun).

The implication: picking the wrong divisor systematically skews your variability estimate in one direction — always verify whether your data represents a full population or a sample first.

What is the best formula for standard deviation?

The “best” formula depends entirely on what you’re measuring. Population standard deviation (σ = √[Σ(xi-μ)² / N]) applies when your dataset represents the entire group you care about. Sample standard deviation (s = √[Σ(xi-x̄)² / (n-1)]) applies when you’re working with a subset and want to estimate the larger population (Macabacus).

Population formula

The population formula uses the Greek letter sigma (σ) and divides by N. This gives you the true spread of your complete dataset. Macabacus provides the formal definition: sqrt(sum((x-mean)^2)/N) (Macabacus).

Sample formula

The sample formula divides by n-1 instead of N. This adjustment — called Bessel’s correction — corrects the underestimation that occurs when using a sample to estimate population variability. Wikipedia traces this correction to the 1810s (Wikipedia).

Vs variance

Variance is simply the standard deviation squared. MathsIsFun defines it as “the average of the squared differences from the mean” (MathsIsFun). Standard deviation is often preferred in practice because it’s expressed in the same units as your original data — variance gives you squared units, which can be harder to interpret.

The catch: variance is mathematically convenient for formulas, but standard deviation stays in your data’s native units — always report SD alongside variance for clarity.

How to calculate standard deviation on Excel?

Excel offers three main functions for standard deviation, and picking the right one matters. Ablebits confirms the distinction clearly: STDEV.P handles population data while STDEV.S handles samples (Ablebits).

STDEV.S function

For sample data, type =STDEV.S(range) into any cell. This uses n-1 in the denominator, giving you an unbiased estimate. UConn demonstrates this via the Insert Function tool, making it accessible for beginners (UConn).

From mean

If you already know your mean and want to calculate the standard deviation, Excel’s advanced users can build the formula manually: =SQRT(SUMSQ(range-mean)/8) for a sample of 9 data points. However, STDEV.S handles all this internally and produces the same result with less room for error.

For test scores {85, 76, 92, 88, 69}, the STDEV.S function returns approximately 9.65 — indicating typical score deviations of about 9.65 points from the mean (Macabacus).

What this means: older Excel files may silently use sample SD even when you intended population SD — audit your function choice before trusting legacy spreadsheets.

How to calculate standard deviation by hand?

Calculating by hand — while slower than using software — builds genuine intuition about what standard deviation actually measures. Scribbr argues this approach deepens understanding in ways that clicking through spreadsheet functions simply cannot (Scribbr).

Example with {5, 5, 9, 9, 9, 10, 5, 10, 10}

Step 1 — Mean: (5+5+9+9+9+10+5+10+10) ÷ 9 = 72 ÷ 9 = 8.00

Step 2 — Deviations: -3, -3, 1, 1, 1, 2, -3, 2, 2

Step 3 — Squared deviations: 9, 9, 1, 1, 1, 4, 9, 4, 4

Step 4 — Sum: 9+9+1+1+1+4+9+4+4 = 42

Step 5 — Divide (sample, n-1): 42 ÷ 8 = 5.25

Step 6 — Square root: √5.25 ≈ 2.29

This dataset’s standard deviation of 2.29 tells you that individual values typically fall about 2.29 points from the mean of 8.00. The relatively small SD compared to the mean indicates fairly tight clustering.

Other datasets

For the dataset {5, 3, 4, 7}, the process works identically:

Mean = (5+3+4+7) ÷ 4 = 4.75

Squared deviations = 7.5625 + 3.0625 + 0.5625 + 5.0625 = 16.25

Sample variance = 16.25 ÷ 3 = 5.4167

Sample SD = √5.4167 ≈ 2.33

For the dataset {2, 4, 4, 4, 5, 5, 7, 9}, following the same steps yields a sample standard deviation of approximately 2.05.

The pattern: students who master the manual process internalize what SD measures in a way that spreadsheet users often miss — this pays dividends when interpreting results rather than just generating numbers.

What is the standard deviation of 5 5 9 9 9 10 5 10 10?

This specific dataset appears frequently in search results, so here’s the complete breakdown. The nine values {5, 5, 9, 9, 9, 10, 5, 10, 10} sum to 72, giving a mean of exactly 8.00 — a clean number that makes the deviations particularly easy to follow.

Step-by-step solution

Deviations from mean (8.00):

• 5 − 8.00 = −3.00
• 5 − 8.00 = −3.00
• 9 − 8.00 = 1.00
• 9 − 8.00 = 1.00
• 9 − 8.00 = 1.00
• 10 − 8.00 = 2.00
• 5 − 8.00 = −3.00
• 10 − 8.00 = 2.00
• 10 − 8.00 = 2.00

Squared deviations: 9, 9, 1, 1, 1, 4, 9, 4, 4 → Sum = 42

Sample variance: 42 ÷ (9−1) = 42 ÷ 8 = 5.25

Sample standard deviation: √5.25 ≈ 2.29

Interpretation

A standard deviation of 2.29 against a mean of 8.00 means that roughly 68% of values fall between 5.71 and 10.29. The coefficient of variation (SD ÷ mean) equals 2.29 ÷ 8.00 = 28.6%, indicating moderate relative variability. In practical terms, this dataset clusters fairly tightly around its center — you wouldn’t expect many extreme values.

For researchers, this SD of 2.29 is useful for comparing against other similar datasets. For students, this dataset demonstrates all six calculation steps cleanly, with no awkward numbers to complicate the process.

How to calculate standard deviation in Python?

Python’s standard approach uses NumPy, which provides straightforward functions for both population and sample calculations. The NumPy documentation specifies numpy.std(ddof=1) for sample standard deviation (NumPy Docs).

Here’s the basic Python code:

import numpy as np

data = [5, 5, 9, 9, 9, 10, 5, 10, 10]

# Sample standard deviation (ddof=1)
sample_sd = np.std(data, ddof=1)
print(f"Sample SD: {sample_sd:.2f}")

# Population standard deviation (ddof=0, default)
pop_sd = np.std(data, ddof=0)
print(f"Population SD: {pop_sd:.2f}")

This outputs approximately 2.29 for the sample SD and 2.16 for the population SD — exactly matching the hand calculations from earlier sections.

The ddof parameter (delta degrees of freedom) controls the divisor. Setting ddof=1 applies Bessel’s correction, dividing by n-1 instead of n. Setting ddof=0 uses the standard population formula.

“We use n-1 only when calculating a sample standard deviation in order to get a closer approximation of the population standard deviation.” — Outlier.org (Educational Resource)

“STDEV.S provides a slightly higher standard deviation, accounting for the uncertainty of not having data for the entire population.” — Macabacus (Excel Tutorial)

Standard deviation is one of those tools that pays compound interest: the more you use it, the more patterns you see. For data analysts, it becomes a daily checkpoint. For students, it’s foundational for everything from psychology research to quality control. The formula’s universal nature — unchanged across regions and applications since Karl Pearson’s formalization — means that mastery transfers everywhere.