Track 2 · Describing Data · Lesson 5 of 11

Quartiles and percentiles

Calculate percentiles, quartiles and the interquartile range with the type 7 method, find a percentile rank, and explain why different software can report slightly different quartiles.

30 minFoundation8 graded exercisesFree

A pay review asks: what salary separates the lowest-paid 10% of staff from the rest, and what does the middle half of employees earn? Percentiles answer exactly that kind of question, and — unlike the mean — they are not dragged around by a handful of very high salaries.

The $p$ -th percentile is a value with (about) $p\%$ of the data at or below it and the rest above it. The median is the 50th percentile. The quartiles split the sorted data into four parts:

$Q_1$ , the first quartile = 25th percentile;
$Q_2$ = the median = 50th percentile;
$Q_3$ , the third quartile = 75th percentile.

The interquartile range $\text{IQR} = Q_3 - Q_1$ is the width of the middle half of the data: a measure of spread that, like the median, ignores the extremes.

With a finite sample, a percentile usually falls between two observations, and there are several reasonable rules for where exactly. This course uses the most common one, Hyndman–Fan type 7 — the default in R, NumPy, and Excel's PERCENTILE.INC / QUARTILE.INC. All graded answers use it unless a question says otherwise.

h = (n - 1)\,p + 1, \qquad Q(p) = x_{(\lfloor h \rfloor)} + \big(h - \lfloor h \rfloor\big)\big(x_{(\lfloor h \rfloor + 1)} - x_{(\lfloor h \rfloor)}\big)

Percentile by the type 7 method (linear interpolation between order statistics)

$x_{(k)}$: the $k$ -th smallest value (the $k$ -th order statistic): sort the data first
$p$: the proportion, e.g. $p = 0.25$ for $Q_1$ , $p = 0.9$ for the 90th percentile
$h$: the position in the sorted list; it runs from 1 (the minimum, $p = 0$ ) to $n$ (the maximum, $p = 1$ )
$\lfloor h \rfloor$: $h$ rounded down; the fractional part $h - \lfloor h \rfloor$ says how far to move towards the next value

Worked example

Quartiles of ten delivery distances

Ten delivery distances (km), already sorted: 12, 15, 17, 20, 22, 25, 28, 31, 40, 55. Find $Q_1$ , $Q_3$ and the IQR.

$Q_1$ : $h = 9 \times 0.25 + 1 = 3.25$ . Start at the 3rd value (17) and go a quarter of the way to the 4th (20): $17 + 0.25 \times 3 = 17.75$ km.
$Q_3$ : $h = 9 \times 0.75 + 1 = 7.75$ . Start at the 7th value (28) and go three quarters of the way to the 8th (31): $28 + 0.75 \times 3 = 30.25$ km.
IQR $= 30.25 - 17.75 = 12.5$ km. The middle half of the deliveries span 12.5 km. The median ( $h = 5.5$ ) is $(22 + 25)/2 = 23.5$ km.

Q1 = 17.75 km, Q3 = 30.25 km, IQR = 12.5 km (type 7).

Empirical cumulative distribution of 600 monthly base salaries (salaries dataset). Read across from 0.25, 0.5 and 0.75 on the vertical axis to find Q1, the median and Q3.

