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.

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 Guide

How to Use the Statistics & Standard Deviation Calculator: Practical Guide & Visual Analysis

Paste or type any list of numbers to compute parametric (mean, sample SD, population SD, Z-score) and non-parametric (median, Q1, Q3, IQR, range) summary statistics. Whether you are researching standard deviation calculator, mean median mode range, variance calculator, iqr quartile calculator, z score calculator, this interactive tool helps you compute sample and population standard deviation, variance, mean, median, quartiles, IQR, Z-score, and a 5-bin histogram from any dataset.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Dataset Values (Comma or Space Separated), Observation x for Z-Score Check, the calculator immediately computes Sample Mean (x̄) & Sample SD (s), Population SD (σ) & Sample Variance (s²), Median & Interquartile Range (IQR), Min, Max, Range & Z-Score and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Global)

In our benchmark scenario (9-Observation Dataset: 12, 15, 18, 22, 22, 25, 29, 34, 41), Mean = 24.2222, Median = 22, Sample SD (s) = 9.3244, Population SD (σ) = 8.7911. Keep in mind that uses linear interpolation between order statistics for Q1 (25th percentile) and Q3 (75th percentile). For related calculations in Mathematics & Statistics, explore Scientific Calculator, Percentage Calculator, GPA & Final Grade Calculator.