Math & Statistics
Statistics Calculator โ Mean, Median, Mode & Standard Deviation
Use our free Statistics Calculator to calculate mean, median, mode, standard deviation, variance, and other statistical measures. Perfect for students, researchers, and data analysis.
Understanding Descriptive Statistics
Descriptive statistics summarize and describe the main features of a dataset. The three pillars are: central tendency (mean, median, mode โ where data clusters), dispersion (range, variance, standard deviation โ how spread out data is), and position (quartiles, percentiles โ where individual values fall).
Enter your numbers below separated by commas, spaces, or line breaks. The calculator computes all key measures instantly, including both population (ฯ) and sample (s) standard deviations โ use population when your data includes every value, and sample when it's a subset of a larger group.
How to Use This Calculator
- Enter your numbers separated by commas, spaces, or line breaks
- Click Calculate Statistics
- View central tendency, dispersion, and position measures
- Use Clear to reset and enter new data
Common Use Cases
- Analyzing test scores
- Research data analysis
- Quality control metrics
- Survey result summaries
- Financial data analysis
- Sports performance stats
Statistical Formulas Quick Reference
| Measure | Formula | What It Tells You |
|---|---|---|
| Mean (xฬ) | ฮฃx / n | Average value โ sensitive to outliers |
| Median | Middle value when sorted | Central value โ resistant to outliers |
| Mode | Most frequent value | Most common observation in the data |
| Range | Max โ Min | Total spread of the data |
| Variance (ฯยฒ) | ฮฃ(x โ xฬ)ยฒ / n | Average squared deviation from mean |
| Std Dev (ฯ) | โ(ฮฃ(x โ xฬ)ยฒ / n) | Typical distance from the mean |
| Sample Variance | ฮฃ(x โ xฬ)ยฒ / (nโ1) | Unbiased estimate for population variance |
| Sample Std Dev (s) | โ(ฮฃ(x โ xฬ)ยฒ / (nโ1)) | Unbiased estimate for population std dev |
Population vs. Sample Statistics
| Feature | Population (N) | Sample (n) |
|---|---|---|
| Definition | Entire group of interest | Subset drawn from the population |
| Notation | ฮผ (mean), ฯ (std dev), N (size) | xฬ (mean), s (std dev), n (size) |
| Variance divisor | Divide by N | Divide by n โ 1 (Bessel's correction) |
| When to use | Census, complete dataset | Surveys, experiments, partial data |
| Example | All students' scores in a school | 30 randomly selected students' scores |
Interpreting Standard Deviation (Empirical Rule)
For normally distributed data, the empirical rule (68-95-99.7 rule) tells you how data clusters around the mean:
| Range | Coverage | Meaning |
|---|---|---|
| xฬ ยฑ 1ฯ | 68.27% | About 2 out of 3 values fall here |
| xฬ ยฑ 2ฯ | 95.45% | Nearly all typical values fall here |
| xฬ ยฑ 3ฯ | 99.73% | Almost everything โ beyond is an outlier |
FAQ โ Statistics Calculator
What is the difference between mean, median, and mode?
The mean is the arithmetic average (sum รท count). The median is the middle value when data is sorted โ it splits the dataset in half. The mode is the most frequently occurring value. For symmetric data, all three are similar. For skewed data, the median is usually the best measure of center because it's not affected by extreme values.
When should I use population vs. sample standard deviation?
Use population standard deviation (ฯ, divide by N) when your data includes every member of the group you're studying. Use sample standard deviation (s, divide by nโ1) when your data is a subset of a larger population. The nโ1 correction (Bessel's correction) compensates for the tendency of samples to underestimate population variability.
What does standard deviation actually tell me?
Standard deviation measures the typical distance of data points from the mean. A small standard deviation means values cluster tightly around the average; a large one means they're spread out. For example, test scores with mean 75 and SD 5 are much more consistent than scores with mean 75 and SD 20.
What if my dataset has no mode?
If every value appears exactly once, there is no mode โ the data is uniformly distributed with no repeated values. Some datasets can also be bimodal (two modes) or multimodal (three or more modes). This calculator reports "No mode" when all values have equal frequency.
How do outliers affect statistical measures?
Outliers heavily affect the mean, variance, standard deviation, and range. They have little effect on the median and mode. For example, the dataset {10, 12, 11, 13, 100} has a mean of 29.2 (pulled up by 100) but a median of 12 (unaffected). When outliers are present, the median is a more reliable measure of center.
What is variance and how does it relate to standard deviation?
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. Variance is useful in mathematical formulas (it's additive for independent variables), while standard deviation is more interpretable because it's in the same units as the original data.
What are quartiles and how are they calculated?
Quartiles divide sorted data into four equal parts. Q1 (25th percentile) is the median of the lower half, Q2 is the overall median (50th percentile), and Q3 (75th percentile) is the median of the upper half. The interquartile range (IQR = Q3 โ Q1) measures the spread of the middle 50% of data.
What is the empirical rule (68-95-99.7)?
For normally distributed data: ~68% of values fall within 1 standard deviation of the mean, ~95% within 2, and ~99.7% within 3. Values beyond 3 standard deviations are considered outliers. This rule helps quickly assess whether a data point is typical or unusual.
Can I use this calculator for grouped or frequency data?
Enter each value the number of times it occurs. For example, if 5 appears 3 times and 10 appears 2 times, enter: 5, 5, 5, 10, 10. The calculator will correctly compute all statistics from the expanded dataset.
What's the difference between descriptive and inferential statistics?
Descriptive statistics summarize your actual data (mean, median, SD). Inferential statistics use sample data to make predictions about a larger population (confidence intervals, hypothesis tests, p-values). This calculator focuses on descriptive statistics.