Statistics Calculator
Comprehensive statistical analysis with descriptive statistics, distribution tests, and data visualization
Data Input
Separate values with commas, spaces, or new lines
Statistics Calculator: Mean, Standard Deviation & Confidence Intervals
Our Statistics Calculator instantly computes the full suite of descriptive statistics — mean, median, mode, standard deviation, variance, skewness, kurtosis, and quartiles — from any dataset. Built for students, researchers, engineers, and anyone working with numerical data.
Key Statistical Measures Reference
| Measure | Formula | Best Used When |
|---|---|---|
| Mean | Σx ÷ n | Symmetric data, no outliers |
| Median | Middle value (sorted) | Skewed data or outliers present |
| Mode | Most frequent value | Categorical or discrete data |
| Variance (s²) | Σ(x−x̄)² ÷ (n−1) | Average squared deviation |
| Std Deviation (s) | √Variance | Spread in original data units |
| IQR | Q3 − Q1 | Middle 50% range, outlier detection |
| Skewness > 0 | Right-tailed distribution | Income, housing prices |
| Kurtosis = 3 | Normal (mesokurtic) | Baseline shape comparison |
How to Use the Statistics Calculator
- Enter your data — paste or type values separated by commas or spaces (e.g., 72, 85, 91, 68, 78).
- Choose population or sample — use sample (divides by n−1) for survey or experimental data; population only when you have every possible observation.
- Click Calculate — all measures update instantly including the five-number summary and histogram.
- Set confidence level — choose 90%, 95%, or 99% to see the confidence interval around the mean.
- Check for outliers — the IQR fence method is applied automatically; flagged values are highlighted.
Example Calculations
Example 1 — Student Test Scores
Data: 72, 85, 91, 68, 78, 95, 84, 76, 89, 62
- Mean: 800 ÷ 10 = 80.0
- Sorted median: (78 + 84) ÷ 2 = 81.0
- Sample std dev ≈ 10.6
- IQR = Q3 − Q1 = 89 − 72 = 17
- 95% CI: 80.0 ± (2.262 × 10.6/√10) = 72.4 to 87.6
Example 2 — Outlier Detection
Data: 10, 12, 11, 13, 9, 250
- IQR = 12.5 − 9.5 = 3 → Upper fence = 12.5 + (1.5 × 3) = 17
- 250 > 17 → 250 is a confirmed outlier
- Mean with outlier: 50.8 vs median: 11.5 — the median is a far better central measure here.
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides by n and is used when you have every member of the group (e.g., all scores in a class of 30). Sample standard deviation (s) divides by n−1 (Bessel's correction) and is used when your data is a subset of a larger group. For most research, surveys, and experiments, use sample standard deviation.
When should I use median instead of mean?
Use median when your dataset contains outliers or is strongly skewed. Income data is a classic example — a few high earners pull the mean upward while the median better represents the typical value. Median is also preferred for ordinal data (rankings, Likert scales) and for any variable where extreme values distort interpretation.
What does a 95% confidence interval actually mean?
A 95% confidence interval means that if you repeated the same sampling procedure 100 times, approximately 95 of those intervals would contain the true population mean. It does not mean there is a 95% probability the true mean lies in this specific interval — the true mean either is or is not contained in it. Wider intervals reflect smaller sample sizes or higher variation.
How do I detect outliers using the IQR method?
Calculate Q1 (25th percentile) and Q3 (75th percentile), then compute IQR = Q3 − Q1. Any value below Q1 − (1.5 × IQR) or above Q3 + (1.5 × IQR) is a mild outlier. Values beyond Q1 − (3 × IQR) or Q3 + (3 × IQR) are extreme outliers. The IQR method is more robust than the Z-score method because it is not distorted by the outliers themselves.
What is skewness and how do I interpret it?
Skewness measures the asymmetry of a distribution. A value near 0 indicates roughly symmetric data. Positive skewness (right skew) means the tail extends to the right — income, housing prices, and response times commonly show this pattern. Negative skewness (left skew) means the tail extends left. Values between −0.5 and 0.5 are generally considered approximately symmetric.
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