What does percentile mean, and how do quartiles work?

Direct answer: A percentile is a position in a ranked list. It says what share of the scores lie below a given value. Illowsky and Dean (2023) put it plainly: a percentile indicates the relative standing of a data value, and percentiles divide a data set into 100 equal parts.

Imagine ranking every student from lowest to highest mark and then cutting the line into 100 equal slices. The percentile is the slice a student lands in. It is a statement about rank, so it depends on who else sat the exam. The same raw mark can be a high percentile in a hard paper and a modest one in an easy paper. Throughout this guide we use one invented set of 20 exam marks out of 100, listed from lowest to highest: 38, 45, 49, 52, 55, 57, 58, 60, 62, 63, 65, 66, 68, 70, 72, 74, 77, 81, 86 and 98. The mean is 64.8 and the median is 64. These figures are made up for teaching and come from no real cohort.

Direct answer: The 70th percentile is the value below which about 70% of the scores lie. In our 20 marks, 14 students scored below 72, so a mark of 72 sits at the 70th percentile by a simple count. The percentage score is a separate fact: 72 out of 100 is 72%.

The difference between percentage score and percentile rank is the one students confuse most often, so it is worth fixing in words. The percentage score compares a mark with the maximum available, while the percentile compares a mark with the other marks. The same mark of 72 is therefore a 72% score and a 70th percentile rank. Illowsky and Dean (2023) count slightly differently for a 20-question test: a score at the 70th percentile means 70% of students scored that value or lower. The two numbers can be far apart. A student with 98% in our set is at the top of the group, while a student with 60% sits at the 35th percentile because 7 of the 20 marks are lower.

The rank used in this table is the count of scores below, divided by 20 and multiplied by 100. Other textbooks, including Illowsky and Dean (2023), count the scores at or below the value (by that rule 72 would be the 75th percentile), and some add half of any tied scores, so state which version you used when you report a percentile rank.

Direct answer: There are two formulas. For a percentile rank, divide the number of scores below a value by the total number of scores and multiply by 100. For a percentile value, find its position with one of several definitions, such as (n + 1) × p, and interpolate between neighbouring scores.

The first formula answers the question "what percentile is this mark?", and our table above applies it: 14 / 20 × 100 = 70. The second answers the opposite question, "which mark sits at the 70th percentile?", and here there is no single universal rule. Hyndman and Fan (1996) observed: "There are a large number of different definitions used for sample quantiles in statistical computer packages" (p. 361). Their paper catalogues nine of them. The position (n + 1) × p belongs to the definition they number 6, which gives 21 × 0.70 = 14.7 for our data. Marks 14 and 15 are 70 and 72, so the 70th percentile is 70 + 0.7 × (72 − 70) = 71.4. The definition behind R's default, which they number 7, uses the position 1 + (n − 1) × p = 14.3 and returns 70.6. Both are correct under their stated rule, and the difference is the subject of a later section.

Direct answer: The 90th percentile is the value that about 90% of scores fall below, and the 95th percentile is the same idea with 95%. In our exam data they are close to 81.5 and 86.6 under the R default definition, so only the top 5% of students scored above the 95th percentile.

Roughly 95% of the marks lie below the 95th percentile and 5% lie above it. The calculation for the 95th percentile follows the same steps as before. With the R default definition, the position is 1 + 19 × 0.95 = 19.05, which lies between the 19th mark (86) and the 20th mark (98). The result is 86 + 0.05 × 12 = 86.6. With the (n + 1) × p definition the position is 19.95, which gives 86 + 0.95 × 12 = 97.4. The gap between 86.6 and 97.4 for the same data shows why high percentiles are usually the most definition-sensitive, because they sit in the thin tail where the neighbouring marks are far apart. In small samples a very high percentile therefore rests on one or two observations and should be reported cautiously. The same thing happens on the 90th percentile: 81.5 under the first definition and 85.5 under the second.

Direct answer: Quartiles cut ordered data into four equal parts. The first quartile Q1 is the 25th percentile, the median Q2 is the 50th, and the 75th percentile is the third quartile Q3. The interquartile range, IQR = Q3 − Q1, measures the spread of the middle half of the data.

Illowsky and Dean (2023) describe the interquartile range as a number that indicates the spread of the middle half, or the middle 50%, of the data. For our 20 marks, with the R default definition, Q1 = 56.5, the median is 64 and Q3 = 72.5, so IQR = 16. Half of the students scored between 56.5 and 72.5. The IQR is the usual companion of the median when a variable is skewed, in the same way that the standard deviation accompanies the mean; the link between the two is explained in standard deviation vs variance.