Show the data as a table

Monthly base salary (SAR)	Monthly base salary
6630	0.00166667
7090	0.00333333
7110	0.005
7130	0.00666667
7270	0.00833333
7280	0.0116667
7300	0.0133333
7380	0.0166667
7400	0.0183333
7420	0.02
7450	0.0216667
7510	0.0233333
7530	0.0266667
7540	0.0283333
7550	0.03
7590	0.0333333
7640	0.035
7720	0.0383333
7740	0.0433333
7780	0.0483333
7790	0.0516667
7810	0.0533333
7820	0.055
7850	0.0583333
7860	0.06
7910	0.0616667
7950	0.065
7980	0.07
8030	0.0716667
8040	0.0733333
8060	0.0783333
8090	0.08
8100	0.0816667
8110	0.085
8130	0.0883333
8140	0.09
8150	0.0916667
8180	0.0933333
8200	0.095
8210	0.0966667
8260	0.0983333
8270	0.1
8290	0.103333
8300	0.108333
8320	0.11
8330	0.113333
8340	0.115
8350	0.116667
8360	0.118333
8380	0.12
8390	0.121667
8400	0.123333
8410	0.125
8430	0.126667
8450	0.128333
8460	0.133333
8470	0.135
8490	0.136667
8500	0.138333
8510	0.14
8540	0.141667
8550	0.143333
8560	0.145
8570	0.146667
8580	0.148333
8590	0.15
8680	0.151667
8690	0.153333
8720	0.156667
8730	0.158333
8740	0.161667
8850	0.163333
8880	0.165
8890	0.166667
8900	0.168333
8930	0.17
8950	0.173333
8980	0.176667
9000	0.18
9010	0.181667
9050	0.183333
9060	0.185
9080	0.186667
9090	0.188333
9150	0.19
9180	0.191667
9190	0.193333
9220	0.198333
9240	0.201667
9270	0.203333
9280	0.205
9340	0.206667
9350	0.208333
9370	0.21
9400	0.211667
9440	0.213333
9460	0.215
9490	0.218333
9530	0.22
9540	0.221667
9550	0.225
9560	0.226667
9570	0.228333
9600	0.231667
9630	0.233333
9680	0.235
9710	0.236667
9730	0.238333
9740	0.24
9780	0.241667
9800	0.245
9810	0.246667
9830	0.248333
9850	0.25
9860	0.251667
9890	0.253333
9900	0.255
9910	0.256667
9930	0.261667
10080	0.263333
10120	0.268333
10140	0.27
10180	0.273333
10300	0.275
10340	0.278333
10360	0.28
10370	0.281667
10400	0.283333
10410	0.285
10460	0.286667
10480	0.288333
10550	0.29
10590	0.291667
10620	0.293333
10630	0.295
10740	0.296667
10770	0.298333
10800	0.3
10850	0.301667
10860	0.306667
10880	0.308333
10900	0.31
10910	0.311667
10920	0.313333
10960	0.316667
10980	0.318333
10990	0.32
11060	0.321667
11080	0.323333
11100	0.325
11130	0.326667
11150	0.328333
11170	0.331667
11210	0.333333
11230	0.338333
11270	0.34
11290	0.341667
11300	0.343333
11330	0.345
11350	0.346667
11380	0.351667
11390	0.353333
11410	0.355
11430	0.356667
11450	0.358333
11490	0.361667
11510	0.363333
11550	0.365
11560	0.368333
11570	0.37
11610	0.371667
11630	0.373333
11660	0.375
11700	0.376667
11710	0.378333
11720	0.38
11770	0.381667
11790	0.386667
11830	0.39
11860	0.391667
11900	0.395
11910	0.396667
11930	0.398333
11950	0.401667
11960	0.405
11980	0.41
11990	0.411667
12000	0.413333
12020	0.415
12070	0.416667
12120	0.42
12170	0.421667
12190	0.423333
12200	0.425
12210	0.426667
12300	0.428333
12310	0.431667
12320	0.433333
12370	0.435
12380	0.436667
12450	0.438333
12460	0.44
12490	0.441667
12540	0.443333
12550	0.445
12590	0.446667
12610	0.451667
12630	0.453333
12640	0.455
12680	0.456667
12690	0.461667
12740	0.463333
12770	0.465
12790	0.466667
12800	0.47
12820	0.471667
12830	0.476667
12840	0.478333
12870	0.48
12900	0.481667
12930	0.485
12970	0.486667
12980	0.488333
13000	0.49
13010	0.491667
13080	0.493333
13090	0.496667
13140	0.5
13150	0.501667
13160	0.503333
13240	0.505
13250	0.51
13260	0.511667
13280	0.513333
13290	0.516667
13310	0.518333
13340	0.521667
13350	0.523333
13420	0.525
13430	0.526667
13440	0.528333
13450	0.53
13460	0.531667
13480	0.533333
13490	0.535
13500	0.536667
13510	0.538333
13540	0.541667
13590	0.543333
13620	0.545
