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Enter your data (comma, space, or newline separated) Accepts integers and decimals. Minimum 4 values recommended.

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About

The five-number summary is a set of descriptive statistics that partitions a dataset into four equal groups. It consists of the minimum value x_min, the first quartile Q₁ (25th percentile), the median Q₂ (50th percentile), the third quartile Q₃ (75th percentile), and the maximum value x_max. These five values define the shape, spread, and center of a distribution without assuming normality. Misidentifying quartile boundaries leads to incorrect interquartile range calculations, which directly corrupts outlier fencing thresholds used in quality control (Six Sigma), clinical trial data screening, and financial risk models.

This calculator uses the exclusive quartile method (Tukey's hinges), where Q₁ and Q₃ are computed as medians of the lower and upper halves of the sorted data, excluding the overall median for odd-sized datasets. This matches the method used by most statistics textbooks and TI-series calculators. Note: results may differ slightly from tools using the inclusive (Moore & McCabe) method or interpolation-based percentile methods. The tool requires a minimum of 4 data points to produce meaningful quartile splits.

Formulas

The five-number summary is computed from a sorted dataset x₁ ≤ x₂ ≤ … ≤ x_n.

Q₂ =

{

x_(n+1)/2 if n is oddx_n/2 + x_n/2+12 if n is even

Q₁ is the median of the lower half (indices 1 to m), and Q₃ is the median of the upper half (indices n − m + 1 to n), where m = floor(n ÷ 2).

IQR = Q₃ − Q₁

LF = Q₁ − 1.5 × IQR

UF = Q₃ + 1.5 × IQR

Where Q₂ = median, IQR = interquartile range, LF = lower fence, UF = upper fence. Any data point x_i where x_i < LF or x_i > UF is classified as an outlier.

Reference Data

Statistic	Symbol	Percentile	Description	Sensitivity to Outliers
Minimum	x_min	0th	Smallest observation in the dataset	Extremely high
First Quartile	Q₁	25th	Median of the lower half	Low
Median	Q₂	50th	Middle value of sorted data	Low
Third Quartile	Q₃	75th	Median of the upper half	Low
Maximum	x_max	100th	Largest observation in the dataset	Extremely high
Interquartile Range	IQR	-	Q₃ − Q₁	Robust (resistant)
Lower Fence	LF	-	Q₁ − 1.5 × IQR	Robust
Upper Fence	UF	-	Q₃ + 1.5 × IQR	Robust
Range	R	-	x_max − x_min	Extremely high
Mean	x	-	Arithmetic average of all values	High
Sample Size	n	-	Total count of observations	None
Mild Outlier	-	-	Between 1.5× and 3× IQR from quartile	-
Extreme Outlier	-	-	Beyond 3× IQR from quartile	-

Frequently Asked Questions

The exclusive method (Tukey's hinges, used here) excludes the median from both halves when computing Q1 and Q3 for odd-sized datasets. The inclusive method includes the median in both halves. For a dataset of n = 7, the exclusive method takes Q1 as the median of the lower 3 values and Q3 as the median of the upper 3 values. The inclusive method would use all 4 values in each half. Results can differ by small amounts, so always verify which method your course or industry standard requires.

A minimum of 4 data points is required to produce meaningful quartile splits. With 1 value, min = max = median. With 2-3 values, quartile computation is ambiguous and method-dependent. The calculator will warn you and still attempt computation, but results should be interpreted with caution for very small samples.

There are at least 9 recognized methods for computing sample quantiles (Hyndman & Fan, 1996). Excel's QUARTILE.EXC uses a linear interpolation method (Type 6), R's default quantile() uses Type 7, and this calculator uses Tukey's method. For large datasets (n > 30), all methods converge. Differences are most noticeable with small or discrete datasets.

A mild outlier falls between 1.5× IQR and 3× IQR beyond Q1 or Q3. An extreme outlier falls beyond 3× IQR. Formally: mild if Q1 − 3×IQR ≤ x < Q1 − 1.5×IQR or Q3 + 1.5×IQR < x ≤ Q3 + 3×IQR. Extreme if x < Q1 − 3×IQR or x > Q3 + 3×IQR. The 1.5 multiplier was chosen by Tukey because it captures approximately 99.3% of normally distributed data.

Yes. Unlike mean and standard deviation, the five-number summary makes no distributional assumptions. It is particularly effective for skewed distributions, bimodal data, or datasets with outliers where parametric summaries are misleading. The box plot derived from it visually reveals skewness: if the median line is not centered in the box, the distribution is skewed.

Whiskers extend from Q1 down to the smallest data point within the lower fence (Q1 − 1.5×IQR) and from Q3 up to the largest data point within the upper fence (Q3 + 1.5×IQR). They do NOT extend to min and max. Points beyond the fences are plotted individually as outlier dots. This is the standard Tukey box plot convention.