Understanding Quartiles and Summary

Overview

This lesson explains how to find quartiles and the five-number summary of a data set using the locator (percentile) method, with examples for both raw lists and frequency tables.

Understanding Quartiles and the Median

• The median (Q2) divides ordered data so half the values are below and half above.
• Quartiles split data into four equal parts: Q1 (first quartile, 25%), Q2 (second quartile, median, 50%), Q3 (third quartile, 75%).
• 25% of data lies between the minimum and Q1; 25% between Q1 and Q2; 25% between Q2 and Q3; 25% between Q3 and the maximum.

The Five-Number Summary

• The five-number summary consists of the minimum, Q1, median (Q2), Q3, and maximum values.
• This summary provides a snapshot of data distribution but is not a measure of variation like standard deviation.

Finding Quartiles: Locator Method

• Order data from smallest to largest before finding quartiles.
• For Q1, compute L = 0.25 × N (N = number of data values).
• If L is a decimal, round up to the next whole number; Q1 is the value at that position.
• If L is whole, Q1 is the average of the Lth and (L+1)th values.
• Repeat for Q3 using L = 0.75 × N.

Example with Raw Data

• Example: For 15 values, the median is at position 8 (Q2).
• Calculate L for Q1: 0.25 × 15 = 3.75 → round up to 4; Q1 is the 4th value.
• Calculate L for Q3: 0.75 × 15 = 11.25 → round up to 12; Q3 is the 12th value.
• Five-number summary: 6, 18, 49, 82, 97.

Example with Frequency Table

• Find total data values (N) by summing frequencies.
• Minimum and maximum are the smallest and largest values in the table.
• For N=30 (even), median is the average of 15th and 16th values.
• Q1: L = 0.25 × 30 = 7.5 → round up to 8th value; find corresponding value.
• Q3: L = 0.75 × 30 = 22.5 → round up to 23rd value; find corresponding value.
• Five-number summary: 30, 40, 50, 70, 80.

Key Terms & Definitions

• Median (Q2) — The middle value of ordered data.
• Quartiles (Q1, Q2, Q3) — Values dividing data into four equal parts.
• Five-number summary — Set: minimum, Q1, median, Q3, maximum.
• Locator method — Technique using percentiles to find quartiles.
• Frequency table — Table listing data values and their counts.

Action Items / Next Steps

• Review five-number summary and locator method steps.
• Prepare to create box plots from the five-number summary in the next lesson.