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You're already familiar with quartiles, which divide data into equal parts, right? Well, percentiles are like quartiles' sibling, but they're even more detailed!
If quartiles divide data into chunks, percentiles divide ordered data into equal chunks. That's a lot, huh? Like dividing a chocolate bar into tiny squares.
Each chunk is separated by a percentile value. There are percentile values, starting from , , , ..., up to .
Percentiles are very useful for seeing the position of a specific value relative to the entire dataset, like test-score rankings in one group or a child's growth compared to peers of the same age.
Just like finding quartiles for grouped data, we also use interpolation to find the value of a percentile () when the data is grouped.
The steps are very similar:
First, we determine which data point corresponds to the i-th percentile. The formula is:
Once we have the position, we look at the cumulative frequency table () to find out which class interval this percentile falls into.
Once we know the class, we use this magic interpolation formula:
Where:
Notice, the formula is very similar to the quartile formula, the only difference is the part (quartiles use ).
For example, let's say we have the math test scores of :
| Test Score | Frequency () | Cumulative Frequency () | Lower Boundary () | Class Width () |
|---|---|---|---|---|
We want to find the value of the th Percentile ().
Find the Position of :
The position of is the -th data point.
Determine the Class of :
Look at the column. Where is the th data point? The class has (not enough). The class has (data points through are here). So, the class is .
Gather Ingredients for the Formula:
Calculate :
So, the th Percentile value is . This means of the students scored or less, and scored above .
Try calculating the value of the th Percentile () from the math test score data above!
Position of :
The position of is the -th data point.
Class of :
Look at . The th data point is in the class (because the previous class's is , and this class's is ).
Formula Ingredients:
Calculate :
The th Percentile value is .
| Total |
