Statistics Reference
Percentiles and Quartiles Explained
Percentiles answer “where does this value sit?” rather than “how big is it?” — which is why they power test scores, growth charts, and box plots. This guide computes them on one small dataset, treats the method disagreements honestly (Excel, R, and your textbook can all differ), and follows the quartiles to their most useful application: outlier fences.
Position, Not Size
The pth percentile is the value below which about p% of the observations fall. Quartiles are the three percentiles that cut ordered data into four equal parts:
| Name | Percentile | Meaning |
|---|---|---|
| Q1 | 25th | A quarter of the data lies below |
| Q2 | 50th | The median — half below, half above |
| Q3 | 75th | Three quarters of the data lies below |
Between Q1 and Q3 sits the middle 50% of the data — the interquartile range, the spread measure that ignores extremes entirely.
Worked Example: Eight Customer Ratings
Sorted data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8), using the median-split method that box plots use.
- Median (Q2): the middle pair is 4 and 5, so Q2 = 4.5.
- Q1: median of the lower half {2, 4, 4, 4}→ (4 + 4)/2 = 4.
- Q3: median of the upper half {5, 5, 7, 9}→ (5 + 7)/2 = 6.
- IQR: 6 − 4 = 2 — the middle half of customers rated within a two-point band.
- Tukey fences: Q1 − 1.5×IQR = 1 and Q3 + 1.5×IQR = 9. No rating falls beyond them, so no outliers — the outlier calculator and box plot generator apply exactly this rule.
These are the same values the quartile calculator and descriptive statistics tool report for this dataset.
The Honest Part: Methods Legitimately Disagree
“The 80th percentile” sounds unique, but on small samples the answer depends on the interpolation rule. Two common methods on the same eight ratings:
- Linear interpolation (Excel PERCENTILE.INC, R type 7, and this site's percentile calculator): position = 1 + 0.80(8 − 1) = 6.6, so interpolate 60% of the way from the 6th value (5) to the 7th (7): 5 + 0.6 × 2 = 6.2.
- Nearest-rank (many textbooks): take the ⌈0.80 × 8⌉ = 7th value: 7.
Both are defensible; they differ by 0.8 here and converge as n grows. Quartiles inherit the same issue — the median-split values above (4 and 6) versus interpolated quartiles (Q1 = 4, Q3 = 5.5 under type 7) can differ too. The professional habit: name the method when the sample is small, and never mix outputs from two tools without checking their conventions.
Percentile Ranks: The Reverse Lookup
Given a value instead of a percentage, the percentile rank reports its position. A common definition counts values below plus half of the ties: for the rating 7, six of eight values lie below and none tie except itself, giving (6 + 0.5)/8 ≈ the 81st percentile. This is the number behind “your child's height is in the 81st percentile” — a position claim, not a measurement. For bell-shaped data, percentile ranks connect directly to z-scores through the z-table.
Try the Percentile Calculator
Find the value at any percentile or a value's percentile rank, using linear interpolation with the working shown.
Frequently Asked Questions
What does a percentile actually mean?
The pth percentile is the value below which about p% of the data falls: scoring at the 80th percentile means outperforming roughly 80% of scores. Percentiles describe position within a distribution rather than the raw value itself, which is what makes them comparable across different tests, scales, and populations.
How are quartiles related to percentiles?
Quartiles are three specific percentiles that split ordered data into four equal parts: Q1 is the 25th percentile, Q2 the 50th (the median), and Q3 the 75th. The distance between Q1 and Q3 - the interquartile range - holds the middle half of the data and is the standard robust measure of spread.
Why do Excel, my calculator, and my textbook give different quartiles for the same data?
Because there are several legitimate definitions. Small samples expose the differences: the median-split (Tukey) method used by box plots, exclusive vs inclusive variants, and interpolation formulas (Excel's PERCENTILE.INC vs .EXC, R's nine types) can each give slightly different Q1/Q3. All converge as samples grow. The fix is not to hunt for the 'right' answer but to state which method you used.
What is a percentile rank, and how is it different from a percentile?
They are inverse lookups. A percentile starts from a percentage and returns the value at that position (the 80th percentile of these scores is 6.2). A percentile rank starts from a value and returns its percentage position (a score of 7 sits at about the 81st percentile in this data). Calculators typically offer both directions.
When should I use median and quartiles instead of mean and standard deviation?
When the data is skewed or contains outliers - incomes, house prices, response times. The mean and SD chase extreme values; the median and IQR do not (moving the largest value from 9 to 900 leaves them unchanged). For roughly symmetric, well-behaved data the mean/SD pair carries more information and feeds inference formulas.
How do quartiles detect outliers?
Tukey's rule flags values beyond the fences Q1 - 1.5xIQR and Q3 + 1.5xIQR - the same rule that draws box plot whiskers. In the worked example (IQR = 2), the fences sit at 1 and 9, so nothing qualifies; stretching the maximum to 12 would flag it. The 1.5 multiplier is a convention: 3xIQR marks 'far out' points.