You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.
Quartiles, percentiles and box plots
Sort the data and locate the y-th percentile at position L = (n + 1) × y ÷ 100, interpolating between neighbors when L is not a whole number. Quartiles cut the data into four parts, quintiles five, deciles ten. The interquartile range (IQR) is the third quartile (Q3) minus the first quartile (Q1), the height of a box plot's box.
Before this: Mean, median, mode and what outliers do to them
On the exam
- Calculate, interpret, and evaluate measures of central tendency and location to address an investment problem
Calculate and interpret quantiles and the interquartile range, and read a box and whisker plot. Expect a percentile that needs interpolation, and an outlier fence built from 1.5 times the interquartile range.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
How do you locate a percentile?
Sort the data from smallest to largest, find a position, and read off (or interpolate) the value there. Quantiles are measures of location : they say where a value sits in the sorted data, not where the data are centered.
A quantile is a cut point that splits sorted data into equal-sized groups. Quartiles make four groups, quintiles five, deciles ten and percentiles one hundred.
- 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
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- 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
The position of the y-th percentile in a list of n sorted values is:
L rarely lands on a whole number. When it falls between two positions, interpolate:
A sorted list holds 12 returns. Using L = (n + 1) × y ÷ 100, the position of the 25th percentile is closest to:
(12 + 1) × 0.25 = 3.25
Position, L3.25
- 12+1=×0.25=the position, counted from the smallest value
- 12ENTER1+0.25×the position, counted from the smallest value
ENTER enter (on the HP, puts the number on the stack)
Try this 0 of 3
- Set the percentile to 75 and check that Q3 is 4.50%
- Drag the largest return past the upper fence so it becomes an outlier
- Switch the position rule to n and read the 25th percentile
Twelve monthly returns, sorted (%), with their positions
- pos 1−6.2
- pos 2−3.1
- pos 3−1.8
- pos 4−0.4
- pos 50.7
- pos 61.5
- pos 72.2
- pos 82.9
- pos 93.6
- pos 104.8
- pos 116.0
- pos 129.5
L = 3.25: between positions 3 and 4 (outlined)
Box and whisker plot (fences at 1.5 × IQR)
Drag the orange dot (the largest return), or focus it and use the arrow keys. A hollow dot is an outlier.
Position rule
L = 3.25 sits 25% of the way from position 3 to position 4.
Show working
Position: L = (n + 1) × y ÷ 100 = 13 × 25 ÷ 100 = 3.25.
Value = X3 + (L − 3) × (X4 − X3) = −1.8 + 0.25 × (−0.4 − (−1.8)) = −1.45%.
Box: Q1 (L = 3.25) = −1.45%, median (L = 6.5) = 1.85%, Q3 (L = 9.75) = 4.50%. IQR = 4.50 − (−1.45) = 5.95.
Fences: −1.45 − 1.5 × 5.95 = −10.375%; 4.50 + 1.5 × 5.95 = 13.425%. Every return lies inside, so there are no outliers by this rule.
- 13×25÷100=L = 3.25
- 0.4+|-+1.8=gap 1.40
- ×0.25=0.35
- −1.8=−1.45%
+|- change sign
- 13ENTER25×100÷L = 3.25
- 0.4CHSENTER1.8CHS−gap 1.40
- 0.25×0.35
- 1.8CHS+−1.45%
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
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.
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.
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 Memorize from Q1 to Q3; its height is the IQR
- Line inside the box the median (Q2)
- Whiskers run out to the fences or to the extreme values
- Fences Use it Q1 − 1.5 IQR and Q3 + 1.5 IQR in the common version
- Points beyond a fence flagged as outliers and drawn as dots
-
In the common version the fences sit 1.5 IQR beyond the box. Any point past a fence is flagged as an outlier.
A return series has Q1 = -0.50% and Q3 = 3.50%. The interquartile range, in percentage points, is closest to:
3.50% − (-0.50%) = 4 percentage points
Interquartile range4.00%
- 3.5−0.5+|-=
+|- change sign
- 3.5ENTER0.5CHS−
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
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.
Go deeper Why different software gives slightly different quartiles
Several conventions exist for placing a percentile. Spreadsheet functions offer more than one, and some textbooks use n instead of n + 1. On a small data set they can disagree in the second decimal. The exam uses the (n + 1) rule shown here, so use it and do not trust a spreadsheet’s default to match.
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.
Memorize this
MemorizeQuantile names
| Name | Parts | Cut points |
|---|---|---|
| Median | 2 | 1 |
| Quartile | 4 | 3 |
| Quintile | 5 | 4 |
| Decile | 10 | 9 |
| Percentile | 100 | 99 |
Memory hook Cut points are always one fewer than the parts: 4 parts, 3 cuts.
Know it cold , then interpolate. . Fences sit 1.5 IQR outside the box.
- 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
- Box Memorize from Q1 to Q3; its height is the IQR
- Line inside the box the median (Q2)
- Whiskers run out to the fences or to the extreme values
- Fences Use it Q1 − 1.5 IQR and Q3 + 1.5 IQR in the common version
- Points beyond a fence flagged as outliers and drawn as dots
-
Quick example
19 sorted returns, 80th percentile. L = (19 + 1) × 0.80 = 16, so the 16th value exactly.
