Algebra 1

Percentiles

Where does one value sit among all the others, and which value sits at a given place in the line?

You need

What percent means: a rate per hundredMath 6 Quartiles and the five number summaryMath 6

Unlocks

Checking whether data look normalAlgebra 2 Normal calculations: areas and percentilesAlgebra 2

Where you are 2 in Range and interquartile range

Statistics7

Center and spread13

Range and interquartile range2

Quartiles and the five number summary

Percentiles

When you reach for it

A marathon finishing time of 252 minutes means little on its own. Was it quick or slow for that race? The useful statement is where it sits: “60 percent of the runners finished in that time or less.” That is a percentile. You reach for percentiles when one value has to be placed among many, and for the reverse question: which time marks the quickest tenth of the field, or the middle?

The Quartiles and the five number summary already do this at three places: the lower quartile Q_1 a quarter of the way up the ordered data, the median half way, the upper quartile Q_3 three quarters of the way, with the interquartile range Q_3 - Q_1 measuring the middle half. Percentiles allow any place, not just those three. They are how a large data set, such as a race with thousands of runners, is summarized, and they are the curves on a baby’s growth chart.

The procedure

What a percentile is

The pth percentile, written P_p, is a value with about p percent of the data at or below it, and so about 100 - p percent above it. Three percentiles already have names:

P_{25} = Q_1, \qquad P_{50} = \text{median}, \qquad P_{75} = Q_3.

The word “about” matters. With 20 values, each value is 5 percent of the data, so the share at or below a point moves in steps of 5 percent and may never equal 42 percent exactly. The rule below says which value to report.

The percentile rank of a value

From a value to its place: the percentile rank of a value x is the share of the data at or below x, written as a percent. With n values,

\text{rank of } x = \frac{\text{number of values at or below } x}{n} \times 100.

A value with rank 60 is “at the 60th percentile”. Ranks are often rounded to a whole number. Some books count only the values strictly below x, or half of the values equal to x; with thousands of values the versions differ by almost nothing, with ten values by several points. This lesson always counts at or below.

The value at a given percentile

From a place to a value, in a list of n values sorted from smallest to largest. Find the position

L = \frac{p}{100} \times n.

If L is not a whole number, round it up and take the value in that position. If L is a whole number, take the mean of the values in positions L and L + 1.

Both cases come straight from the definition. Take n = 20 and p = 42, so L = 8.4. The 8th value has 8 of 20, or 40 percent, at or below it: not enough. The 9th has 45 percent, the first value to reach 42 percent, and rounding 8.4 up finds exactly that value. Now take p = 30, so L = 6. Every number from the 6th value up to just short of the 7th has exactly 6 values, 30 percent, at or below it, so all of them qualify; the rule reports the middle of that gap, the same choice the median makes for an even count.

The rule always gives the usual median. It gives the same quartiles as the method used in Comparing box plots (medians of the lower and upper halves, the median left out of both halves when n is odd) for every n except one more than a multiple of 4, such as 9 or 13, where the two can differ by part of the gap between neighboring values. For quartiles, keep using the halves. With small data sets, books, calculators and spreadsheet programs use slightly different percentile rules and can give slightly different answers, each a fair reading of “about p percent”. With large data sets they agree closely.

Reading a cumulative relative frequency graph

A large data set is rarely listed value by value. A cumulative relative frequency graph plots, against each value x, the share of the data at or below x: the running total of a relative frequency table. That height is the percentile rank of x, so the graph works both ways. To find the rank of a value, go up from it to the graph and across to the share axis. To find P_p, go across from p percent to the graph and down to the value axis. The graph joins its known points with straight segments, which assumes the values are spread evenly inside each interval, so a reading between points is an estimate.

A percentile is a position

A percentile says how much of the data sits at or below a value, nothing more. It is not a percent of the largest value, not a percent of some full amount, and not a mark of good or bad. A baby whose length is at the 90th percentile is not 90 percent of any length: about 90 percent of babies of the same age in the chart’s reference group are that length or shorter. In a race, times are ranked from smallest up, so a higher percentile is a slower time.

Worked examples

All data in this lesson are made up.

Example 1

The value at a percentile in a list

Twenty runners in a small marathon finished in these times, in minutes, already sorted:

\begin{aligned}&198,\ 205,\ 211,\ 219,\ 224,\ 228,\ 233,\ 236,\ 241,\ 245,\\ &248,\ 252,\ 257,\ 263,\ 270,\ 276,\ 284,\ 295,\ 308,\ 331.\end{aligned}

Find P_{30}, P_{42} and P_{85}.

