Statistics Calculator with Steps
Calculate mean, median, mode, standard deviation, skewness, kurtosis, and more step-by-step. Visualize with box plots, histograms, violin plots, and detect outliers automatically. Perfect for GCSE, A-Level, and university statistics.
Separate numbers with commas, spaces, or newlines.
Enter your data and press Calculate to see step-by-step questions with interactive visualizations.
How helpful was this?
Help other students find great tools
What is Descriptive Statistics?
Descriptive statistics summarize and describe the main features of a dataset without making inferences about a larger population. They help us understand two fundamental properties: where data clusters (central tendency) and how data varies (spread or dispersion).
Unlike inferential statistics (which make predictions), descriptive statistics simply describe what the data shows. They are the foundation of all statistical analysis and are essential for GCSE, A-Level, and university-level mathematics.
Pillar 1: Central Tendency
Where does the data cluster? Mean, median, and mode answer this question.
Pillar 2: Spread
How varied is the data? Range, IQR, variance, and standard deviation measure this.
Key Formulas
Types of Data
Quantitative
Numerical data that can be measured or counted. Heights, weights, test scores.
Heights: 165cm, 172cm, 180cm
Qualitative
Categorical data that describes qualities. Colours, names, categories.
Eye colour: blue, brown, green
Discrete
Countable values — can only take specific values with gaps between them.
Number of siblings: 0, 1, 2, 3
Continuous
Measurable values — can take any value within a range, including decimals.
Temperature: 18.5°C, 20.3°C
Measures of Central Tendency
Mean (Average)
x̄ = Σx / n
When to use: Symmetric data without extreme outliers
+ Uses all data values; mathematically useful
− Heavily affected by outliers
Median (Middle)
Middle value of sorted data
When to use: Skewed data or data with outliers
+ Resistant to extreme values
− Does not use all data values
Mode (Most Frequent)
Most common value
When to use: Categorical data or finding the most typical value
+ Works with non-numeric data; can have multiple modes
− May not exist or may not be unique
| Scenario | Mean | Median | Mode |
|---|---|---|---|
| Symmetric data (no outliers) | Best | Good | OK |
| Skewed data | Misleading | Best | OK |
| Data with extreme outliers | Misleading | Best | Good |
| Categorical data | N/A | N/A | Best |
| Need for further calculations | Best | Limited | Limited |
Measures of Spread
Range
Range = Max − Min
The simplest measure of spread. Easy to calculate but highly sensitive to outliers. One extreme value can dramatically change the range.
Interquartile Range (IQR)
IQR = Q3 − Q1
Measures the spread of the middle 50% of data. Much more resistant to outliers than the range. Used in box plots and outlier detection.
Variance
σ² = Σ(x − x̄)² / n
The average of squared deviations from the mean. Gives squared units (e.g., cm²), which is why we often prefer standard deviation.
Standard Deviation
σ = √variance
The most commonly used measure of spread. Same units as the original data. A small SD means data clusters tightly around the mean.
Population vs Sample: When to Use Each
Population (÷ n)
Use when your data IS the entire population.
- • Test scores of your entire class
- • All employees in a small company
- • Every item in a batch
- • GCSE typically uses this formula
Sample (÷ n−1)
Use when your data is a SAMPLE from a larger population.
- • Survey of 100 people from a city
- • Testing 50 items from a production line
- • Research study participants
- • A-Level and university use this formula
Understanding Standard Deviation: The 68-95-99.7 Rule
For data that follows a normal (bell-shaped) distribution, the standard deviation tells us exactly what proportion of data falls within each range:
This means that if exam scores have a mean of 70 and a standard deviation of 10:
- About 68% of students scored between 60–80
- About 95% of students scored between 50–90
- About 99.7% of students scored between 40–100
Values beyond 3σ from the mean are extremely rare (0.3% chance) and are often considered outliers. Our calculator shows this visually in the Distribution tab.