- 19+1=×0.8= 16
- 19ENTER1+0.8× 16
ENTER enter (on the HP, puts the number on the stack)
Quick example
L = 6.4 sits between the 6th value, 1.2, and the 7th, 1.7. 1.2 + 0.4 × (1.7 − 1.2) = 1.40%.
- 1.7−1.2=×0.4+1.2= 1.4
- 1.7ENTER1.2−0.4×1.2+ 1.4
ENTER enter (on the HP, puts the number on the stack)
Quick example
Q1 = −0.8% and Q3 = 2.2%. IQR = 2.2 − (−0.8) = 3.0 percentage points.
- 2.2−0.8+|-= 3
+|- change sign
- 2.2ENTER0.8CHS− 3
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Quick example
Q1 = −0.6% and Q3 = 2.4%, so IQR = 3.0. Upper fence 2.4 + 1.5 × 3.0 = 6.90%; lower fence −0.6 − 4.5 = −5.10%.
- 2.4−0.6+|-=×1.5+2.4= 6.9
+|- change sign
- 2.4ENTER0.6CHS−1.5×2.4+ 6.9
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
How it is asked
-
Percentile that needs interpolation
15 sorted returns. The 6th is 1.2% and the 7th is 1.7%. The 40th percentile?
L = 16 × 0.40 = 6.4, then 1.2 + 0.4 × 0.5 = 1.40%. Trap: using n (15 × 0.4 = 6.0).
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Upper fence
Q1 = −0.5% and Q3 = 3.5%. Upper fence using 1.5 × IQR?
IQR = 4.0, so 3.5 + 6.0 = 9.50%. Trap: adding one IQR (7.50%).
-
What the box shows
In a box and whisker plot, the height of the box represents…
The interquartile range, not the range or the standard deviation.
Exam traps
- Using n instead of n + 1 for the position.
- Rounding L to a whole position instead of interpolating.
- Reading the whisker length as the interquartile range. The IQR is the box.
- Forgetting to sort first, or counting positions from the largest value.
Checkpoint
4 exam-style questions. About 90 seconds each.
1A sorted list of 15 returns has 1.2% in the 6th position and 1.7% in the 7th. Using L = (n + 1) × y ÷ 100, the 40th percentile is closest to:
2In a box and whisker plot, the height of the box represents the:
3A data set has Q1 = −0.5% and Q3 = 3.5%. Using the 1.5 × IQR rule, the upper fence is closest to:
4A sorted list holds 19 returns. Using the (n + 1) rule, the 80th percentile sits at position:
Key takeaways
- Position L = (n + 1) × y ÷ 100; if it is not a whole number, interpolate.
- Quartiles split into 4, quintiles 5, deciles 10, percentiles 100.
- IQR = Q3 − Q1 is the height of the box.
- Fences 1.5 × IQR beyond the box flag outliers.
Quick example
19 sorted returns, 80th percentile. L = (19 + 1) × 0.80 = 16, so the 16th value exactly.
- 19+1=×0.8= 16
- 19ENTER1+0.8× 16
ENTER enter (on the HP, puts the number on the stack)
Quick example
L = 6.4 sits between the 6th value, 1.2, and the 7th, 1.7. 1.2 + 0.4 × (1.7 − 1.2) = 1.40%.
- 1.7−1.2=×0.4+1.2= 1.4
- 1.7ENTER1.2−0.4×1.2+ 1.4
ENTER enter (on the HP, puts the number on the stack)
Quick example
Q1 = −0.8% and Q3 = 2.2%. IQR = 2.2 − (−0.8) = 3.0 percentage points.
- 2.2−0.8+|-= 3
+|- change sign
- 2.2ENTER0.8CHS− 3
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Quick example
Q1 = −0.6% and Q3 = 2.4%, so IQR = 3.0. Upper fence 2.4 + 1.5 × 3.0 = 6.90%; lower fence −0.6 − 4.5 = −5.10%.
- 2.4−0.6+|-=×1.5+2.4= 6.9
+|- change sign
- 2.4ENTER0.6CHS−1.5×2.4+ 6.9
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Questions about this lesson?
Where this comes back
- Range, mean absolute deviation, variance and standard deviation Statistical Measures of Asset Returns
- Introduction to Risk Management Portfolio Construction
Questions candidates ask
How do you calculate a percentile in CFA Level 1?
Sort the data and find the position L = (n + 1) × y ÷ 100. If L is a whole number, take that value. If not, interpolate: lower value plus the fractional part of L times the gap to the next value. The exam uses n + 1, not n.
What is the interquartile range?
The distance between the third and first quartiles, Q3 − Q1. It measures the spread of the middle half of the data and ignores the extremes. It is the height of the box in a box and whisker plot, and the basis for the 1.5 × IQR outlier fences.
How do you read a box and whisker plot?
The box runs from Q1 to Q3 with a line at the median. The whiskers extend to the fences or to the extreme values. In the common version the fences sit 1.5 IQR beyond the box, and points past them are drawn as dots and treated as outliers.
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