Solution

Here n = 20, and since the list is sorted, a time’s position is its place in the line.

For P_{30}, L = 0.30 \times 20 = 6, a whole number, so take the mean of the 6th and 7th times:

P_{30} = \frac{228 + 233}{2} = 230.5 \text{ minutes}.

Check: 6 of the 20 times, 30 percent, are at or below 230.5.

For P_{42}, L = 0.42 \times 20 = 8.4 is not whole, so round up to 9: P_{42} = 241 minutes. At or below 241 sit 9 times, 45 percent, while at or below the 8th time, 236, sit only 40 percent. So 241 is the first time that reaches 42 percent.

For P_{85}, L = 0.85 \times 20 = 17 is whole, so

P_{85} = \frac{284 + 295}{2} = 289.5 \text{ minutes}.

Of this field, 85 percent finished within 289.5 minutes, just under 4 hours 50 minutes.

Example 2

Percentile rank of a time

Use the twenty times of Example 1. Zelda finished in 252 minutes and Domitilla in 205. Find each one’s percentile rank, then compare Zelda’s time with P_{60}.

Solution

Count the times at or below each. Zelda’s 252 is the 12th time, so

\frac{12}{20} \times 100 = 60,

and Zelda finished at the 60th percentile. Domitilla’s 205 is the 2nd time, and \displaystyle \frac{2}{20} \times 100 = 10, the 10th percentile. Times are ranked from smallest up, so Domitilla’s lower percentile is the quicker run.

For P_{60}, L = 0.60 \times 20 = 12 is whole, so \displaystyle P_{60} = \frac{252 + 257}{2} = 254.5, not 252. There is no clash: every time from 252 up to just under 257 has exactly 60 percent of the field at or below it, and the rule reports the middle of that gap. Zelda’s time is its left end.

Example 3

Ties and an odd count

A running club’s fifteen members finished a marathon in these times, in minutes:

\begin{aligned}&212,\ 219,\ 224,\ 224,\ 230,\ 236,\ 241,\ 241,\\ &241,\ 250,\ 258,\ 263,\ 270,\ 284,\ 302.\end{aligned}

Find the percentile rank of 241, then P_{90}, the median and the quartiles.

Solution

Three members share 241, in positions 7, 8 and 9. “At or below” counts all three, so 9 of the 15 times are at or below 241:

\frac{9}{15} \times 100 = 60.

All three of those runners are at the 60th percentile. The rank jumps from \displaystyle \frac{6}{15} \times 100 = 40 at 236 straight to 60 at 241: ties make the rank move in bigger steps.

For P_{90}, L = 0.90 \times 15 = 13.5 rounds up to 14, so P_{90} = 284 minutes. For the median, L = 0.50 \times 15 = 7.5 rounds up to 8, so the median is 241, the middle of 15 values as usual.

For Q_1, L = 0.25 \times 15 = 3.75 rounds up to 4, giving 224. For Q_3, L = 0.75 \times 15 = 11.25 rounds up to 12, giving 263. The halves method agrees: with the median left out, the lower half is the first seven times, whose middle value is 224, and the upper half the last seven, whose middle value is 263.

Example 4

Reading the race graph both ways

A larger made-up marathon had 500 finishers. The graph shows, for each time, the share of runners who finished at or below it. Its plotted points are (180, 0\%), (210, 10\%), (240, 30\%), (270, 60\%), (300, 85\%), (330, 95\%) and (360, 100\%).

Figure 1Made-up data. The share of 500 marathon runners who finished at or below each time. The dashed paths read the median and the percentile rank of 300 minutes.

Estimate the median and P_{90}, the percentile rank of 300 minutes, and how many runners finished within 5 hours.

Solution

For the median, go across from 50 percent to the graph. The level meets the segment from (240, 30\%) to (270, 60\%), which climbs 30 points in 30 minutes, one point a minute. Since 50 is 20 points above 30,

\text{median} \approx 240 + 20 = 260 \text{ minutes}.

For P_{90}, the level 90 percent meets the segment from (300, 85\%) to (330, 95\%). Since 90 is half way from 85 to 95, P_{90} \approx 300 + 15 = 315 minutes.

For the rank of 300 minutes, go up from 300 to the graph and across: the height is 85 percent, so 300 minutes is at the 85th percentile. Five hours is 300 minutes, so about 0.85 \times 500 = 425 runners finished within 5 hours.

