Statistics Calculator
Comprehensive descriptive statistics with a dynamic box-and-whisker plot. Computes mean, quartiles, IQR, standard deviation, variance, and highlights mathematical outliers.
Statistics
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The Formula
A box-and-whisker plot (Tukey box plot) visualizes the spread of data through five key numbers: minimum, Q1 (first quartile), median (Q2), Q3 (third quartile), and maximum. The interquartile range (IQR) measures the spread of the middle 50% of data. Data points beyond 1.5×IQR from the quartiles are considered mathematical outliers.
Variable Definitions
First Quartile (25th percentile)
The median of the lower half of the data. 25% of values fall below Q1. Also called the lower quartile.
Median (50th percentile)
The middle value of the sorted data set. Same as Q2 in the five-number summary.
Third Quartile (75th percentile)
The median of the upper half of the data. 75% of values fall below Q3. Also called the upper quartile.
Interquartile Range
The range of the middle 50% of data: Q3 − Q1. Resistant to outliers, unlike the full range.
Standard Deviation
A measure of data spread. σ is used for population, s for sample (uses n−1 for unbiased estimate).
How to Use This Calculator
- 1
Enter your data set as numbers separated by commas, spaces, or semicolons.
- 2
Select whether the data represents a sample (most common) or the full population.
- 3
View the comprehensive summary statistics including quartiles, IQR, minimum, maximum, mean, and standard deviation.
- 4
The box-and-whisker plot visually shows the data distribution with the median, quartiles, and any highlighted outliers.
- 5
Outliers beyond 1.5×IQR from the quartiles are flagged for investigation.
A box plot visualizes data spread through the five-number summary and IQR
Understanding the Concept
Descriptive statistics summarize and organize data so it can be easily understood. The five-number summary (minimum, Q1, median, Q3, maximum) provides a complete picture of data spread with just five values. The box plot was invented by John Tukey in 1970 and has become the standard way to visualize statistical distributions. Outliers — data points more than 1.5×IQR from the quartiles — are flagged because they may represent measurement errors, data entry mistakes, or genuinely unusual values that warrant investigation. The difference between sample and population standard deviation is critical: sample standard deviation uses n−1 (Bessel's correction) to provide an unbiased estimate of the population parameter. Without this correction, the sample standard deviation would tend to underestimate the true population value, especially for small samples. The IQR is a robust measure of spread because it ignores the extreme 25% of values on each side, making it resistant to outliers. A small IQR relative to the range indicates that most data is concentrated in the middle, while a large IQR indicates wide dispersion in the central portion of the data.
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