Percentile Values Versus Percentile Ranks

Two Questions That Sound Similar

A report might say that a measurement is at the 60th percentile, or that a particular value has a percentile rank of 60. Those statements are related, but they answer opposite-direction questions. A percentile value is a cutoff measured in the original units of the data. A percentile rank is a percentage describing the relative position of a value in the data.

For example, if exam scores are measured in points, the 60th percentile value is a score in points. The percentile rank of a score is a percentage. To avoid mixing them up, first ask: Am I given a percentage and looking for a data value, or given a data value and looking for a percentage?

Definition: The percentile value at percentile \(p\) is a data-scale value chosen to represent the \(p\)th percentile under a stated percentile rule. The percentile rank of a value \(x\) is the percentage position assigned to \(x\) in a data set under a stated percentile-rank rule. A percentile value has the same units as the observations; a percentile rank is expressed as a percentage.

The rule matters. Different software or conventions can produce different percentile values, especially for small data sets. Percentile ranks can also be defined in more than one way when there are ties. In the examples here, percentile values use the linear interpolation position \(h=1+(n-1)p\), where \(p\) is written as a proportion, and percentile ranks use the midrank rule: observations below the value count fully, and observations tied with it count halfway.

One Ordered Data Set, Two Directions

Suppose a hypothetical group of ten students receives these quiz scores, already ordered from smallest to largest:

2,\ 4,\ 4,\ 7,\ 9,\ 12,\ 12,\ 15,\ 18,\ 20

Write the ordered observations as \(x_{(1)},x_{(2)},\ldots,x_{(n)}\). Here \(n=10\), so, for instance, \(x_{(5)}=9\) and \(x_{(6)}=12\). The subscript in parentheses indicates a position in the ordered list, not a power.

Worked Example: Finding the 60th Percentile Value

Find the 60th percentile value for the ten quiz scores, using the stated linear interpolation rule.

The question gives a percentage, \(p=0.60\), and asks for a score. Calculate the position:

$$ h=1+(n-1)p=1+(10-1)(0.60)=6.4 $$

Position 6.4 is 0.4 of the way from position 6 to position 7. The scores at those positions are both 12, so interpolation gives:

P_{60}=x_{(6)}+0.4\bigl(x_{(7)}-x_{(6)}\bigr) =12+0.4(12-12)=12

Under this rule, the 60th percentile value is 12 quiz points. The result is a score, not a percentage rank.

Now reverse the direction: start with a score and ask for its percentile rank. Using the midrank convention, let \(L\) be the number of observations below the score and \(E\) the number equal to it. The rank is calculated as:

\text{Percentile rank of }x =100\left(\frac{L+\frac{1}{2}E}{n}\right)\%

The formula gives each observation below \(x\) full credit toward its position and each observation tied at \(x\) half credit. Dividing by \(n\) converts that position to a proportion of the data; multiplying by 100 expresses it as a percentage.

Worked Example: Finding the Percentile Rank of 12

Find the percentile rank of a score of 12 in the same data set, using the midrank rule.

Five scores are below 12: 2, 4, 4, 7, and 9. Two scores equal 12. Thus \(L=5\), \(E=2\), and \(n=10\). Substitute:

100\left(\frac{L+\frac{1}{2}E}{n}\right)\% =100\left(\frac{5+\frac{1}{2}(2)}{10}\right)\% =100\left(\frac{6}{10}\right)\%=60\%

The score of 12 has a percentile rank of 60% under this convention. The answer is a relative position, not a score in quiz points.

Why the Two Calculations Are Not Exact Inverses

It is tempting to assume that the 60th percentile value must always have percentile rank 60%, or that a value with rank 60% must be the 60th percentile value. The two worked calculations happen to match: the 60th percentile value is 12 points, and the rank of 12 is 60%. But that match is not guaranteed.

One reason is that a percentile value may be interpolated between observed scores. A value between two observations need not have the same empirical position as the percentage used to select it. Another reason is that several observations can be tied at one value. A tie creates a block of observations at a shared location, so a rank convention must decide how that block contributes.

Worked Example: An Interpolated Percentile Value and Its Rank

Find the 75th percentile value for the same ten scores, then calculate the percentile rank of that resulting value using the midrank rule.

