Percentiles

Overview

A percentile indicates the relative standing of a data point within a data set. It is the value below which a given percentage of observations fall. Percentiles help to interpret the position of a value relative to the rest of the data and are widely used in assessments, grading, and statistics.

Key Definitions

Percentile:

• The p$$-th percentile ($0 \leq p \leq 100$) is the value below which $p$% of the data falls.
• For example, the 25$$th percentile is the value below which 25% of the data lies.

Quartiles:

• Q1 (25th percentile): The value below which 25% of the data lies.
• Q2 (50th percentile or median): The value below which 50% of the data lies.
• Q3 (75th percentile): The value below which 75% of the data lies.

Interpreting Percentiles:

• Percentiles divide the data into 100 equal parts, helping compare individual data points within a distribution.

Calculating Percentiles

Rank the Data: Arrange the data points in ascending order.

Determine the Position:

Use the formula:

$$P = \frac{p}{100} \times (n + 1)$$

Where:

• $P$: Position of the percentile in the ordered data.

• $p$: Desired percentile (e.g., $p = 25$ for the 25th percentile).

• $n$: Number of data points. Interpolate if Necessary:

• If $p$ is not an integer, take the average of the two closest data points.

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Worked Examples

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Summary

• Percentiles show the relative standing of a value in a data set.
• Quartiles are special percentiles:
  • $Q1$: 25th percentile.
  • $Q2$: 50th percentile (median).
  • $Q3$: 75th percentile.
• To calculate percentiles:
  1. Arrange data in ascending order.
  2. Use $P = \frac{p}{100} \times (n + 1)$ to find the position.
  3. Interpolate if $P$ is not an integer.
• Percentiles are valuable for comparing data and understanding distributions.