Original examples

One chosen percentile.
Every step visible.

These use the Type 7 sample convention only. Enter sample values on separate rows, retain duplicates and choose one percentile.

Seventy-fifth percentile between observations

10
20
40
100

Choose 75. There are n=4 observations, so h=(4-1)×0.75=2.25. The lower zero-based index is2, visible sorted position 3, value 40. The upper value 100 is at position 4. Weight0.25 gives40+0.25×(100-40)=55. The reduced rational result is55, although55 is not an input observation.

The same sample's median

Choose 50 for the four values above. Rank is1.5, lower/upper positions2 and 3 contain20 and 40, and weight 0.5 gives30. This example uses the same convention as the75th-percentile calculation rather than switching to a nearest-rank rule.

Endpoints

For that sample,0 returns minimum10 and 100 returns maximum100. At0 the rank and upper weight are zero. At100 the rank is3 and both displayed bracket positions are4, so no nonexistent fifth observation is used.

Exact tenths

0.1
0.2

Choose 25. Rank and upper weight are0.25. The result is0.1+0.25×0.1=0.125, exactly1/8. No floating-point residue is rounded away to create that answer.

A singleton

-3.25

Choose 91.234567, or any other accepted percentile. A sample of one always returns its sole value, -3.25, reduced fraction -13/4. Both bracket positions are1 and rank/weight are zero. This says nothing about a population distribution.

Repeated observations

0
0
100

Choose 50. Rank is1, the lower value is0 at sorted position 2 and upper weight is zero, so the result is0. Both zeros remain in the sample. No values are strictly below0; the word percentile does not guarantee a strict-below fraction for this tied small sample.

A result with thirteen fractional places

0
0.000000000001

Choose 50. The result is0.0000000000005, exactly1/2000000000000. It has thirteen fractional places even though each input accepts at most twelve. The output keeps its exact terminating decimal.

A result with twenty fractional places

Use the same two values and choose 0.000001. This percentile means probability0.00000001, not0.000001. The result is0.00000000000000000001, exactly1/100000000000000000000. The extra division by100 is included explicitly.

A large signed span near the upper endpoint

-1000000000000
1000000000000

Choose 99.999999. The upper interpolation weight is0.99999999, yielding999999980000 exactly. The output stays within the sample minimum and maximum, and no rank tolerance snaps it to the maximum.

Invalid rows remain visible errors

An interior blank row, a comma-separated pair, NaN or a value with a unit does not become an omitted observation. An invalid percentile such as100.000001 does not clamp to 100. The whole calculation stops with an error and old usable results clear. Read the method or try the calculator.