Measures of Spread: Quantiles and Percentiles
Theory
Quantiles cut ordered data into equal-sized groups: quartiles into four, deciles into ten and percentiles into a hundred. This Year 10A maths page (Victorian Curriculum 10A) shows how to find them from lists, box plots and cumulative frequency graphs, how to find a percentile rank, and how the interquartile range, the mean absolute deviation and the standard deviation compare, especially when there is an outlier. In the Victorian Curriculum this is Level 10A content; in the Australian Curriculum it is optional content beyond Year 10. Curriculum: Year 10A (Victorian Curriculum 10A: VC2M10AST02); Australian Curriculum: optional content beyond Year 10.
Quantiles are values that cut ordered data into groups of equal size. The quartiles \(Q_1\), \(Q_2\), \(Q_3\) cut it into four, the deciles \(D_1, \ldots, D_9\) into ten, and the percentiles \(P_1, \ldots, P_{99}\) into a hundred. The median is \(Q_2 = D_5 = P_{50}\); also \(Q_1 = P_{25}\) and \(Q_3 = P_{75}\).
For a list, \(Q_1\) is the median of the lower half and \(Q_3\) the median of the upper half. With an odd number of scores, leave the median out of both halves. About \(10\%\) of scores lie below \(D_1\) and about \(90\%\) below \(D_9\).
The cumulative frequency of a value is the number of scores up to and including it (a running total). A cumulative frequency graph (ogive) plots cumulative frequency against the upper end of each class, starting from \(0\) at the lower end of the first class, with the points joined by straight lines.
The percentile rank of a score is the percentage of scores below it. A score at the \(85\)th percentile is higher than about \(85\%\) of the scores.
Measures of spread: the range, the interquartile range \(Q_3 - Q_1\) (middle \(50\%\)), the interdecile range \(D_9 - D_1\) (middle \(80\%\)), the mean absolute deviation (MAD, the mean distance of the scores from the mean) and the standard deviation.
| Measure | Uses | Effect of an outlier |
|---|---|---|
| range | the two extreme scores | very large |
| IQR | the middle \(50\%\) | little or none |
| interdecile range | the middle \(80\%\) | little |
| MAD | every score (distances) | noticeable |
| standard deviation | every score (squared distances) | largest of these |
Quantiles from a list
- Order the scores and count them (\(n\)).
- Quartiles: find the median, then the median of each half (leave the median out when \(n\) is odd).
- Percentiles and deciles: work out the position \(\dfrac{p}{100}(n + 1)\). If it is a whole number, take that score; if it ends in \(.5\), go halfway between the two scores either side.
- Answer with values, not positions, and give units.
From a cumulative frequency graph
- Build the table: add a running total column. The last running total is \(n\).
- Plot each running total above the upper end of its class, start at \(0\) at the lower end of the first class, and join the points with straight lines.
- Read across from \(\tfrac{n}{4}\), \(\tfrac{n}{2}\), \(\tfrac{3n}{4}\) (or \(\tfrac{p}{100} \times n\)) to the graph, then down to the horizontal axis.
Comparing spread
- Calculate the measures asked for. A table with and without an unusual score shows the effect of an outlier clearly.
- Choose the IQR (or interdecile range) when there are outliers or the data are skewed; the standard deviation suits roughly symmetric data with no outliers.
- Unknown scores: write the quartiles or median in terms of the unknowns, use the mean for a total, and solve the equations.
| Hours, \(t\) | Frequency |
|---|---|
| \(0 \le t < 5\) | \(10\) |
| \(5 \le t < 10\) | \(20\) |
| \(10 \le t < 15\) | \(30\) |
| \(15 \le t < 20\) | \(30\) |
| \(20 \le t < 25\) | \(20\) |
| \(25 \le t < 30\) | \(10\) |
(ii) Estimate the median, the quartiles and the IQR.
(iii) Estimate the \(90\)th percentile.
(iv) At what percentile is a screen time of \(12\) hours?
(i) Running totals: \(10,\ 30,\ 60,\ 90,\ 110,\ 120\). Plot \((0, 0)\), \((5, 10)\), \((10, 30)\), \((15, 60)\), \((20, 90)\), \((25, 110)\) and \((30, 120)\).
(ii) Read across from \(\tfrac{120}{4} = 30\), \(\tfrac{120}{2} = 60\) and \(\tfrac{3 \times 120}{4} = 90\):
| \(Q_1\) | \(\approx\) | \(10 \text{ h}, \quad \text{median} \approx 15 \text{ h}, \quad Q_3 \approx 20 \text{ h}\) |
| \(\text{IQR}\) | \(\approx\) | \(20 - 10 = \mathbf{10} \text{ h}\) |
(iii) \(90\%\) of \(120\) is \(108\). That is \(18\) of the \(20\) students in the \(20\)–\(25\) class, so go \(\tfrac{18}{20}\) of the way along it:
| \(P_{90}\) | \(\approx\) | \(20 + \tfrac{18}{20} \times 5 = \mathbf{24.5} \text{ h}\) |
(iv) \(12\) hours is \(\tfrac{2}{5}\) of the way through the \(10\)–\(15\) class, so about \(30 + \tfrac{2}{5} \times 30 = 42\) students are below it.
| \(\dfrac{42}{120} \times 100\) | \(=\) | \(35\) |
A screen time of \(12\) hours is at about the \(\mathbf{35}\)th percentile.
(i) the median, \(Q_1\), \(Q_3\) and the IQR
(ii) \(D_1\), \(D_9\) and the interdecile range
(iii) the percentile rank of a time of \(30\) minutes.