13660	0.546667
13690	0.548333
13760	0.551667
13780	0.553333
13800	0.555
13840	0.556667
13850	0.558333
13900	0.56
13950	0.561667
13960	0.563333
14020	0.565
14040	0.566667
14150	0.57
14160	0.571667
14180	0.573333
14220	0.575
14240	0.576667
14310	0.578333
14320	0.58
14350	0.581667
14360	0.583333
14370	0.585
14400	0.586667
14410	0.588333
14440	0.59
14470	0.593333
14520	0.595
14540	0.596667
14760	0.598333
14790	0.6
14820	0.601667
14950	0.603333
14980	0.605
15100	0.608333
15110	0.61
15130	0.611667
15160	0.613333
15250	0.615
15260	0.616667
15350	0.618333
15370	0.62
15380	0.623333
15390	0.625
15400	0.626667
15420	0.628333
15450	0.63
15460	0.631667
15490	0.633333
15500	0.635
15520	0.636667
15600	0.638333
15650	0.64
15720	0.641667
15740	0.643333
15790	0.645
15830	0.646667
15840	0.648333
15860	0.65
16200	0.651667
16250	0.653333
16410	0.655
16540	0.656667
16560	0.658333
16610	0.66
16740	0.661667
16770	0.663333
16790	0.665
16850	0.666667
16860	0.668333
16880	0.671667
16890	0.673333
16900	0.675
16920	0.676667
16930	0.678333
16940	0.68
16950	0.681667
16960	0.685
17000	0.688333
17040	0.69
17140	0.691667
17170	0.693333
17230	0.695
17260	0.696667
17360	0.698333
17430	0.7
17460	0.701667
17630	0.703333
17750	0.705
17790	0.706667
17820	0.708333
17910	0.71
17940	0.711667
17970	0.713333
18030	0.715
18040	0.716667
18090	0.718333
18170	0.72
18180	0.721667
18320	0.723333
18430	0.725
18490	0.726667
18550	0.728333
18580	0.73
18680	0.731667
18730	0.733333
18810	0.735
18820	0.736667
18930	0.738333
18970	0.741667
18980	0.743333
19030	0.745
19100	0.746667
19150	0.748333
19280	0.751667
19300	0.753333
19370	0.755
19400	0.756667
19470	0.758333
19550	0.76
19590	0.761667
19610	0.765
19630	0.766667
19720	0.768333
19730	0.771667
19810	0.773333
19850	0.775
19930	0.776667
19970	0.778333
20110	0.78
20120	0.781667
20160	0.783333
20260	0.785
20360	0.786667
20390	0.788333
20480	0.79
20490	0.791667
20540	0.793333
20560	0.795
20690	0.796667
20840	0.798333
20880	0.8
20930	0.801667
20940	0.803333
21000	0.805
21120	0.806667
21280	0.808333
21480	0.81
21510	0.811667
21570	0.813333
21610	0.815
21760	0.816667
21780	0.818333
21860	0.82
22040	0.821667
22050	0.823333
22110	0.825
22130	0.826667
22160	0.828333
22170	0.83
22370	0.831667
22540	0.833333
22610	0.835
22780	0.836667
23310	0.838333
23530	0.84
23620	0.841667
23710	0.843333
23800	0.845
24160	0.846667
24360	0.848333
25060	0.85
25330	0.851667
25410	0.853333
25440	0.855
25590	0.856667
25600	0.858333
25670	0.86
25820	0.863333
25860	0.865
26610	0.866667
26920	0.868333
27120	0.87
27180	0.871667
27260	0.873333
27560	0.875
27720	0.878333
27730	0.88
27750	0.881667
27790	0.883333
27840	0.885
27910	0.886667
28030	0.888333
28290	0.89
28370	0.891667
28400	0.893333
28430	0.895
28460	0.896667
28590	0.898333
28880	0.9
28960	0.901667
28980	0.903333
29020	0.905
29170	0.906667
29280	0.908333
29350	0.91
29440	0.911667
29450	0.913333
29550	0.915
29680	0.916667
30090	0.918333
30120	0.921667
30330	0.923333
30370	0.925
30470	0.926667
30740	0.928333
30780	0.93
30960	0.931667
31270	0.933333
31690	0.935
32350	0.936667
32390	0.938333
32840	0.94
33170	0.941667
33740	0.943333
34370	0.945
35230	0.946667
35240	0.948333
37750	0.95
38710	0.951667
39460	0.953333
40550	0.955
42320	0.956667
42390	0.958333
44050	0.96
44160	0.961667
44700	0.963333
44740	0.966667
44870	0.968333
45460	0.97
45820	0.971667
46020	0.973333
46590	0.975
46860	0.976667
46870	0.978333
47070	0.98
47310	0.981667
48310	0.983333
48520	0.985
49110	0.986667
50090	0.988333

