Converting magnitude into position makes it possible to compare values across different distributions.
To decide whether a score of 82 is high, you need to know the test difficulty and how everyone else scored. If most students scored above 90, 82 is relatively low; if most scored between 50 and 70, it may be very high.
A raw score tells you the magnitude of a value. A percentile tells you where that value stands within a reference distribution.
Key question
How can we express the position of one value within the full set of observations?
What does a percentile mean?
The 90th percentile means that about 90% of the observations in the reference group are at or below that value. It places the value near the top 10%; it does not mean that the raw score is 90.
| Expression | Meaning |
|---|---|
| 90% | a proportion of 90 out of 100 |
| 90 points | the original measured score |
| 90th percentile | a position at or above roughly 90% of the reference group |
Percentile value vs. percentile rank
Some textbooks and software distinguish the value at a given percentile from the percentile rank of an observed value.
How do quartiles divide the data?
| Measure | Position | Meaning |
|---|---|---|
| Q1 | 25th percentile | about 25% are at or below this value |
| Q2 | 50th percentile | the median |
| Q3 | 75th percentile | about 75% are at or below this value |
| IQR | Q3−Q1 | width of the middle 50% |
Software packages use slightly different conventions for calculating quartiles. In small samples, Q1 and Q3 may differ across methods, so reporting the method can be helpful.
Why is the IQR resistant to extremes?
IQR=Q3−Q1 does not use the minimum and maximum directly. As a result, one extremely large observation usually changes it much less than it changes the range.
Potential-outlier rule
Boxplots commonly flag values below Q1−1.5×IQR or above Q3+1.5×IQR as potential outliers.
Not an automatic deletion rule
A value beyond 1.5×IQR is an observation to investigate, not automatically an error that should be removed.
How do raw score and position differ?
| Reference distribution | Interpretation of 82 |
|---|---|
| Most scores are 90–100 | low percentile |
| Most scores are 50–70 | high percentile |
| Scores are widely dispersed | possibly moderately high |
Percentiles are useful for comparing relative position across tests, growth charts, or income distributions with different units and shapes. They do not preserve the actual distance between values.
Move the score
Start with a value near the middle and inspect its percentile rank. Then move it up and down to see why equal score changes do not always produce equal rank changes.
Q1
55
Q2
70
Q3
82
Percentile rank
53%
8 values are at or below 72.
What to notice
In a dense part of the distribution, a small score change can move the percentile substantially. In a sparse region, a larger score change may barely change the position.