Quartiles, percentiles and box plots

What is a quantile?

A quantile is the value at or below which a stated fraction of the data lies. The 30th percentile is the value with 30% of the observations at or below it.

Quantiles are measures of location : they say where a value sits in the sorted data, not where the data are centered. Quartiles make four groups, quintiles five, deciles ten and percentiles one hundred. The second quartile, the 50th percentile and the median are the same value.

Quantilea measure of location; a cut point in sorted data

• Median Memorize 2 equal parts, 1 cut point (the 50th percentile)
• Quartiles Memorize 4 parts, 3 cut points: Q1, Q2 (median), Q3
  • Interquartile range Q3 − Q1, the spread of the middle half
• Quintiles Use it 5 parts, 4 cut points (20th, 40th, 60th, 80th percentiles)
• Deciles Use it 10 parts, 9 cut points
• Percentiles Memorize 100 parts, 99 cut points
• Labels by field Know the idea fund rankings: first quartile = best; size deciles: first = smallest

How do you read quantiles from a frequency table?

Use it

Find the first bin whose cumulative percentage reaches the percentile you want. That bin holds it.

A cumulative frequency table lists each bin with a running total of the share of observations at or below its upper edge.

How do you place a percentile by hand?

Software does this for you, and the reading itself uses spreadsheet functions. By hand, one common method sorts the data, finds a position and interpolates. It is a convention, not the only one, so a question that needs a position will tell you which method to use.

The position of the y-th percentile in a list of n sorted values, by the (n + 1) method, is:

$$L_y = (n + 1)\,\frac{y}{100}$$

L rarely lands on a whole number. When it falls between two positions, interpolate:

$$X_y = X_{\text{lower}} + (L - \text{lower})\,(X_{\text{upper}} - X_{\text{lower}})$$

The (n + 1) method from every side:

Move the Percentile y slider and watch the position move. Position first, then interpolate.

Exam tip Check the position before you interpolate. If L comes out as exactly 16.0, the answer is the 16th value and there is nothing to interpolate.

In a spreadsheet

Know the idea

Use QUARTILE.INC(range, 1) and QUARTILE.INC(range, 3) for Q1 and Q3. PERCENTILE.INC(range, K) takes K as a fraction: 0.2 for the first quintile cut, 0.4 for the second, 0.3 for the 30th percentile. The .INC functions include the smallest and largest values as the 0th and 100th percentiles; .EXC excludes them. Google Sheets’ QUARTILE and PERCENTILE work like .INC.

What are the interquartile range and a box plot?

The interquartile range (IQR) is the distance from the first to the third quartile. It covers the middle half of the data, so extreme values cannot move it.

$$IQR = Q_3 - Q_1$$

A box and whisker plot draws that information. The box runs from Q1 to Q3 with a line at the median, and the whiskers reach out toward the extremes.

Box and whisker plota picture of the quartiles

• Box Memorize from Q1 to Q3; its height is the IQR
• Line inside the box the median (Q2)
• Whiskers to the furthest points inside the fences (or to the min and max)
• Fences Use it Q1 − 1.5 IQR and Q3 + 1.5 IQR: a convention
  • Points beyond a fence flagged as outliers and drawn as dots

In the common version the outlier fences sit 1.5 IQR beyond the box. Any point past a fence is flagged as an outlier.

$$F_U = Q_3 + 1.5\,IQR, \qquad F_L = Q_1 - 1.5\,IQR$$

The box can mark the median or the mean. Whiskers end at the most extreme points still inside the fences; anything beyond is drawn as a dot. Some charts simply run the whiskers to the minimum and maximum. The 1.5 × IQR fence is a practical convention rather than a law, so treat flagged points as candidates to examine, not as errors.

Where are these used?

• Fund rankings. Managers are ranked by quintile or decile of past return.
• Factor investing. Stocks are sorted into decile buckets on a characteristic and the top and bottom buckets are compared.
• Risk. The 5th percentile of a return distribution is a loss figure that a risk manager reads as a “bad day” threshold.

Which end is “first” depends on the field. In a spreadsheet the 0th percentile is the minimum and the 100th the maximum. In manager rankings the first quartile holds the best returns and the fourth the worst. In research on company size the first decile usually holds the smallest companies. Read the convention before you read the number.

Go deeper Why spreadsheets can disagree with the hand method

The (n + 1) method gives the same answers as Excel’s .EXC functions. The .INC functions place the y-th percentile at position 1 + (n − 1) × y ÷ 100 instead. For the 12 returns above, .INC gives Q1 = −0.75% and Q3 = 3.90% (IQR 4.65) against −1.45% and 4.50% (IQR 5.95) by the (n + 1) method; the median is 1.85% either way. On small samples the methods differ noticeably; on large ones they converge.

Go deeper Percentile ranks in manager selection

A fund’s “percentile rank” tells you the share of peers it beat. Being in the 80th percentile of its peer group over three years means it beat about 80% of them. Ranks are easy to compare across groups, but they hide how far apart the funds are: the gap between the 80th and 90th percentile can be tiny or huge depending on the group.