Mean, Median, Mode, Range Calculator
Compute central tendencies, dispersion, quartiles, and standard deviation with complete step-by-step math.
Detailed Statistical Summary
Step-by-Step Derivations
1. Ascending Sorted Data
2. Mean Calculation
Sum = 613, Count = 7. Mean = 613 / 7 = 87.5714.
3. Median Calculation
Odd count (n = 7). Median is at position (7 + 1) / 2 = 4th element: 88.
4. Mode & Frequency
The number 85 appears 2 times (all other numbers appear 1 time). Unimodal dataset.
Measures of Central Tendency
In descriptive statistics, central tendency represents a central or typical value for a probability distribution or dataset:
For deeper analysis and related planning, you can also explore our Area Calculator and Average Calculator.
- Mean ($\bar{x}$): The arithmetic sum of all values divided by total count $n$: $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$ While mathematically intuitive, the mean is sensitive to extreme values (outliers).
- Median: The physical midpoint separating the higher half from the lower half of data. For an odd count $n$, the median is the value at index $\frac{n+1}{2}$. For an even count $n$, it is the arithmetic average of values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.
- Mode: The value(s) that appear with greatest frequency. Unlike mean and median, a dataset can have no mode, one mode (unimodal), two modes (bimodal), or multiple modes.
Measures of Dispersion & Spread
Central tendency alone does not describe how clustered or spread out numbers are. Dispersion metrics provide critical context:
- Range: The absolute difference between maximum and minimum values ($\text{Range} = \max - \min$).
- Quartiles & Interquartile Range (IQR): Quartiles divide sorted data into four quarters. $Q_1$ marks the 25th percentile, and $Q_3$ marks the 75th percentile. $\text{IQR} = Q_3 - Q_1$ captures the middle 50% of data, forming the basis of Tukey box plots.
- Sample Variance ($s^2$) & Sample Standard Deviation ($s$): Measures average squared deviation from the mean, utilizing Bessel's correction ($n-1$ denominator) to provide an unbiased estimator of population variance: $$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$$
Frequently Asked Questions
What is the difference between Mean, Median, and Mode?
The Mean is the arithmetic average (sum divided by count). The Median is the exact middle value when data is sorted in ascending order. The Mode is the number that occurs most frequently in the dataset.
Can a dataset have more than one mode or no mode at all?
Yes. If all numbers in a dataset appear with equal frequency (for example, once each), there is no mode. If two numbers tie for the highest frequency, the dataset is bimodal; if three or more tie, it is multimodal.
When should I use median instead of mean?
The median is preferable when your data has extreme outliers or is heavily skewed (such as household income, housing real estate prices, or net worth), because outliers heavily distort the arithmetic mean while leaving the median resistant.