Lecture on Percentiles and Measures of Data Dispersion

Introduction to Percentiles

• Percentile is different from percentage.
• Definition: Data value where at least 100p% of the data are ≤ it and at least 100(1-p)% are ≥ it.
• Example: 50th percentile is the median (50% of data ≤ value and 50% of data ≥ it).

Example of Percentile Calculation

• 99th Percentile: 100 * 0.99 = 99% of data ≤ value.
• 1st Percentile: 1% of data ≤ value.
• Algorithm for Percentile Calculation:
  1. Arrange data in ascending order.
  2. Calculate n * p.
  3. If np is not an integer, find the smallest integer ≥ np (this value is the percentile).
  4. If np is an integer, average the values at positions np and np + 1.

Detailed Examples

• n = 5, p = 0.5: np = 2.5 → 3rd value (median).
• n = 6, p = 0.5: np = 3 → Average of values at 3rd and 4th positions.

Using Google Sheets for Percentile Calculation

• Percentile Function in Google Sheets:
  • Formula: =PERCENTILE(data, p).
  • Example: Percentile for data (B2:B11) and p (C2).
  • Google Sheets uses a slightly different algorithm.

Google Sheets Algorithm for Percentile Calculation

1. Arrange data in ascending order.
2. Calculate rank: p * (n - 1) + 1.
3. Split rank into integer and fractional parts.
4. Compute ordered data value for integer part.
5. Calculate percentile value: xi + fractional part * (xi+1 - xi).

Importance of Percentiles in Data Analysis

• Quartiles:
  • 25th percentile (Q1): First quartile
  • 50th percentile (Q2): Median or second quartile
  • 75th percentile (Q3): Third quartile
• Five Number Summary: Minimum, Q1, Median, Q3, Maximum.

Quartiles and Interquartile Range (IQR)

• Breaks the data set into four parts.
• Interquartile Range (IQR): Difference between Q3 and Q1, a measure of dispersion.

Conclusion

• Covered definitions and computations of percentiles, quartiles, and measures of dispersion.
• Example calculations and use of Google Sheets.
• Introduction to the next topic: Association between two variables.