"Five-number summary and interquartile range"

Overview of the Five-Number Summary and Interquartile Range

• The five-number summary is a descriptive statistic that provides a quick overview of a dataset. It consists of the following components:
  • Minimum: Lowest data value.
  • Lower Quartile (Q1): The median of the lower half of the data (25th percentile).
  • Median (Q2): Middle value of the ordered data set (50th percentile).
  • Upper Quartile (Q3): The median of the upper half of the data (75th percentile).
  • Maximum: Highest data value.

Steps to Calculate Five-Number Summary

1. Order the Data: Start with organizing the data from least to greatest.

   Example Data: $[51, 51, 53, 57, 57, 63, 64, 70, 73, 76, 77, 79, 81, 83]$

2. Determine the Minimum and Maximum:

   • Minimum: $51$
   • Maximum: $83$

3. Calculate the Median:

   • For an ordered list with an odd number of values, the median is the middle value.
   • For even numbers, average the two middle numbers.
   • Given the data above, the median is $70$

4. Find Lower and Upper Quartiles:

   • Lower Quartile (Q1): Median of the lower half (values left of the median).
     • Values: $[51, 51, 53, 57, 57, 63, 64]$
     • Q1 Calculation: $\frac{57 + 57}{2} = 57$
   • Upper Quartile (Q3): Median of the upper half (values right of the median).
     • Values: $[73, 76, 77, 79, 81, 83]$
     • Q3 Calculation: $\frac{79 + 81}{2} = 80$

5. Calculate the Interquartile Range (IQR):

   • IQR = Q3 - Q1
   • IQR Calculation: $80 - 57 = 23$

Summary of Results:

• Minimum: $51$
• Lower Quartile (Q1): $57$
• Median (Q2): $70$
• Upper Quartile (Q3): $80$
• Maximum: $83$
• Interquartile Range (IQR): $23$