Quartiles, Deciles, and Percentiles

Introduction

• Quartiles divide data into four equal parts.

  • Think of quarters (4 quarters = $1).
  • Number line: divided into four equal parts (Q1, Q2, Q3, Q4).
  • Percentile representation:
    • 0 percentile = 0
    • Q1 = 25th percentile
    • Q2 = 50th percentile (median)
    • Q3 = 75th percentile
    • 100th percentile = 100

• Percentiles divide data into 100 equal parts.

  • Percent represent total.

• Deciles divide data into 10 equal parts.

  • Analogy: decimeter (1/10 of a meter).
  • D1 to D10 correspond to each decile.
  • Decile percentile correspondences:
    • D1 = 10th percentile
    • D2 = 20th percentile, etc.
    • D5 = 50th percentile (Q2)
    • D9 = 90th percentile

Understanding Percentiles

• Percentile Meaning: data point where X% of the data is less than or equal.
  • Example: 70th percentile means 70% of data is <= data point.
  • Provides a ranking system (e.g., SAT score example).

Calculating Quartiles

1. Finding Quartiles in Data List:

   • Example: 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 19
   • Q2: median of the data set (10)
   • Q1: median of the lower half (5)
   • Q3: median of the upper half (15)

2. Using Percentiles to Find Quartiles:

   • Formula: ( \frac{K}{100} \times (n + 1) )
   • Example: 25th percentile (Q1)
   • Calculate location in the data list.
   • Verify by locating specific data points and averaging if necessary.

3. Handling Even Number Data Sets:

   • Example: calculate quartiles for even-numbered data sets by averaging.
   • Use same percentile formula to identify locations.

Calculating Deciles

• Example: Find the 60th percentile by averaging relevant data points.
• Useful for estimating within smaller data sets.

Calculating Percentile from Value

• Given a value, find corresponding percentile using:
  • Formula: ( \frac{X + 0.5Y}{n} \times 100 )
  • Example: Finding percentile for value 12 in a list.

Cumulative Relative Frequency Table

• Purpose: Helps visualize and calculate percentiles and deciles.

• Steps:

  1. List values and frequencies.
  2. Calculate relative frequency (frequency/total).
  3. Calculate cumulative relative frequency (add up relative frequencies).

• Using the Table:

  • Identify deciles/percentiles based on cumulative frequency.
  • Average numbers when the percentile is between two cumulative frequencies.

Summary

• Differentiation between quartiles, deciles, and percentiles.
• Techniques for calculating each.
• Use of frequency tables for visualizing and determining data percentiles.

The lecture provides a comprehensive understanding of how quartiles, deciles, and percentiles break down data and how to calculate them accurately. The illustrative examples and formulas provide key tools for working with statistical data sets.