First, find the position for \(p=0.75\):

$$ h=1+(10-1)(0.75)=7.75 $$

Position 7.75 is 0.75 of the way from the seventh score, 12, to the eighth score, 15:

P_{75}=x_{(7)}+0.75\bigl(x_{(8)}-x_{(7)}\bigr) =12+0.75(15-12)=14.25

The interpolated 75th percentile value is 14.25 quiz points. There are seven observed scores below 14.25 and none tied with it. Therefore:

100\left(\frac{7+\frac{1}{2}(0)}{10}\right)\%=70\%

Under the midrank rule, 14.25 has percentile rank 70%, not 75%. The percentile value was located by interpolating a position between observations, whereas the percentile rank describes the placement of that value relative to the actual observations.

This difference does not mean either calculation is incorrect. Each answers a different question according to its stated rule. A percentile value selected by interpolation is not necessarily an observed score, and its rank under a count-based rule need not equal the percentile used to find it.

A Short Justification of the Midrank Rule

The midrank formula gives a precise way to handle tied values. If \(L\) observations are below \(x\), all \(L\) occupy positions before the tied group. If \(E\) observations equal \(x\), they occupy the next \(E\) positions. The middle position of that tied block is \(L+\frac{E+1}{2}\). Subtracting half a position gives \(L+\frac{E}{2}\), the midrank position used in the formula. Dividing by \(n\) and multiplying by 100 expresses that position as a percentage of the data set.

This is a convention for describing relative standing, not the only possible convention. For example, a report that defines rank as the percentage of observations at or below a value uses a different calculation when ties occur. When interpreting a published percentile rank, check how it was defined rather than assuming every source uses the midrank rule.

Worked Example: Finding a Rank and Comparing It With a Percentile Value

Find the percentile rank of a score of 9. Then find the 45th percentile value using linear interpolation, and compare the results.

There are four observations below 9 and one observation equal to 9. Thus \(L=4\), \(E=1\), and \(n=10\):

100\left(\frac{4+\frac{1}{2}(1)}{10}\right)\% =100\left(\frac{4.5}{10}\right)\%=45\%

So 9 points has percentile rank 45% under the midrank rule. For the 45th percentile value, use \(p=0.45\):

$$ h=1+(10-1)(0.45)=5.05 $$

Position 5.05 lies between \(x_{(5)}=9\) and \(x_{(6)}=12\). Therefore:

P_{45}=9+0.05(12-9)=9.15

The 45th percentile value is 9.15 points, while the observed score of 9 has rank 45%. The rank is a percentage attached to a score; the percentile value is a score selected from a percentage. These answers are close here, but they are not the same kind of quantity.

Choosing the Right Question and Avoiding a Common Mistake

Use the wording of the question to decide which quantity is needed. Phrases such as “the score at the 80th percentile” ask for a percentile value. Phrases such as “what percentile rank does a score of 72 have?” ask for a percentile rank. The units offer a useful check: a score cutoff is reported in score units, while a rank is reported as a percentage.

A common mistake is to write, “The 75th percentile is 75%,” or to interpret a percentile value of 14.25 points as a rank of 14.25%. The correction is to name the quantity and its units: “The 75th percentile value is 14.25 quiz points,” or “The percentile rank is 70% under the midrank convention.” A percentage used to locate a percentile value is not itself the resulting score.

Before reporting either result, state or identify the rule being used. For a percentile value, specify the position rule if it matters. For a percentile rank, explain how ties are counted. This makes the answer reproducible and prevents a difference in conventions from being mistaken for an arithmetic error.

Key takeaway: A percentile value maps a percentile to a data-scale cutoff; a percentile rank maps a data value to a percentage position. They describe related aspects of a distribution, but ties and interpolation mean they are not generally exact inverses.

Check Your Understanding

For each question, identify whether it asks for a percentile value or a percentile rank before calculating or explaining.

A coach asks for the time, in seconds, at the 80th percentile of a set of race times. Which quantity is requested, and what units should the answer have?
A student earns 72 points on a quiz. What additional information and convention would you need to calculate that score’s percentile rank?
In the worked data set, why is 14.25 points the 75th percentile value under the stated interpolation rule even though its midrank percentile rank is 70%?
Explain how the midrank formula handles a score shared by several observations.
A report says a score has percentile rank 65%. Is 65% a score on the original measurement scale? Explain.