Lecture on Quartiles, Deciles, and Percentiles

Introduction

• Quartiles, Deciles, and Percentiles are methods to divide data into equal parts.
• Useful for understanding data distribution.

Quartiles

• Quartiles divide data into four equal parts.
  • Think of quarters in a dollar.
• Number Line Example:
  • Divide from 0 to 100 into four parts: Q1, Q2 (median), Q3.
  • Percentile representation: Q1 (25th percentile), Q2 (50th percentile), Q3 (75th percentile).
• Q2: Median of entire data set.
• Q1: Median of lower half of data.
• Q3: Median of upper half of data.

Deciles

• Deciles divide data into ten equal parts.
  • Think of a decimeter, one-tenth of a meter.
• Number Line Example:
  • D1, D2, ..., D10.
  • D5 represents the 50th percentile.
  • D4 represents the 40th percentile.

Percentiles

• Percentiles divide data into 100 equal parts.
  • Represented in percentage: 100% is the total.
• Understanding Percentiles:
  • Example: 70th percentile means 70% of data is less than or equal to the point.

Calculating Quartiles

• Example Dataset: 2, 3, 5, 7, 8, 10, 11, 13, 15, 16, 19.
• Finding Q2 (Median):
  • Eliminate pairwise from ends to find the middle number.
  • Q2 = 10.
• Finding Q1:
  • Median of lower half: Q1 = 5.
• Finding Q3:
  • Median of upper half: Q3 = 15.

Calculating Percentiles using Formula

• Formula: Location = (K/100) * (n + 1)
  • K: Percentile desired (e.g., 25 for 25th percentile)
  • n: Total number of data points.
• Example Calculations:
  • For Q1 (25th percentile):
    • Location = (25/100) * (11 + 1) = 3rd position.
  • Similar process for Q2 and Q3 using 50th and 75th percentiles.

Examples with Even Number of Items

• Dataset Example:
  • Two equal parts; median is average of two middle numbers.
  • Calculate Q1, Q2, Q3 as averages of middle numbers.
• Use Formula for Location:
  • Adjust for decimal results by averaging adjacent item values.

Calculating Deciles

• Example to Find 6th Decile (60th percentile):
  • Apply formula: Location between 10th and 11th item.
  • Average these values for precise answer.

Understanding Percentile Values

• Given Data Value, Determine Percentile:
  • Formula: P = (X + 0.5Y) / n * 100
  • Use for finding percentile of specific value (e.g., 12).

Cumulative Relative Frequency Table

• Steps to Create:
  • List values, calculate frequency, relative frequency, cumulative frequency.
• Use to Calculate Deciles:
  • Identify percentile location between cumulative frequencies.
  • Match or average values as necessary.

Conclusion

• Key Understanding:
  • Quartiles divide data into 4 parts, deciles into 10, percentiles into 100.
  • Calculations involve understanding positions and averages in datasets.
• Practical Application:
  • Useful in data analysis, exam scores, and performance metrics.

End of Lecture