5 Number Summary
Print PageA 5 Number Summary Calculator (also known as a Five-Number Summary Generator, Box Plot Quartile Utility, IQR & Outlier Boundary Calculator, or Exploratory Order Statistics Analyzer) computes the five key descriptive order statistics of a numerical dataset: Minimum, First Quartile (Q1), Median (Q2), Third Quartile (Q3), and Maximum.
In data science, software engineering latency tracking, financial risk management, and clinical trial reporting, the 5-number summary provides a robust, outlier-resistant snapshot of a dataset’s center, spread, range, and skewness, forming the mathematical foundation for Tukey Box-and-Whisker Plots.
Our free online 5 Number Summary Calculator provides instant calculations across all order statistic parameters:
- Five-Number Summary Vector:
S = { Minimum, Q1, Median (Q2), Q3, Maximum }. - Minimum (Min): The smallest data point in the ordered dataset.
- First Quartile (Q1 / 25th Percentile): The median of the lower half of the dataset.
- Median (Q2 / 50th Percentile): The middle value dividing the dataset into two equal halves.
- Third Quartile (Q3 / 75th Percentile): The median of the upper half of the dataset.
- Maximum (Max): The largest data point in the ordered dataset.
- Interquartile Range (IQR):
IQR = Q3 - Q1(Middle 50% data dispersion). - Total Range:
Range = Maximum - Minimum. - Lower Outlier Fence (LFB):
LFB = Q1 - ( 1.5 · IQR )(Values below LFB are mild outliers). - Upper Outlier Fence (UFB):
UFB = Q3 + ( 1.5 · IQR )(Values above UFB are mild outliers). - Extreme Outlier Fences:
Q1 - ( 3.0 · IQR )andQ3 + ( 3.0 · IQR ).
Master 5-Number Summary Reference Table (API Server Latency: n = 20 Response Times in ms)
The table below displays the 5-number summary, quartiles, IQR, and outlier boundary fences for 20 API server response times (n = 20 Latency Measurements in Milliseconds):
| 5-Number Summary Parameter | Calculated Value (Milliseconds) | Percentile / Rank Definition | Box Plot Visual Component | Dataset Distribution Interpretation |
|---|---|---|---|---|
| Minimum (Min) | 120.0 ms | 0th Percentile (Smallest data point) | Lower Whisker End | Fastest API response time recorded |
| First Quartile (Q1) | 152.5 ms | 25th Percentile (Lower 25% boundary) | Bottom Edge of Box | 25% of server requests take ≤ 152.5 ms |
| Median (Q2 / Center) | 177.5 ms | 50th Percentile (Exact dataset middle) | Center Line in Box | 50% of requests are faster/slower than 177.5 ms |
| Third Quartile (Q3) | 205.0 ms | 75th Percentile (Upper 25% boundary) | Top Edge of Box | 75% of server requests take ≤ 205.0 ms |
| Maximum (Max – Raw) | 420.0 ms | 100th Percentile (Largest data point) | Plotted Dot (Outlier) | Confirmed Upper Outlier (> 283.75 ms) |
| Interquartile Range (IQR) | 52.5 ms (205.0 – 152.5) | Middle 50% Spread (Q3 – Q1) | Height of Central Box | Middle 50% of requests span 52.5 ms |
| Upper Outlier Fence (UFB) | 283.75 ms (205 + 1.5·52.5) | Tukey 1.5·IQR Boundary Limit | Upper Whisker Cap (260 ms) | Any latency exceeding 283.75 ms is an outlier |
Step-by-Step Server Latency 5-Number Summary Calculation
To calculate the 5-number summary for 20 server response times (120, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 210, 220, 240, 260, 420 ms):
Step 1 (Order Dataset): Dataset is already sorted in ascending order (n = 20)
Step 2 (Identify Minimum & Maximum): Minimum = 120.0 ms, Maximum = 420.0 ms
Step 3 (Calculate Median Q2): Average of 10th and 11th values &implies; Q2 = (175 + 180) ÷ 2 = 177.5 ms
Step 4 (Calculate First Quartile Q1): Median of lower 10 values (5th & 6th values) &implies; Q1 = (150 + 155) ÷ 2 = 152.5 ms
Step 5 (Calculate Third Quartile Q3): Median of upper 10 values (15th & 16th values) &implies; Q3 = (200 + 210) ÷ 2 = 205.0 ms
Step 6 (Calculate Interquartile Range IQR): IQR = Q3 - Q1 = 205.0 - 152.5 = 52.5 ms
Step 7 (Calculate Upper Outlier Fence UFB): UFB = Q3 + (1.5 · IQR) = 205.0 + (1.5 · 52.5) = 205.0 + 78.75 = 283.75 ms
Step 8 (Identify Outliers): 420.0 ms > 283.75 ms &implies; 420.0 ms is a confirmed extreme latency outlier
Thus, the 5-number summary is { 120.0 ms, 152.5 ms, 177.5 ms, 205.0 ms, 420.0 ms }, with a middle 50% spread (IQR) of 52.5 ms and an upper outlier boundary at 283.75 ms.
Descriptive Statistics Comparison: 5-Number Summary vs. Mean & SD vs. Percentiles
Below is a comparative reference chart detailing when to use the 5-Number Summary versus alternative summary metrics:
| Summary Method | Included Core Parameters | Outlier Resistance & Skew Robustness | Primary Practical Application |
|---|---|---|---|
| Five-Number Summary | Min, Q1, Median (Q2), Q3, Max | HIGHLY ROBUST (Un-affected by extreme outliers) | Skewed distributions, income levels, latency tracking, box plots. |
| Mean & Standard Deviation | Arithmetic Mean (μ) and SD (σ) | LOW ROBUSTNESS (Heavily distorted by outliers) | Symmetric normal bell curves & parametric hypothesis testing. |
| Deciles / Centiles (Percentiles) | P10, P25, P50, P75, P90, P99 | HIGH ROBUSTNESS | Pediatric growth charts, SLA 99th percentile compliance. |
History & Mathematics: 1970 Arthur Bowley to 1977 John W. Tukey
1970 Arthur Bowley & Quartile Measures
In the 1920s, English statistician Arthur Bowley introduced five-point summary concepts in exploratory statistics, advocating quartile-based measures of skewness.
1977 John W. Tukey & Exploratory Data Analysis
In 1977, Princeton mathematician John Wilder Tukey formalized the 5-number summary and invented the **Box-and-Whisker Plot** in his landmark book Exploratory Data Analysis, establishing 1.5 · IQR as the standard boundary rule for detecting data outliers.
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Frequently Asked Questions (FAQ)
What are the five values in a 5-number summary?
The five values are: Minimum, First Quartile (Q1), Median (Q2), Third Quartile (Q3), and Maximum.
How do you calculate the Interquartile Range (IQR) from a 5-number summary?
Subtract the First Quartile (Q1) from the Third Quartile (Q3): IQR = Q3 - Q1.
How do you identify outliers using a 5-number summary?
Calculate Tukey’s outlier fences: Lower Fence = Q1 - ( 1.5 · IQR ) and Upper Fence = Q3 + ( 1.5 · IQR ). Any data point outside these fences is an outlier.