Statistics & Standard Deviation Calculator

Calculate sample SD (s), population SD (σ), variance, mean, median, IQR, and Z-score with a frequency histogram.

Sample Mean (x̄) & Sample Standard Deviation (s)
x̄ = 24.2222 • s = 9.2436
n = 9 observations • Median = 22 • Population σ = 8.715
Median & IQR (Q3 − Q1) 22 (IQR: 11)
Sample Variance (s²) 85.4444
Min / Max (Range) 12 to 41 (29)
Z-Score for x = 29 0.517 σ
Inputs
Enter at least 2 numeric observations separated by commas or spaces.
Computes how many sample standard deviations x is from the mean.
Results & Graph
Sample Mean (x̄) & Sample Standard Deviation (s)
x̄ = 24.2222 • s = 9.2436
n = 9 observations • Median = 22 • Population σ = 8.715
Median & IQR (Q3 − Q1)
22 (IQR: 11)
Sample Variance (s²)
85.4444
Min / Max (Range)
12 to 41 (29)
Z-Score for x = 29
0.517 σ

Dataset Frequency Histogram (5 Equal-Width Bins)

Distribution of 9 observations from 12 to 41
Graph 1 • Sorted Dataset Observations vs. Sample Mean (x̄) n = 9 observations
0 0.8 1.5 2.3 3 12–17.817.8–23.623.6–29.429.4–35.235.2–41
Graph 2 • Normal Distribution Bell Curve (x̄ ± 1s, ±2s, ±3s) 68.2% within ±1s, 95.4% within ±2s, 99.7% within ±3s
Mean x̄ = 24.22 (s = 9.24) −1s (15.0) x̄ (24.2) +1s (33.5) −2s (5.7) +2s (42.7)
Graph 3 • 5-Number Summary Box-and-Whisker Plot (Min, Q1, Median, Q3, Max) Visualizes median, quartile spread (IQR), and skewness
Min: 12 Q1: 18 Median: 22 Q3: 29 Max: 41 Interquartile Range (IQR = Q3 − Q1) = 11.00 • Range = 29.00
Summary: Dataset of 9 values has mean 24.22, median 22, and sample SD 9.24.
Summary: Across n = 9 values (sum = 218), the arithmetic mean is 24.2222 and the median is 22. The sample standard deviation is s = 9.2436 (population σ = 8.715), and the interquartile range (Q3 − Q1) is 11.
⚡ Send result to: Scientific →Percentage →GPA & Grade →
Interactive chart & math engine powered by CalculatorGraph

Formula & Mathematical Methodology: Sample (Bessel-Corrected) vs. Population Standard Deviation

x̄ = 1n n∑i=1 xi,   s = n∑i=1 (xi − x̄)2n − 1,   σ = n∑i=1 (xi − x̄)2n,   Z = x − x̄s
x̄ = (1/n) Σ x_i; s = √[ (1/(n−1)) Σ (x_i − x̄)² ]; σ = √[ (1/n) Σ (x_i − x̄)² ]; Z = (x − x̄) / s
  • n: Number of observations in the dataset
  • s: Unbiased sample standard deviation using Bessel’s correction (n − 1)
  • σ: Population standard deviation using denominator n

Rounding: Summary statistics round to 4 decimal places.

Worked Example: 9-Observation Dataset: 12, 15, 18, 22, 22, 25, 29, 34, 41

Compute mean, median, sample SD, and population SD for the 9 values.

  1. Sum = 218 → Mean x̄ = 218 ÷ 9 = 24.2222.
  2. Sorted middle (5th) observation → Median = 22.
  3. Sum of squared deviations Σ(x_i − x̄)² = 695.5556.
  4. Sample SD s = √(695.5556 ÷ 8) = 9.3244; Population SD σ = √(695.5556 ÷ 9) = 8.7911.
Result: Mean = 24.2222, Median = 22, Sample SD (s) = 9.3244, Population SD (σ) = 8.7911.

Frequently Asked Questions

When should I use Sample Standard Deviation (s) vs. Population Standard Deviation (σ)?

Use Sample SD (s, dividing by n − 1) whenever your numbers represent a sample drawn from a larger population (the standard in scientific research and quality control). Use Population SD (σ, dividing by n) only when your dataset contains every member of the entire population.

Mathematics & Statistics · Descriptive Statistics & Distributions

Statistics & Standard Deviation Calculator: inputs, method, and how to read the result

Calculate sample SD (s), population SD (σ), variance, mean, median, IQR, and Z-score with a frequency histogram.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Observation x for Z-Score Check: Computes how many sample standard deviations x is from the mean.

What the calculation includes

With the default values shown, Sample Mean (x̄) & Sample Standard Deviation (s) is x̄ = 24.2222 • s = 9.2436. The calculation follows Sample (Bessel-Corrected) vs. Population Standard Deviation; the formula and variable definitions above show how the inputs combine.

A question to settle before using the estimate

When should I use Sample Standard Deviation (s) vs. Population Standard Deviation (σ)? Use Sample SD (s, dividing by n − 1) whenever your numbers represent a sample drawn from a larger population (the standard in scientific research and quality control). Use Population SD (σ, dividing by n) only when your dataset contains every member of the entire population.

Where to use caution

  • Uses linear interpolation between order statistics for Q1 (25th percentile) and Q3 (75th percentile).
  • Requires at least n = 2 observations so sample variance (dividing by n − 1) is well-defined.

Scope: Global Check the assumptions and limitations against your situation before relying on the estimate.

More in Mathematics & Statistics: Scientific Calculator, Percentage Calculator, GPA & Final Grade Calculator.