What is standard deviation and what does it measure?

Standard deviation measures how spread out the values in a dataset are around the mean. A small standard deviation means the values are clustered closely together; a large one means they are spread widely. It is expressed in the same units as the original data, making it easier to interpret than variance. For dataset 2, 4, 4, 6, 8, 8, 8, 10 (mean = 6.25): • Population std dev σ = 2.5495 — on average, values are about 2.55 units from the mean • If all values were identical (e.g. 5,5,5,5), std dev = 0 • If values are very spread (e.g. 1 and 100), std dev is large Real-world uses: • Finance: measuring investment volatility (risk) • Quality control: ensuring products are within tolerance • Education: understanding score distribution on exams • Science: reporting measurement uncertainty • Weather: variability in temperature or rainfall

What is the difference between population and sample standard deviation?

Use population standard deviation (σ, divides by n) when your data IS the entire population you are studying. Use sample standard deviation (s, divides by n−1) when your data is a sample drawn from a larger population and you want to estimate the population's spread. Population formula: σ = √(Σ(x−μ)² / n) Sample formula: s = √(Σ(x−x̄)² / (n−1)) Why n−1 for samples (Bessel's correction): • A sample tends to underestimate the true population variance • Dividing by n−1 instead of n corrects for this bias • The sample mean x̄ is calculated from the data, which 'uses up' one degree of freedom • As n gets large, the difference between σ and s becomes negligible • Rule of thumb: if you collected the data yourself as a sample → use s. If you have the complete population → use σ • Most statistics textbooks default to sample standard deviation

How is standard deviation calculated step by step?

Step 1: Find the mean (x̄ = Σx/n). Step 2: Subtract the mean from each value to get the deviations (x − x̄). Step 3: Square each deviation: (x − x̄)². Step 4: Sum all squared deviations: Σ(x − x̄)². Step 5: Divide by n (population) or n−1 (sample) to get the variance. Step 6: Take the square root to get the standard deviation. For 2, 4, 6, 8 (mean = 5): • Deviations: −3, −1, 1, 3 • Squared: 9, 1, 1, 9 → sum = 20 • Population variance = 20/4 = 5 → σ = √5 ≈ 2.2361 • Sample variance = 20/3 = 6.667 → s ≈ 2.5820 This calculator shows the full deviation table with every step when you click 'Show Step-by-Step'.

What is the empirical rule (68-95-99.7 rule)?

The empirical rule states that for a normal (bell-shaped) distribution: • ~68.27% of values fall within 1 standard deviation of the mean (μ ± σ) • ~95.45% fall within 2 standard deviations (μ ± 2σ) • ~99.73% fall within 3 standard deviations (μ ± 3σ) For a dataset with mean=50 and std dev=10: • 68% of values: 40 to 60 • 95% of values: 30 to 70 • 99.7% of values: 20 to 80 Important caveats: • This rule only applies to approximately normal (bell-shaped) distributions • Skewed or bimodal data will not match these percentages • This calculator shows the actual percentages for your data alongside the theoretical values — if they differ significantly, your data is not normally distributed • In quality control: ±3σ is the basis for the '6-sigma' quality standard (3.4 defects per million)

What is a z-score and how is it calculated?

A z-score (standard score) measures how many standard deviations a value is from the mean: z = (x − μ) / σ. A z-score of 0 means the value equals the mean. A z-score of +2 means the value is 2 standard deviations above the mean. A z-score of −1.5 means 1.5 standard deviations below. For mean=6.25, σ=2.5495: • x=2: z=(2−6.25)/2.5495 = −1.67 (1.67 std devs below mean) • x=8: z=(8−6.25)/2.5495 = +0.69 (0.69 std devs above mean) • x=6.25: z=0 (exactly the mean) Why z-scores matter: • Compare values across different datasets or scales • Find probabilities using the standard normal table • Identify outliers (|z| > 2 is unusual, |z| > 3 is very unusual) • Standardise features in machine learning (z-score normalisation) • Convert test scores to percentile rankings

Standard Deviation Calculator — Population & Sample

The Standard Deviation Calculator computes population and sample standard deviation, variance, z-scores, and the empirical rule for any dataset. Enter numbers separated by commas or spaces — toggle between population (÷n) and sample (÷n−1) mode with a single click. The step-by-step table shows every...

