"Percentiles"

Introduction to Percentiles

• A percentile is a measurement that indicates the value below which a given percentage of observations in a group of observations falls.
• Essential for understanding how data compares relative to a larger dataset.

Experimental Setup

• An experiment is conducted where subjects have their reaction times (in ms) recorded when a louder sound interrupts a constant soft noise.

Data Set

• Reaction times (in milliseconds):
  • 40, 164, 171, 175, 186, 193, 202, 212, 222, 229, 231, 235, 250, 261, 269, 271, 273, 296, 311

Finding Percentiles

Different textbooks define methods for finding percentiles. Here are two common methods:

Method A

1. Order the Observations: Start with the data sorted from smallest to largest.

   • Ordered Data: 40, 164, 171, 175, 186, 193, 202, 212, 222, 229, 231, 235, 250, 261, 269, 271, 273, 296, 311

2. Calculate the Index: Use the formula $n \times \frac{p}{100}$ where:

   • $n$ = total number of observations.
   • $p$ = desired percentile.

3. Determine the pth Percentile:

   • If the index is an integer, find the average of the value at this index and the next position.
   • If the index is not an integer, round up to the next integer and use that value directly.

Example Calculations

• 40th Percentile (
  $n = 18$, $p = 40$)

  • Index = $n \times \frac{40}{100} = 7.2$ (round to 8)
  • 8th observation = 212

• 75th Percentile (
  $n = 18$, $p = 75$)

  • Index = $n \times \frac{75}{100} = 13.5$ (average the 13th and 14th values)
  • $\text{Average} = \frac{250 + 261}{2} = 255.5$

Method B

1. Order the Observations: Same as Method A, start with sorted data.

2. Calculate the Index: Use the formula $n + 1 \times \frac{p}{100}$ where:

   • $n$ = total number of observations.
   • $p$ = desired percentile.

3. Determine the pth Percentile:

   • If the index is an integer, directly use the observation value at that index.
   • If the index is not an integer (represented as $k + d$), find a weighted average between the values at the $k$th and $(k+1)$th positions.

Example Calculations

• 40th Percentile (
  $n = 18$, $p = 40$)

  • Index = $n + 1 \times \frac{40}{100} = 7.6$ (interpolate between 7th and 8th)
  • $40th \text{ Percentile} = 212 + 0.6 \times (222 - 212) = 218$

• 75th Percentile (
  $n = 18$, $p = 75$)

  • Index = $n + 1 \times \frac{75}{100} = 14.25$
  • $75th \text{ Percentile} = 269 + 0.25 \times (271 - 269) = 269.5$

Results

• Using Method A:

  • 40th Percentile: 212 ms
  • 75th Percentile: 255.5 ms

• Using Method B:

  • 40th Percentile: 218 ms
  • 75th Percentile: 269.5 ms

Conclusion

• Both methods yield different results for percentiles, but both are valid.
• Understanding how to compute percentiles is essential for data analysis in statistics.