Percentiles and quartiles are important ideas in statistics that help us understand and interpret data sets. Even though they both help us see how data is spread out, they do this in different ways.

Percentiles

Percentiles divide a data set into 100 equal parts. This allows us to rank data points and see where they stand compared to others.

For any given percentile, like the p-th percentile, it shows the value below which p percent of the data falls.

For example, if a student scored in the 85th percentile on a test, it means that the student did better than 85% of all the other test-takers.

Here’s how you calculate the p-th percentile:

1. Order the Data: Put the data in order from lowest to highest.

2. Determine the Rank: Calculate the rank (R) using this formula: $R = \frac{p}{100} \times (N + 1)$ Here, N is the total number of data points.

3. Find the Percentile Value:

   • If R is a whole number, then the p-th percentile is the value at that position in the ordered list.
   • If R is not a whole number, round it up to the next whole number. Then, take the average of the values at those two positions.

Quartiles

Quartiles split a data set into four equal parts instead of 100. The first quartile (Q1) is the 25th percentile, the second quartile (Q2) is the median (or 50th percentile), and the third quartile (Q3) is the 75th percentile.

Each quartile helps us see how data is spread out in different sections.

Here’s how you find quartiles:

1. Order the Data: Arrange the data from smallest to largest.
2. Find Q1, Q2, and Q3:
   • Q1 (25th Percentile): Find the median of the first half of the data.
   • Q2 (Median): Find the median of the entire data set.
   • Q3 (75th Percentile): Find the median of the second half of the data.

When you look at quartiles, they give you a broader view of the data, showing how it divides into sections rather than focusing on exact positions.

Key Differences

1. Granularity:

   • Percentiles provide a detailed look at the data by dividing it into 100 parts.
   • Quartiles give a general overview by dividing the data into four parts.

2. Interpretation:

   • Percentiles show specific points, allowing us to see how one score compares to the rest. For example, if a score of 90 is in the 95th percentile, it shows that it is much higher than most.
   • Quartiles help us understand how the data is spread out, like seeing if most of the data is below the middle point (the median).

3. Usefulness:

   • Percentiles are useful when we need to see how well someone performed compared to others, like in school tests.
   • Quartiles help us identify trends in data, like grouping people by income in studies about society.

Conclusion

In short, both percentiles and quartiles help us understand how data is organized and spread out. Percentiles give detailed comparisons of individual data points, while quartiles show a broader view of how data is divided into sections. Each of these statistical tools has its own strengths, and they work together to help us better understand statistics in a school setting.