Mean, Median & Mode Calculator
Calculate central tendency measures (mean, median, mode), range, sample and population variance, and standard deviation for any dataset.
Calculate Summary Statistics & Central Tendency
Enter numbers separated by commas, spaces, or newlines, and select sample vs population variance.
Calculation Results
Calculated using standard mathematical descriptive statistics equations: \bar{x} = \frac{\sum x_i}{n}, \quad s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
Quick Summary
The Mean, Median & Mode Calculator computes central tendency ($\bar{x} = \frac{\sum x}{n}$), middle median value, most frequent mode(s), dataset range, sample variance ($s^2$), and standard deviation ($s$).
Formula Explanation
\text{Arithmetic Mean } \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
\text{Sample Variance } s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}
\text{Sample Standard Deviation } s = \sqrt{s^2}How It Works
Descriptive statistics summarize the central location and dispersion of a numerical dataset. The Mean, Median & Mode Calculator sorts your data, computes central tendency measures, and evaluates sample variance ($n-1$ Bessel's correction) versus population variance ($N$).
Step-by-Step Worked Example
Practical Problem: Given the sample dataset: 12, 15, 18, 15, 20, 25, 15, 30, 22 ($n = 9$). Calculate mean, median, mode, range, sample variance, and sample standard deviation.
- Step 1: Sort dataset in ascending order:
12, 15, 15, 15, 18, 20, 22, 25, 30. - Step 2: Calculate Arithmetic Mean ($\bar{x}$): $\text{Sum} = 172$; $\bar{x} = 172 / 9 = \mathbf{19.111}$.
- Step 3: Identify Median ($5\text{th}$ element of 9): The middle value is $\mathbf{18.000}$.
- Step 4: Identify Mode: The number
15appears 3 times $\rightarrow \mathbf{15.000\text{ (count 3)}}$. - Step 5: Calculate Sample Variance & Standard Deviation: $\text{SS} = \sum (x_i - 19.111)^2 = 250.889$; $s^2 = 250.889 / 8 = \mathbf{31.361}$; $s = \sqrt{31.361} = \mathbf{5.600}$.
Real-World Calculation Examples
Scenario 1: Student Exam Scores (n = 9)
Parameters: 12, 15, 18, 15, 20, 25, 15, 30, 22
Result: Mean 19.11 (Median 18.00, Mode 15.00, StdDev 5.60).
Scenario 2: Symmetrical Distribution (10, 20, 30, 40, 50)
Parameters: 10, 20, 30, 40, 50
Result: Mean 30.00 & Median 30.00 (Perfect symmetry, No Mode).
Scenario 3: Skewed Real Estate Prices
Parameters: $200k, $220k, $230k, $250k, $1,500k
Result: Mean $480k vs Median $230k (Median reflects typical home price).
Scenario 4: Bimodal Customer Ratings (1, 1, 2, 5, 5, 5)
Parameters: 1, 1, 1, 5, 5, 5
Result: Bimodal Modes 1 & 5 (Identifies split customer sentiment).
Key Benefits of Using This Calculator
Complete Descriptive Statistics
Calculates mean, median, mode, range, variance, and standard deviation simultaneously.
Multimodal & No Mode Detection
Handles datasets with multiple modes (bimodal/trimodal) or unique datasets with no mode.
Sample vs Population Variance
Supports both Sample ($n-1$ Bessel's correction) and Population ($N$) formulas.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is the difference between Mean, Median, and Mode?
Mean is the arithmetic average ($\sum x / n$), Median is the middle score when sorted, and Mode is the most frequently occurring value in the dataset.
When should I use Median instead of Mean?
Use Median for skewed distributions (such as household income or home values) containing extreme outliers that distort the arithmetic mean.
Why is sample variance divided by (n - 1) instead of n?
Dividing by $n - 1$ (Bessel's correction) corrects the downward bias in estimating population variance from a small sample.
What does standard deviation measure?
Standard deviation measures the average distance or dispersion of data points away from the mean.
Can a dataset have more than one mode?
Yes — a dataset can be bimodal (2 modes), trimodal (3 modes), or multimodal if multiple numbers share the highest frequency.
How is median calculated when a dataset has an even number of values?
When $n$ is even, the median is the average of the two middle numbers ($\text{Median} = \frac{x_{n/2} + x_{(n/2)+1}}{2}$).
What is range in statistics?
Range is the absolute difference between the maximum and minimum values in a dataset ($\text{Range} = \text{Max} - \text{Min}$).
What is the Empirical Rule (68-95-99.7 Rule)?
In a normal bell-curve distribution: 68% of data falls within $\pm 1\sigma$, 95% falls within $\pm 2\sigma$, and 99.7% falls within $\pm 3\sigma$.
What is coefficient of variation (CV %)?
CV is relative standard deviation expressed as percentage ($\text{CV \%} = \frac{s}{\bar{x}} \times 100$).
Can mean and median be negative numbers?
Yes — mean, median, and mode can be negative, zero, or positive depending on the input data.