Common Mistakes to Avoid
Worked Examples
Calculate mean, median, and mode
Data: 4, 7, 7, 9, 12
- 1.Sort (already sorted): 4, 7, 7, 9, 12
- 2.Mean = (4 + 7 + 7 + 9 + 12) / 5 = 39 / 5 = 7.8
- 3.Median = middle value (3rd) = 7
- 4.Mode = most frequent = 7 (appears twice)
Find Q1, Q3, IQR and identify outliers
Data: 3, 5, 7, 8, 9, 10, 11, 13, 15, 40
- 1.n = 10, Q1 position = 25th percentile = 5.75, Q1 = 7.25
- 2.Q3 position = 75th percentile = 13.5, Q3 = 13.5
- 3.IQR = 13.5 − 7.25 = 6.25
- 4.Lower fence = 7.25 − 1.5 × 6.25 = −2.125 → no lower outliers
- 5.Upper fence = 13.5 + 1.5 × 6.25 = 22.875 → 40 is an outlier!
Population vs sample standard deviation
Data: 10, 12, 14, 16, 18
- 1.Mean = (10 + 12 + 14 + 16 + 18) / 5 = 14
- 2.Squared deviations: 16, 4, 0, 4, 16 → Sum = 40
- 3.Population σ² = 40 / 5 = 8, so σ = √8 ≈ 2.828
- 4.Sample s² = 40 / 4 = 10, so s = √10 ≈ 3.162
- 5.The sample SD is larger because it corrects for sample bias (Bessel's correction).
Interpreting skewness and data shape
Data: 2, 3, 4, 4, 5, 5, 5, 6, 8, 15
- 1.Mean = 5.7, Median = 5
- 2.Mean > Median → positive (right) skewness
- 3.The value 15 is pulling the mean upward
- 4.Skewness ≈ 1.2 (moderately right-skewed)
- 5.For this distribution, the median (5) is a better representative than the mean (5.7).
When to Use Each Measure: A Decision Guide
Is your data symmetric?
No outliers, roughly bell-shaped
Is your data skewed?
Has outliers or a long tail
Is your data categorical?
Non-numeric or finding most common
Frequently Asked Questions
What is the difference between mean, median, and mode?
Mean is the average (sum/count). Median is the middle value when sorted. Mode is the most frequent value. Each is useful for different types of data — mean for symmetric, median for skewed, mode for categorical.
How do I calculate standard deviation?
Find the mean, subtract it from each value and square the result, find the average of squared differences (÷n for population, ÷n−1 for sample), then take the square root.
Population vs sample standard deviation?
Population SD (σ) divides by n — use when you have ALL the data. Sample SD (s) divides by n−1 — use when your data is a sample from a larger population. GCSE uses population; A-Level often uses sample.
How do I find the median with an even number of values?
Sort the data, find the two middle values (at positions n/2 and n/2+1), and calculate their average. For {2, 4, 6, 8}: median = (4+6)/2 = 5.
What are quartiles and IQR?
Quartiles divide sorted data into four equal parts. IQR (Q3−Q1) measures the middle 50% spread and is resistant to outliers — ideal for skewed data.
What is skewness?
Skewness measures asymmetry. Positive (right) skew: mean > median, long right tail. Negative (left) skew: mean < median, long left tail. Near zero: symmetric.
How do I detect outliers?
Tukey method: Lower fence = Q1 − 1.5×IQR, Upper fence = Q3 + 1.5×IQR. Values outside these fences are mild outliers. Beyond ±3×IQR are extreme outliers.
What is coefficient of variation?
CV = (σ / |μ|) × 100%. It lets you compare variability between datasets with different units. Below 15% = low variability, 15-30% = moderate, above 30% = high.
When should I use median instead of mean?
Use the median for skewed data, data with outliers, or when you want a resistant measure. Common examples: house prices, salary data, and any dataset with extreme values.
Is this suitable for GCSE and A-Level statistics?
Yes! Covers all required topics including mean, median, mode, standard deviation (population and sample), quartiles, IQR, box plots, histograms, cumulative frequency, stem-and-leaf, and outlier detection with step-by-step working.
Explore More Free Tools
All our tools are 100% free with step-by-step learning