Five-Number Summary

Created:August 11, 2024

This calculator helps you analyze the distribution of your data through its key percentiles. It calculates the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values, providing a comprehensive overview of your dataset's spread and central tendency. The calculator also generates both a professional box plot and an annotated histogram to visualize your data's distribution, automatically detects and highlights outliers, and provides detailed interpretation explaining whether your data is symmetric or skewed.

Quick Calculator

Need a quick calculation? Enter your numbers below:

Enter numbers (separated by commas or spaces):

Calculator

1. Load Your Data

2. Select Columns & Options

Select Numeric Column:

Exclude Outliers

Related Calculators

Box Plot Calculator

Percentile and Quartile Calculator

Mean, Median, Mode Calculator

Range, Variance, Standard Deviation Calculator

Learn More

Understanding the Five-Number Summary

Definition

The Five-Number Summary provides a comprehensive overview of a dataset's distribution through five key values: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

Components

Minimum ( $x_{min}$ ): Smallest value in the dataset
First Quartile ( $Q_1$ ): 25th percentile
Median ( $Q_2$ ): 50th percentile
Third Quartile ( $Q_3$ ): 75th percentile
Maximum ( $x_{max}$ ): Largest value in the dataset

Key Metrics

Range: $Range = x_{max} - x_{min}$

Interquartile Range (IQR): $IQR = Q_3 - Q_1$

Important Considerations

Not sensitive to the exact values of outliers
May not fully capture multimodal distributions
Should be used alongside other descriptive statistics

Practical Example with Box Plot

Understanding Five-Number Summary Through Box Plots

Consider a dataset of student test scores. Here's how to read the five-number summary from a box plot:

Five-Number Summary

Minimum: 10 points
Q1: 15 points (25th percentile)
Median: 20 points (50th percentile)
Q3: 25 points (75th percentile)
Maximum: 30 points

Box Plot Components

The box spans from Q1 to Q3
The line inside the box shows the median
The whiskers extend to min and max values
Points beyond whiskers represent outliers
The IQR is the height of the box (Q3 - Q1 = 10 points)

Interpretation

• The middle 50% of students scored between 15 and 25 points
• The median score of 20 points indicates typical performance
• The distribution is symmetric (median in center of box)
• Two outliers exist at 8 and 32 points
• The total range (excluding outliers) is 20 points

Creating Your Own Box Plot

To create a box plot from your data:

Arrange your data in ascending order
Find the median (Q2) by locating the middle value
Find Q1 (median of lower half) and Q3 (median of upper half)
Calculate the IQR (Q3 - Q1)
Identify outliers (values beyond Q1 - 1.5×IQR or Q3 + 1.5×IQR)
Draw the box spanning Q1 to Q3 with a line at the median
Add whiskers extending to the min/max (excluding outliers)
Plot any outliers as individual points

You can use our Box Plot Calculator to create box plots automatically from your data.

How to calculate Five Number Summary in R

Use summary() for basic statistics orquantile() for more control over percentiles.

# Sample numerical data
data <- c(12, 15, 18, 20, 22, 25, 28, 30, 35, 40, 45, 50, 55, 60, 65)

# Method 1: Quick summary (includes mean and other stats)
summary(data)

# Method 2: Exact five number summary
five_num <- quantile(data, probs = c(0, 0.25, 0.5, 0.75, 1))
names(five_num) <- c("Minimum", "Q1", "Median", "Q3", "Maximum")
print(five_num)

# Method 3: Detailed analysis with additional statistics
library(tidyverse)

five_number_summary <- tibble(
  Statistic = c("Minimum", "Q1", "Median", "Q3", "Maximum", "Range", "IQR"),
  Value = c(
    min(data),
    quantile(data, 0.25),
    median(data),
    quantile(data, 0.75),
    max(data),
    max(data) - min(data),
    IQR(data)
  )
)
print(five_number_summary)

# Visualization: Box plot
boxplot(data, 
        main = "Five Number Summary Box Plot",
        ylab = "Values",
        col = "lightblue",
        border = "darkblue")

# Add labels to the box plot
text(1.3, quantile(data, c(0, 0.25, 0.5, 0.75, 1)), 
     labels = c("Min", "Q1", "Median", "Q3", "Max"), 
     cex = 0.8)

How to calculate Five Number Summary in Python

Use describe() method in pandas or numpy.percentile() for precise control.

Python

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

# Sample data
data = [12, 15, 18, 20, 22, 25, 28, 30, 35, 40, 45, 50, 55, 60, 65]
df = pd.DataFrame({'values': data})

# Method 1: Using pandas describe (includes additional stats)
print("Pandas describe():")
print(df['values'].describe())

# Method 2: Manual calculation of five number summary
five_number = {
    'Minimum': np.min(data),
    'Q1': np.percentile(data, 25),
    'Median': np.median(data),
    'Q3': np.percentile(data, 75),
    'Maximum': np.max(data)
}

# Additional statistics
five_number['Range'] = five_number['Maximum'] - five_number['Minimum']
five_number['IQR'] = five_number['Q3'] - five_number['Q1']

# Display as DataFrame
summary_df = pd.DataFrame(list(five_number.items()), 
                         columns=['Statistic', 'Value'])
print("\nFive Number Summary:")
print(summary_df)

# Method 3: Using numpy quantile for exact percentiles
percentiles = np.percentile(data, [0, 25, 50, 75, 100])
labels = ['Minimum', 'Q1', 'Median', 'Q3', 'Maximum']

for label, value in zip(labels, percentiles):
    print(f"{label}: {value}")

# Visualization: Box plot
plt.figure(figsize=(8, 6))
plt.boxplot(data, tick_labels=['Data'])
plt.title('Five Number Summary Box Plot')
plt.ylabel('Values')
plt.grid(True, alpha=0.3)
plt.show()

# Histogram with quartile lines
plt.figure(figsize=(10, 6))
plt.hist(data, bins=10, alpha=0.7, color='lightblue', edgecolor='black')
for i, (label, value) in enumerate(zip(labels, percentiles)):
    color = ['red', 'orange', 'green', 'orange', 'red'][i]
    plt.axvline(value, color=color, linestyle='--', linewidth=2, label=label)
plt.xlabel('Values')
plt.ylabel('Frequency')
plt.title('Distribution with Five Number Summary')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

How to calculate Five Number Summary in Excel

Use MIN, QUARTILE, and MAX functions, or create a box plot for visualization.

Excel

=MIN(A1:A100)    # Minimum
=QUARTILE(A1:A100,1)  # Q1 (First Quartile)
=MEDIAN(A1:A100)      # Q2 (Median)
=QUARTILE(A1:A100,3)  # Q3 (Third Quartile)
=MAX(A1:A100)         # Maximum
=MAX(A1:A100)-MIN(A1:A100)  # Range
=QUARTILE(A1:A100,3)-QUARTILE(A1:A100,1)  # IQR

Method 1: Using Formulas

=MIN(A1:A100) - Find the minimum value

=QUARTILE(A1:A100,1) - Calculate Q1 (25th percentile)

=MEDIAN(A1:A100) - Find the median (Q2)

=QUARTILE(A1:A100,3) - Calculate Q3 (75th percentile)

=MAX(A1:A100) - Find the maximum value

Method 2: Data Analysis ToolPak

Go to Data → Data Analysis
Select Descriptive Statistics
Choose your data range
Check Summary statistics
Click OK to get comprehensive statistics including quartiles

Method 3: Box Plot Visualization

Select your data range
Go to Insert → Charts → Statistical → Box and Whisker
Excel will automatically create a box plot showing your five number summary
The box shows Q1, median, and Q3; whiskers show min and max