The same two numbers drive the box plot rule for outliers. Illowsky and Dean (2023) state that a value is suspected to be a potential outlier if it is less than (1.5)(IQR) below the first quartile or more than (1.5)(IQR) above the third quartile. For our data, 1.5 × 16 = 24, so the fences are 56.5 − 24 = 32.5 and 72.5 + 24 = 96.5. The mark of 38 lies above the lower fence of 32.5, so it is not flagged, but 98 lies above the upper fence, so a box plot drawn with these quartiles marks 98 as a potential outlier.

Direct answer: They use different sample quantile definitions, which place the quartile at a different position between neighbouring scores. For our 20 marks Q1 is 55.5 under one definition and 56.5 under another. The differences are small, but they can change whether a borderline value is flagged as an outlier.

R's documentation states that its default is type 7, while type 6 is used by Minitab and by SPSS (R Core Team, n.d.). Hyndman and Fan (1996) wrote that the SPSS frequencies command "appears to use Q6(p), although this is nowhere documented" (p. 364), and in the same paragraph they add that Splus used Q7(p). That 1996 observation concerns an old version of the program, so treat it as history. A current IBM Community answer on the Explore procedure lists several percentile methods and notes that SPSS and Excel might use different definitions (Peck, 2024). A second IBM Community answer adds that the Frequencies procedure gives no choice of the calculation method, while Explore lets you specify both the percentiles and the method (IBM Community Support Admin, 2021).

Excel adds another layer. Microsoft documents PERCENTILE.INC, which accepts k from 0 to 1 inclusive, and PERCENTILE.EXC, which accepts k from 0 to 1 exclusive and returns an error outside that range (Microsoft, n.d.-a; n.d.-b). The pages do not publish the interpolation formula, so check the documentation of your Excel version before you state which definition it follows.

The last column is the practical warning. The same exam marks, the same 1.5 × IQR rule and the same top score of 98 lead to different outlier decisions only because of the quantile definition. Hyndman and Fan (1996) recommended the type 8 definition, saying "we propose that Q8(p) is the best choice" (p. 364), and R's documentation describes the type 8 estimates as approximately median-unbiased regardless of the distribution (R Core Team, n.d.). Which one is "right" matters less than reporting which one you used.

Direct answer: In SPSS, go to Analyze, then Descriptive Statistics, then Frequencies, open Statistics and tick Quartiles, Cut points or Percentiles. If you need to choose the calculation method, use Analyze, Descriptive Statistics, Explore instead, or the EXAMINE syntax with a method keyword.

Kent State University Libraries (2026) describe the Frequencies route: the Statistics dialogue offers quartiles (25th, 50th and 75th), cut points for equal groups, such as 20th, 40th, 60th and 80th for five groups, and any percentiles you enter as whole numbers. To suppress the long frequency table, untick Display frequency tables. The IBM Community example for the Explore route is EXAMINE VARIABLES=salary BY educ /PLOT NONE /PERCENTILES(5,10,25,50,75,90,95,97.5) HAVERAGE. (IBM Community Support Admin, 2021), and Peck (2024) notes that HAVERAGE is the default when PERCENTILES is specified without a keyword. Its position, (w + 1)p, is the same position as the type 6 definition above. In R, quantile(x, probs = c(.25, .5, .75), type = 7) makes the choice explicit, and changing type reproduces the table above.

Direct answer: State the software, the version and the quantile definition, then give the quartiles with the IQR and the sample size. Write, for example, "Q1 = 56.5, Q3 = 72.5, IQR = 16, n = 20 (R, type 7)". Say whether you mean a percentile rank or a percentage score.

Three habits protect you from marker queries. First, never write "the 70th percentile" when you mean "70%". Second, if you apply the 1.5 × IQR rule, say which quartile definition produced the fences, because the outlier list can change as in the table above. Third, when you move between programs, for example by checking an SPSS result in R, set the definition to match before you compare the numbers. A short methods sentence is enough: "Quartiles were computed in R 4.x using the default type 7 definition (Hyndman & Fan, 1996), and potential outliers were flagged beyond 1.5 times the interquartile range from the quartiles." Replace the version with the one you really used. For the shape of the distribution that the quartiles describe, see kurtosis explained.

Most students can compute quartiles themselves once they know which definition their software applies, and building that skill is worth the effort. Help is useful when two programs give different values, when a supervisor questions an outlier decision, or when the methods text needs to match a reporting standard. MAAS has supported students since 2015 with a network of more than 100 experts. Through SPSS, R and Python help in your discipline, a MAAS expert works with you on your own data, explains the output and reviews how you report it, so you can defend each figure in your viva or marking. Our guide to thesis statistics help explains which kinds of support a university normally allows and which parts of the analysis must stay your own.