Five Number Summary Calculator

Online calculator for calculating the five number summary of a data series

Five Number Summary Calculator

Five Number Summary

The five number summary is a statistical method that describes the spread of data through five key metrics.

Enter Data
Numbers separated by spaces, semicolons, or one number per line
Five Number Summary Results
Minimum:
Lower Quartile (Q1):
Median (Q2):
Upper Quartile (Q3):
Maximum:
Five Number Summary Properties

Description: The five metrics describe the statistical spread and distribution of the data

Min ≤ Q1 ≤ Median ≤ Q3 ≤ Max Robust Statistics Box-Plot Foundation

Visualization

The five number summary forms the foundation for box plots.
It shows the distribution and spread of the data.

Min Q1 Median Q3 Max IQR (Interquartile Range)

Box-plot representation of the five number summary

What is the Five Number Summary?

The five number summary (Five Number Summary) is a fundamental concept in descriptive statistics:

  • Definition: Describes distribution through five characteristic values
  • Components: Minimum, Q1, Median, Q3, Maximum
  • Robustness: Insensitive to outliers
  • Visualization: Foundation for box plots (Boxplot-Diagramme)
  • Application: Data exploration, quality control, comparisons
  • Interpretation: Shows center, spread, and symmetry of data
  • Quartiles: Divide data into four equal parts
  • IQR: Interquartile range = Q3 - Q1

The Five Metrics in Detail

Each of the five metrics has a specific meaning:

Minimum (Min)
  • Meaning: Smallest value in the dataset
  • Position: 0% of data lies below it
  • Interpretation: Lower boundary of the data
Lower Quartile (Q1)
  • Meaning: 25th percentile of the data
  • Position: 25% of data lies below it
  • Calculation: Position = ¼(n+1)
Median (Q2)
  • Meaning: Middle value (50th percentile)
  • Position: 50% of data lies below it
  • Calculation: Position = ½(n+1)
Upper Quartile (Q3)
  • Meaning: 75th percentile of the data
  • Position: 75% of data lies below it
  • Calculation: Position = ¾(n+1)
Maximum (Max)
  • Meaning: Largest value in the dataset
  • Position: 100% of data lies below it
  • Interpretation: Upper boundary of the data
Interquartile Range (IQR)
  • Meaning: Span of the middle 50% of data
  • Calculation: IQR = Q3 - Q1
  • Usage: Measure of spread

Applications of the Five Number Summary

The five number summary is used in many fields:

Science & Research
  • Experimental data analysis
  • Quality control in laboratories
  • Clinical studies and patient data
  • Environmental data and measurement series
Business & Finance
  • Financial market analysis
  • Income distributions
  • Revenue statistics
  • Risk assessment
Education & Social Sciences
  • Exam results and grade distribution
  • Survey data and polling
  • Demographic analysis
  • Performance comparisons
Industry & Engineering
  • Process monitoring and quality assurance
  • Production statistics
  • Error analysis
  • Lifetime testing

Formulas for the Five Number Summary

Minimum and Maximum
\[\text{Minimum} = x_{\min} = \min\{x_1, x_2, \ldots, x_n\}\] \[\text{Maximum} = x_{\max} = \max\{x_1, x_2, \ldots, x_n\}\]

Smallest and largest value of the sorted data series

Quartile Positions
\[\text{Position Q1} = \frac{1}{4}(n+1)\] \[\text{Position Q2} = \frac{2}{4}(n+1) = \frac{1}{2}(n+1)\] \[\text{Position Q3} = \frac{3}{4}(n+1)\]

Positions of quartiles in sorted data series (n = count)

Quartile Calculation (integer position)
\[Q_k = x_i\]

When position i is a whole number, the quartile equals the value at position i

Quartile Calculation (non-integer position)
\[Q_k = \frac{x_i + x_{i+1}}{2}\]

When position is between i and i+1: average of the two neighboring values

Interquartile Range (IQR)
\[IQR = Q_3 - Q_1\]

The IQR is a robust measure of spread for the middle 50% of data

Outlier Boundaries (Tukey's Fences)
\[\text{Lower Boundary} = Q_1 - 1.5 \times IQR\] \[\text{Upper Boundary} = Q_3 + 1.5 \times IQR\]

Values outside these boundaries are considered potential outliers

Step-by-Step Example Calculation

Given
2, 5, 4, 8, 3, 7, 9, 3, 1, 6

Calculate the five number summary for this data series

1. Sort Data and Count
\[\text{Sorted: } 1, 2, 3, 3, 4, 5, 6, 7, 8, 9\] \[n = 10\]

Sort data in ascending order and determine count

2. Minimum and Maximum
\[\text{Minimum} = 1\] \[\text{Maximum} = 9\]

First and last value of the sorted series

3. Calculate Lower Quartile (Q1)
\[\text{Position Q1} = \frac{1}{4}(10+1) = 2.75\] \[\text{Lies between position 2 and 3:}\] \[Q_1 = \frac{x_2 + x_3}{2} = \frac{2 + 3}{2} = 2.5\]