ENTER YOUR DATA

Commas, spaces, or new lines

n = 8

POPULATION STD DEV (σ)

2.537223

POPULATION VARIANCE (σ²)

6.4375

MEAN (μ)

6.25

COUNT (n)

SUM (Σ)

Population σ (÷n)

2.537223

var = 6.4375

Sample s (÷n−1)

2.712405

var = 7.357143

EMPIRICAL RULE (68–95–99.7)

Within 1σ

6/8 (75%)theory: ~68.3%

Within 2σ

8/8 (100%)theory: ~95.5%

Within 3σ

8/8 (100%)theory: ~99.7%

Z-SCORES (z = (x−μ)/σ)

x=2

-1.68

x=4

-0.89

x=4

-0.89

x=6

-0.1

x=8

0.69

x=8

0.69

x=8

0.69

x=10

1.48

QUICK EXAMPLES

BELL CURVE

POPULATION STATISTICS

Std deviation2.53722289

Variance6.4375

Mean ± 1σ3.7128 to 8.7872

Mean ± 2σ1.1756 to 11.3244

Coefficient of variation40.6%

Sum of sq. deviations51.5

FORMULA USED

σ = √( Σ(x−x̄)² / n )
= √( 51.5 / 8 )
= √6.4375
= 2.537223

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter your numbers in the text box, separated by commas, spaces, or new lines. The calculator accepts integers, decimals, and negative numbers. At least 2 values are required.
2
Toggle between Population (÷n) and Sample (÷n−1) using the buttons. Use population when you have complete data; use sample when your data is a subset of a larger group.
3
The standard deviation and variance appear instantly. Both population and sample values are shown side by side for comparison.
4
Click Show Step-by-Step to see the full deviation table: each value x, its deviation from the mean (x−x̄), and the squared deviation (x−x̄)². The sum of squared deviations and final formula are shown below.
5
Check the bell curve — your data points appear as red dots along the baseline, and the three shaded regions show the ±1σ, ±2σ, and ±3σ ranges. The empirical rule panel compares your actual data distribution to the theoretical 68-95-99.7% rule.

WORKED EXAMPLE

Dataset: 2, 4, 4, 6, 8, 8, 8, 10 (n=8, mean=6.25). Deviations²: 18.0625, 5.0625, 5.0625, 0.0625, 3.0625, 3.0625, 3.0625, 14.0625. Sum=52. Population: σ²=52/8=6.5, σ=√6.5≈2.5495. Sample: s²=52/7≈7.4286, s≈2.7261. Z-scores: x=2→z=−1.67, x=8→z=+0.69.

FORMULAS

STANDARD DEVIATION FORMULAS

NAME	FORMULA	DESCRIPTION
Population std dev	σ = √(Σ(x−μ)² / n)	Use when data is the entire population
Sample std dev	s = √(Σ(x−x̄)² / (n−1))	Use when data is a sample; n−1 is Bessel correction
Population variance	σ² = Σ(x−μ)² / n	Average squared deviation from mean
Sample variance	s² = Σ(x−x̄)² / (n−1)	Bessel-corrected variance for sample data
Deviation	d = x − x̄	Distance of each value from the mean
Z-score	z = (x − μ) / σ	Number of std devs from the mean
Empirical rule	μ ± σ: ~68%, μ ± 2σ: ~95%, μ ± 3σ: ~99.7%	For normal distributions only
Coefficient of variation	CV = (σ/μ) × 100%	Relative spread as percentage of mean

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.