Interpolated percentile value and interpretation
The percentile value is a threshold on the data’s measurement scale. The one-based position can be fractional because the method interpolates between adjacent sorted observations.
To compare the arithmetic mean, median, and mode of observations, use the Mean, Median & Mode Calculator and follow its own input conventions.
The formula
Sort the list, calculate position 1 + (n − 1)p, and interpolate the two neighboring entries. Here p is the selected percentile divided by 100; this is the inclusive type-seven convention.
Verified worked example
For 10, 20, 30, 40 and the 25th percentile, h = 1 + 3 × 0.25 = 1.75. Interpolating 75% of the way from 10 to 20 gives 17.5.
How to use this calculator
- Enter the dataset in any order, keeping repeated observations.
- Enter the desired percentile as a number between 0 and 100.
- Read the threshold value and interpolation position, and compare percentile conventions when using other software.
Percentile input conventions
A percentile value is a threshold in an ordered set, not the percentage rank of a particular supplied score. The tool sorts observations and uses the inclusive h = 1 + (n − 1)p convention, also known as quantile type seven. The 0th percentile is the minimum, the 100th is the maximum, and halfway corresponds to the median under this convention.
Interpreting interpolated percentile value
Several defensible sample-percentile definitions exist. A nearest-rank table or an exclusive spreadsheet function may therefore produce a different answer, especially in a small sample. State the convention when reporting a result or comparing software. Repeated observations remain in the sorted list and influence the percentile by their frequency. This is an empirical description of entered values, not an estimate of a theoretical distribution without sampling assumptions.
Assumptions & limitations
What this calculation assumes
- Empirical percentiles use inclusive linear interpolation, also known as quantile type seven.
What to keep in mind
- Inclusive interpolated sample quantiles only; no weighted percentiles or percentile ranks.
Common questions
Why is the percentile between observations?
Linear interpolation allows a fractional sorted position. It combines the neighboring values rather than always choosing one observed entry.
Does this find my test-score rank?
No. It finds the value at a requested percentile of a dataset. Finding the fraction of scores below a specific score is a separate percentile-rank task.
What do the 0th and 100th percentiles return?
They return the minimum and maximum of the entered list respectively. A one-observation dataset returns that same value for every selected percentile.
Sources & further reading
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