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Five-Number Summary

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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 (xminx_{min}): Smallest value in the dataset
  • First Quartile (Q1Q_1): 25th percentile
  • Median (Q2Q_2): 50th percentile
  • Third Quartile (Q3Q_3): 75th percentile
  • Maximum (xmaxx_{max}): Largest value in the dataset

Key Metrics

Range: Range=xmax−xminRange = x_{max} - x_{min}

Interquartile Range (IQR): IQR=Q3−Q1IQR = Q_3 - Q_1

Applications

  • Detecting outliers using the IQR method
  • Creating box plots for visual data representation
  • Comparing distributions across different groups

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:

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

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

Related Links

Box Plot Calculator

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Percentile and Quartile Calculator

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Mean, Median, Mode Calculator

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Range, Variance, Standard Deviation Calculator

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