Position 2.75 → Average of 2 (position 2) and 3 (position 3)

4. Calculate Median (Q2)
\[\text{Position Q2} = \frac{2}{4}(10+1) = 5.5\] \[\text{Lies between position 5 and 6:}\] \[Q_2 = \text{Median} = \frac{x_5 + x_6}{2} = \frac{4 + 5}{2} = 4.5\]

Position 5.5 → Average of 4 (position 5) and 5 (position 6)

5. Calculate Upper Quartile (Q3)
\[\text{Position Q3} = \frac{3}{4}(10+1) = 8.25\] \[\text{Lies between position 8 and 9:}\] \[Q_3 = \frac{x_8 + x_9}{2} = \frac{7 + 8}{2} = 7.5\]

Position 8.25 → Average of 7 (position 8) and 8 (position 9)

6. Five Number Summary - Complete Result
Minimum = 1.00
Q1 = 2.50
Median (Q2) = 4.50
Q3 = 7.50
Maximum = 9.00

IQR (Interquartile Range): \(IQR = Q_3 - Q_1 = 7.5 - 2.5 = 5.0\)

7. Interpretation
  • Distribution: Data ranges from 1 to 9
  • Center: Median is at 4.5 (middle value)
  • Spread: IQR of 5.0 shows moderate spread of middle 50% of data
  • Symmetry: Data is slightly right-skewed (median closer to Q1 than Q3)
  • 25% of values lie below 2.5 and 25% above 7.5

Mathematical Foundations of the Five Number Summary

The five number summary is a fundamental concept in exploratory data analysis and was popularized by American statistician John Tukey. It provides a robust method for describing the location and spread of data.

Basic Principles and Properties

The five number summary is based on fundamental statistical concepts:

  • Order Statistics: Based on sorted data, independent of distribution assumptions
  • Robustness: Insensitive to outliers and extreme values
  • Percentiles: Q1 (25%), Median (50%), Q3 (75%) divide data into four equal parts
  • Completeness: Captures location, spread, and extreme values in compact form
  • Visualizability: Forms the foundation for box-plot representations

Box-Plot and Visual Representation

The five number summary is typically visualized in a box plot:

Box-Plot Components

The "box" ranges from Q1 to Q3 and contains the middle 50% of data. The line in the box marks the median.

Whiskers (Antennae)

Lines from minimum to Q1 and from Q3 to maximum show the spread of data outside the IQR.

Outlier Detection

Values outside Q1 - 1.5×IQR and Q3 + 1.5×IQR are marked as potential outliers.

Comparability

Box-plots enable direct visual comparison of multiple data sets or groups.

Interpretation Possibilities

The five number summary allows various interpretations:

Symmetry of Distribution

If median is exactly midway between Q1 and Q3, indicates symmetric distribution. If closer to Q1 (or Q3), distribution is right- (or left-) skewed.

Spread and Variability

IQR shows spread of middle 50% of data. The larger the IQR, the more variable the data in its center.

Comparison of Datasets

Comparing five number summaries can identify differences in location and spread between different groups or time periods.

Outlier Detection

The 1.5×IQR rule (Tukey's Fences) provides a standardized method for identifying potential outliers in data.

Advantages and Disadvantages

The five number summary has specific strengths and limitations:

Advantages
  • Robustness: Insensitive to outliers and extreme values
  • Simplicity: Easy to calculate and interpret
  • Distribution-free: No assumptions about underlying distribution needed
  • Visualization: Direct graphical representation through box-plots possible
  • Comparability: Enables simple comparisons between groups
Limitations
  • Information Loss: Reduces n data points to 5 metrics
  • No Details: Doesn't show exact shape of distribution
  • Multimodality: Multiple peaks in data are not captured
  • Small Samples: Less informative with few data points
  • Quartile Definition: Different calculation methods can yield slightly different results

Practical Application Guidelines

Data Exploration

Five number summary is ideal for a first overview of new datasets and identifying special features.

Quality Control

In production, it helps monitor process variation and detect deviations from targets.

Research and Reporting

Compact presentation of study results and summarizing large datasets for reports and publications.

Comparative Analysis

Effective comparison of multiple groups, time periods, or conditions through side-by-side box-plots.

Summary

The five number summary is an indispensable tool of exploratory data analysis. Its combination of simplicity, robustness, and informative power makes it a standard in descriptive statistics. It forms the foundation for box-plots and enables quick insights into data structure without requiring detailed distribution assumptions. Combined with other statistical measures, it provides a complete picture of data distribution.

More statistics functions

Arithmetic Mean • Contraharmonic Mean • Covariance • Empirical distribution CDF • Deviation • Five-Number Summary • Geometric Mean • Harmonic Mean • Inverse Empirical distribution CDF • Kurtosis • Log Geometric Mean • Lower Quartile • Median • Pooled Standard Deviation • Pooled Variance • Skewness (Statistische Schiefe) • Upper Quartile • Variance