The dashed paths in the figure trace the first and third readings. The answers are estimates, since the straight segments assume times are spread evenly inside each 30-minute interval.

Example 5

A growth chart

The made-up chart shows three curves for babies aged 0 to 12 months: the lengths at the 10th, 50th and 90th percentiles of a reference group of babies of each age.

Figure 2A made-up growth chart. Each curve gives the length at one percentile for babies of each age.

Ottoline’s daughter measured 61 cm at 3 months and 68 cm at 9 months. Find her percentile at each age and say what changed. What does the 90th percentile curve say at 6 months?

Solution

At 3 months, 61 cm lies on the 50th percentile curve: about half of 3-month-olds in the reference group are 61 cm or shorter. At 9 months, 68 cm lies on the 10th percentile curve: only about 10 percent of 9-month-olds are 68 cm or shorter.

She grew 68 - 61 = 7 cm, yet her percentile fell from 50 to 10. Both hold because each percentile compares her with babies of her own age, and those babies grew more: the 50th percentile rose from 61 to 71 cm over the same six months. The percentile tracks her position, not her size.

At 6 months the 90th percentile curve is at 70 cm, so about 90 percent of 6-month-olds in the reference group measure 70 cm or less, and about 10 percent measure more.

Example 6

From a percentile rank to a count

In a marathon with 250 finishers, all times different, Wilfred’s time is at the 80th percentile and Zenobia’s at the 34th. The slowest time was 400 minutes and Wilfred’s was 280. How many runners finished in Wilfred’s time or less, and in Zenobia’s? Does \displaystyle \frac{280}{400} tell his percentile?

Solution

A rank of 80 means 80 percent of the field is at or below Wilfred’s time:

0.80 \times 250 = 200.

So 200 runners, Wilfred included, finished in his time or less, and 250 - 200 = 50 took longer. For Zenobia, 0.34 \times 250 = 85 runners, her included.

Wilfred’s time is \displaystyle \frac{280}{400} = 0.7, or 70 percent, of the slowest time. That compares two times; it says nothing about how many runners finished at or below his, which is what a percentile counts. Here the count gives 80, not 70.

Boundaries

Ordered numbers only. The data must be values that can be sorted. A percentile of eye colors means nothing.

Strictly between 0 and 100. The rule for P_p needs 0 < p < 100. For p = 100 it would ask for a value in position n + 1, which does not exist; the largest value has rank 100 and that is the end of the scale. For a small p, L can be below 1 and rounds up to 1, the smallest value.

Small data sets depend on the rule. This lesson’s rule, the halves method for quartiles and the functions in calculators and spreadsheets can disagree by part of a gap between neighboring values. Say which rule was used.

A rank belongs to one group. The same length is a different percentile among 3-month-olds than among 9-month-olds, and the same time is a different percentile in a fast field than in a slow one.

Graph readings are estimates whenever they fall between plotted points.

Where this goes next

The Cumulative frequency graphs and percentiles lesson builds graphs like the one in Example 4 from a table, and a Relative frequency histograms shows the same kind of data as bars. The quartiles found here feed the Box plots that show outliers, which uses them to flag unusual values. A z-score describes position another way, in standard deviations from the mean, and later work reads percentiles from a smooth model of a whole population rather than from a list (percentiles from a model curve).

Exercises

All data are made up. Exercises 1 to 6 use a running club’s sixteen marathon times, in minutes:

\begin{aligned}&205,\ 214,\ 218,\ 226,\ 231,\ 235,\ 239,\ 244,\\ &247,\ 253,\ 259,\ 266,\ 272,\ 281,\ 293,\ 316.\end{aligned}

Percentile rank in a list

Find the percentile rank of 239 minutes.
Find the percentile rank of 272 minutes.
Find the percentile ranks of the quickest time, 205, and the slowest, 316.

The value at a percentile

Find the median, Q_1 and Q_3 of the club times with the rule, and check the quartiles against the halves method.
Find P_{10} and P_{90} of the club times.
Find P_{60} of the club times, and check that at least 60 percent of the times are at or below it.
For the data 12, 15, 15, 18, 20, 20, 20, 23, 27, 30, 34, find P_{30}, the median, and the percentile ranks of 15 and 20.

Reading the race graph

Exercises 8 to 12 use the graph of Example 4.

Estimate P_{10} and P_{95}.
Estimate the percentile ranks of 240 and 330 minutes.
Estimate Q_3 and P_{20}.
About how many of the 500 runners finished between 240 and 300 minutes?
About how many runners took longer than 270 minutes?

