Last updated: 2026-06-05

A percentile is a value in a dataset at or below which a given percentage of observations fall. The 75th percentile, for instance, is the value at or below which 75% of the sorted data lies. Quartiles are the three percentiles — Q1 (25th), Q2 (50th), and Q3 (75th) — that divide the dataset into four equal parts. This calculator finds any percentile and all three quartiles for a comma-separated list of numbers, along with the interquartile range (IQR).

The linear interpolation method

Several competing conventions exist for computing percentiles. This calculator uses the inclusive linear interpolation method — Method 7 in R, and identical to Excel's PERCENTILE.INC function and Python's numpy.percentile with the default linear interpolation.

The algorithm proceeds in three steps:

1. Sort the n values in ascending order.
2. Compute the fractional index: position = (n − 1) × p / 100.
3. Let lo = ⌊position⌋ and hi = ⌈position⌉, and let frac = position − lo. The percentile equals sorted[lo] + frac × (sorted[hi] − sorted[lo]).

When position is an integer, the interpolation fraction is zero and the result is exactly sorted[position]. When p = 0 the result is the minimum; when p = 100 it is the maximum.

Worked example

Consider the dataset 4, 8, 15, 16, 23, 42 — six values sorted in ascending order.

Q1 (p = 25): position = (6 − 1) × 0.25 = 1.25. The value at index 1 is 8; at index 2 it is 15. Q1 = 8 + 0.25 × (15 − 8) = 8 + 1.75 = 9.75.

Q2 / Median (p = 50): position = 5 × 0.50 = 2.5. Values at indices 2 and 3 are 15 and 16. Q2 = 15 + 0.5 × (16 − 15) = 15.5.

Q3 (p = 75): position = 5 × 0.75 = 3.75. Values at indices 3 and 4 are 16 and 23. Q3 = 16 + 0.75 × (23 − 16) = 16 + 5.25 = 21.25.

IQR = Q3 − Q1 = 21.25 − 9.75 = 11.5. The middle half of this dataset spans a range of 11.5 units.

To find the 10th percentile: position = 5 × 0.10 = 0.5. Values at indices 0 and 1 are 4 and 8. P10 = 4 + 0.5 × (8 − 4) = 6.0.

Quartiles and the box plot

The five-number summary — minimum, Q1, Q2, Q3, and maximum — is the basis of the box plot (also called a box-and-whisker plot). In a standard box plot, the box spans from Q1 to Q3, the line inside the box marks the median (Q2), and the whiskers extend to the most extreme values that still fall within 1.5 × IQR of the box edges.

Values outside the whiskers are flagged as outliers. For the example dataset, the lower fence is Q1 − 1.5 × IQR = 9.75 − 17.25 = −7.5, and the upper fence is Q3 + 1.5 × IQR = 21.25 + 17.25 = 38.5. The value 42 exceeds the upper fence, so it would be marked as a mild outlier.

IQR as a robust measure of spread

The IQR is the most commonly used robust measure of statistical dispersion. Unlike the standard deviation, which squares and sums every deviation from the mean, IQR considers only the middle 50% of sorted values. A single extreme observation does not change Q1 or Q3, so IQR is insensitive to outliers.

This property makes IQR the preferred spread measure for skewed distributions — household income, property prices, hospital waiting times — where the standard deviation can be dominated by a small number of extreme values and give a misleading picture of typical variability.

Different percentile conventions

The R programming language documents nine distinct percentile methods, and various scientific fields have established their own conventions. The main differences arise in how fractional indices are handled and whether the percentile of the minimum (rank 1) is placed at 0% or at 1/n × 100%.

• Method 7 (this calculator, R default, Excel PERCENTILE.INC, numpy default): position = (n − 1) × p/100. Gives p = 0 at the minimum and p = 100 at the maximum.
• Method 6 (Excel PERCENTILE.EXC, SPSS default): position = n × p/100. The minimum and maximum are not reachable percentiles, so the valid range is 1/(n+1) to n/(n+1).
• Methods 1–3: Nearest-rank methods that return an actual observed value rather than an interpolated one. Method 1 (ceiling rank) is the convention used in some educational statistics textbooks.

For datasets larger than a few hundred observations, all methods converge to essentially the same answer. Differences are most visible in small samples. When comparing percentile outputs between software tools, confirm which method each uses.

Relationship to the z-score

A percentile is a purely empirical rank: it describes the position of a value within the observed dataset without assuming any distribution. The 90th percentile is simply the value at or below which 90% of the data falls.

A z-score, by contrast, measures how many standard deviations a value sits above or below the mean, and is meaningful primarily when the data is approximately normally distributed. Under a perfect normal distribution, a z-score of 1.28 corresponds to the 90th percentile. In a heavily skewed or multimodal dataset, the same z-score may correspond to a very different empirical percentile.

Percentiles are therefore more appropriate for data whose distribution is unknown or non-normal, while z-scores are preferred when normality is a reasonable assumption and scale-independent comparisons are needed.

Minimum sample size

The linear interpolation method requires at least two data points. With exactly two values, the quartiles are computed by interpolation across the single interval, and all values from the minimum to the maximum are reachable percentiles.

For reliable percentile estimates in practice, larger samples are necessary. The uncertainty in a sample percentile is inversely related to sample size: a 95% confidence interval for the true population 90th percentile is substantially wider with n = 20 than with n = 200.