Here \(n + 1 = 20\), so every position below is a whole number.
| \(\text{median}\) | \(:\) | \(\tfrac{50}{100} \times 20 = 10\text{th score} = \mathbf{24}\) |
| \(Q_1\) | \(:\) | \(\tfrac{25}{100} \times 20 = 5\text{th score} = \mathbf{18}\) |
| \(Q_3\) | \(:\) | \(\tfrac{75}{100} \times 20 = 15\text{th score} = \mathbf{32}\) |
| \(\text{IQR}\) | \(=\) | \(32 - 18 = \mathbf{14}\) |
(These match the medians of the lower and upper halves of \(9\) scores each.)
| \(D_1\) | \(:\) | \(\tfrac{10}{100} \times 20 = 2\text{nd score} = \mathbf{14}\) |
| \(D_9\) | \(:\) | \(\tfrac{90}{100} \times 20 = 18\text{th score} = \mathbf{40}\) |
| \(D_9 - D_1\) | \(=\) | \(40 - 14 = \mathbf{26}\) |
(iii) \(13\) of the \(19\) times are below \(30\) minutes.
| \(\dfrac{13}{19} \times 100\) | \(=\) | \(68.4\ldots\) |
A time of \(30\) minutes is at about the \(\mathbf{68}\)th percentile.
(i) Find the mean absolute deviation, the population standard deviation and the IQR.
(ii) Repeat without the outlier \(38\).
(iii) Which measure changes most? Explain.
(i) \(\bar{x} = \dfrac{108}{6} = 18\). Distances from the mean: \(8, 6, 5, 3, 2, 20\).
| \(\text{MAD}\) | \(=\) | \(\dfrac{44}{6} = 7.33\ldots\) |
| \(\sigma\) | \(=\) | \(\sqrt{\dfrac{64 + 36 + 25 + 9 + 4 + 400}{6}}\) |
| \(\) | \(=\) | \(\sqrt{\dfrac{538}{6}} = 9.47\ldots\) |
| \(\text{IQR}\) | \(=\) | \(20 - 12 = 8\) |
(ii) Now \(\bar{x} = \dfrac{70}{5} = 14\). Distances: \(4, 2, 1, 1, 6\). With \(5\) scores, leave out the median \(13\): \(Q_1 = \dfrac{10 + 12}{2} = 11\), \(Q_3 = \dfrac{15 + 20}{2} = 17.5\).
| \(\text{MAD}\) | \(=\) | \(\dfrac{14}{5} = 2.8\) |
| \(\sigma\) | \(=\) | \(\sqrt{\dfrac{58}{5}} = 3.41\ldots\) |
| \(\text{IQR}\) | \(=\) | \(17.5 - 11 = 6.5\) |
| MAD | \(\sigma\) | IQR | |
|---|---|---|---|
| with \(38\) | \(7.33\) | \(9.47\) | \(8\) |
| without \(38\) | \(2.8\) | \(3.41\) | \(6.5\) |
(iii) The standard deviation falls the most (by about \(6.06\)). It squares each distance, so the large distance of \(38\) from the mean (\(20^2 = 400\)) dominates. The IQR hardly changes because it ignores the extreme quarters.
(ii) About how many took between \(20\) and \(35\) minutes?
(iii) Find the range and the IQR.
(iv) Every time is increased by \(20\%\). Find the new median and the new IQR.
(v) Describe the shape of the distribution.
The five-number summary is \(12,\ 20,\ 26,\ 35,\ 58\). Each section of the plot holds about a quarter of the \(200\) parcels, that is \(50\).
| \(\text{(i) above } Q_3\) | \(:\) | \(\text{about } \mathbf{50}\) |
| \(\text{(ii) } Q_1 \text{ to } Q_3\) | \(:\) | \(\text{about } \mathbf{100}\) |
| \(\text{(iii) range}\) | \(=\) | \(58 - 12 = \mathbf{46}\) |
| \(\text{IQR}\) | \(=\) | \(35 - 20 = \mathbf{15}\) |
(iv) Multiplying every time by \(1.2\) multiplies every quantile, and the IQR, by \(1.2\).
| \(\text{median}\) | \(=\) | \(1.2 \times 26 = \mathbf{31.2}\) |
| \(\text{IQR}\) | \(=\) | \(1.2 \times 15 = \mathbf{18}\) |
(v) The upper quarter (\(35\) to \(58\)) is spread much more widely than the lower quarter (\(12\) to \(20\)), and the median is closer to \(Q_1\) than to \(Q_3\): the data are positively skewed.
Common pitfalls
Frequently asked questions
What is the difference between a quartile, a decile and a percentile?
They are all quantiles. Quartiles split ordered data into four equal groups, deciles into ten and percentiles into a hundred. The median is the 2nd quartile, the 5th decile and the 50th percentile.
How do you find a percentile from a list of scores?
Order the scores and work out the position p over 100 times (n + 1). If the position is a whole number, take that score; if it is halfway between two whole numbers, take the value halfway between the two scores.
How do you read the median from a cumulative frequency graph?
Go to half the total on the cumulative frequency axis, read across to the graph, then down to the horizontal axis. Use a quarter and three quarters of the total for the lower and upper quartiles.
What does percentile rank mean?
It is the percentage of scores below a given score. A percentile rank of 60 means the score is higher than about 60 percent of the scores.
Which measure of spread is best when there is an outlier?
The interquartile range, because it uses only the middle half of the data. The range, the mean absolute deviation and especially the standard deviation all change a lot when an outlier is included.