First 500 of 507 rows.

P10	Q1 (P25)	Median (P50)	Q3 (P75)	P90	IQR
8 288	9 857.5	13 145	19 280	28 888	9 422.5

Percentiles of monthly base salary (SAR) in the synthetic salaries dataset, type 7

Read it like this: 10% of employees earn 8 288 SAR or less a month, and 10% earn 28 888 SAR or more; the middle half earn between 9 857.5 and 19 280 SAR. The gap from the median up to Q3 (6 135 SAR) is much larger than from Q1 to the median (3 287.5 SAR) — the percentiles reveal the right skew directly.

A percentile rank runs the other way: given a value, what percentage of the data lies below it? About 78% of these employees earn less than 20 000 SAR, so 20 000 SAR has a percentile rank of about 78. (Definitions differ slightly when the value itself occurs in the data — some count values at or below, some count half of the ties; with no ties, they agree.)

Quick check

For 600 salaries, Q1 = 9 857.5 SAR and Q3 = 19 280 SAR. Roughly how many employees earn between them?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

0 of 8 exercises correct (not saved — sign in to keep your progress)

Exercise 1 · Guided, difficulty 1 of 3

First quartile by interpolation

Not started

Twelve trainees' assessment scores are:

46, 79, 83, 53, 29, 38, 37, 34, 86, 32, 63, 69

Find the first quartile $Q_1$ using the type 7 method (position $h = (n - 1)p + 1$ ). Give your answer to 2 decimal places.

Exercise 2 · Practice, difficulty 2 of 3

Quartiles and IQR

Not started

A pharmacy logged the time (minutes) to fill ten prescriptions:

68, 18, 98, 81, 41, 47, 42, 88, 34, 64

Using the type 7 method, find $Q_1$ , $Q_3$ and the interquartile range. Give each to 2 decimal places.

Exercise 3 · Practice, difficulty 2 of 3

The 90th percentile

Not started

A help desk measured the resolution time (minutes) of 15 tickets:

395, 137, 173, 409, 121, 145, 376, 436, 205, 258, 420, 303, 371, 457, 282

The service target is set at the 90th percentile. Find it with the type 7 method, to 1 decimal place.

Exercise 4 · Application, difficulty 2 of 3

The salary distribution in percentiles

Not started

Using base_salary_sar in the salaries dataset (600 employees), calculate the 10th percentile, the median and the 90th percentile (type 7), and the ratio $P_{90}/P_{10}$ — a common measure of pay dispersion. Give the percentiles in SAR to the nearest whole riyal and the ratio to 2 decimal places.

Data: salaries · Open in the lab to explore and calculate.

Exercise 5 · Practice, difficulty 2 of 3

Percentile rank

Not started

Twenty applicants took an aptitude test. Their scores were:

75, 92, 81, 97, 74, 96, 45, 68, 65, 78, 41, 70, 69, 44, 43, 82, 87, 86, 42, 83

A new applicant scores 72.5. Compared with these twenty, what is the percentile rank of that score — the percentage of the twenty scores that are below it?

Exercise 6 · Interpretation, difficulty 2 of 3

Two answers for the same quartile

Not started

An analyst computes $Q_1$ of eleven invoice amounts in R and gets 412.5 SAR. A colleague uses Excel's QUARTILE.EXC on the same data and gets 395 SAR. What is the best explanation?

Exercise 7 · Practice, difficulty 1 of 3

Percentile vocabulary

Not started

Match each term to its meaning.

Q_1

Median

Q_3

IQR

Percentile rank of a value

Exercise 8 · Interpretation, difficulty 2 of 3

Explaining a pay percentile

Not started

An HR letter tells an employee: “Your base salary is at the 75th percentile for your job level.” The employee asks whether this means they earn 75% of the top salary. In two or three sentences, explain what the statement does and does not mean.

What you can now do

Calculate any percentile, the quartiles and the IQR with the type 7 method
Find the percentile rank of a value
Explain why R, Excel .INC/.EXC and other tools may report different quartiles
Describe a skewed distribution with percentiles in plain language