Reading the growth chart

Exercises 13 to 15 use the chart of Example 5.

At 12 months, what length is the 50th percentile? About what share of 12-month-olds in the reference group measure 77 cm or less?
Zbigniew measures 64 cm at 6 months. At what percentile is he?
Honorine’s son measured 58 cm at 3 months and 74 cm at 12 months. Find his percentile at each age and how much he grew.

Reasoning

Explain why the rule for P_{50} always gives the usual median, whether n is odd or even.
For 3, 5, 6, 8, 9, 11, 12, 14, 17, find Q_1 by the halves method and P_{25} by the rule. They differ. Explain why both are fair answers to “about 25 percent at or below”.
Explain why, counting at or below, the largest value of any data set has percentile rank 100 but the smallest never has rank 0. What is the smallest value’s rank if n = 10 and the two smallest values are equal?

Answers

43.75, about the 44th percentile. 239 is the 7th time, and \displaystyle \frac{7}{16} \times 100 = 43.75.
81.25, about the 81st percentile. 272 is the 13th time, and \displaystyle \frac{13}{16} \times 100 = 81.25.
6.25, about the 6th percentile, and 100. One time of 16 is at or below 205, and all 16 are at or below 316.
Median 245.5, Q_1 = 228.5, Q_3 = 269. Median: L = 8, the mean of 244 and 247. Q_1: L = 4, the mean of 226 and 231. Q_3: L = 12, the mean of 266 and 272. The halves method agrees: the lower eight times have middle pair 226 and 231, the upper eight 266 and 272.
P_{10} = 214 and P_{90} = 293. L = 1.6 rounds up to 2; L = 14.4 rounds up to 15.
P_{60} = 253. L = 0.6 \times 16 = 9.6 rounds up to 10. At or below 253 sit 10 of 16 times, 62.5 percent; at or below the 9th, 247, only 56.25 percent.
P_{30} = 18, median 20, rank of 15 about 27, rank of 20 about 64. With n = 11, L = 3.3 rounds up to 4 and L = 5.5 to 6. Then \displaystyle \frac{3}{11} \times 100 \approx 27.3 and \displaystyle \frac{7}{11} \times 100 \approx 63.6.
P_{10} \approx 210 and P_{95} \approx 330 minutes. Both levels meet the graph at plotted points.
30 and 95, read at plotted points.
Q_3 \approx 288 and P_{20} \approx 225 minutes. 75 is 15 of the 25 points from 60 to 85, so \displaystyle 270 + \frac{15}{25} \times 30 = 288; 20 is half way from 10 to 30, so 210 + 15 = 225.
About 275. 85 - 30 = 55 percent finished in that window, and 0.55 \times 500 = 275.
About 200. 60 percent finished within 270 minutes, so 40 percent took longer, and 0.40 \times 500 = 200.
74 cm, and about 90 percent: 77 cm lies on the 90th percentile curve at 12 months.
The 10th percentile: 64 cm lies on the 10th percentile curve at 6 months.
The 10th percentile at 3 months and the 50th at 12 months; he grew 74 - 58 = 16 cm.
\displaystyle L = \frac{50}{100} \times n = \frac{n}{2}. If n is odd, \displaystyle \frac{n}{2} is not whole and rounds up to \displaystyle \frac{n+1}{2}, the middle position. If n is even, \displaystyle \frac{n}{2} is whole and the rule averages positions \displaystyle \frac{n}{2} and \displaystyle \frac{n}{2} + 1, the two middle values. Both are the usual median.
Q_1 = 5.5 and P_{25} = 6. The median, 9, is left out, so the lower half is 3, 5, 6, 8, with middle pair 5 and 6. The rule gives L = 2.25, rounded up to 3, so the value 6. At or below 5 sit \displaystyle \frac{2}{9} of the data, about 22 percent; at or below 6, \displaystyle \frac{3}{9}, about 33 percent. The share jumps straight past 25 percent between 5 and 6, so no value has exactly 25 percent at or below, and every point from 5 to 6 is a fair reading of “about 25 percent”. The rule takes 6, the first value to reach it; the halves method takes the middle of the gap.
All n values are at or below the largest, so its rank is \displaystyle \frac{n}{n} \times 100 = 100. The smallest value has at least itself at or below it, so its rank is at least \displaystyle \frac{1}{n} \times 100, which is more than 0. With n = 10 and the two smallest equal, both count: \displaystyle \frac{2}{10} \times 100 = 20.