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CFA Level I Exam · Statistical Characteristics of Asset Returns

Quantiles, Percentiles and Data Visualization for CFA Level I

Updated 7 October 2026 · Fact-checked

A quantile splits sorted data into equal-sized parts: quartiles into 4, quintiles into 5, deciles into 10, percentiles into 100. To find the y-th percentile, sort the data and locate position (n + 1) × y ÷ 100. If it is not a whole number, interpolate between neighbours.

Understand Quantiles, Percentiles and Data Visualization

A quantile is a cut point that divides ordered data into equal-sized groups. Think of lining up 100 return observations from lowest to highest and marking places along the line.

Common quantiles: quartiles divide data into four parts, quintiles into five, deciles into ten, and percentiles into one hundred. The median is the 2nd quartile, the 5th decile and the 50th percentile. The interquartile range (IQR) is Q3 − Q1. It measures the spread of the middle half of the data and ignores extreme values.

To find a quantile, sort the data first. Then use the position formula. The position is often not a whole number. Then you interpolate: take the lower value and add the fractional part times the gap to the next value. The CFA curriculum uses this (n + 1) approach for the position, so use it.

For visualization, a frequency distribution groups data into intervals (bins) and counts how many observations fall in each. A relative frequency is the count divided by the total. A histogram draws these as adjacent bars: the x-axis shows return intervals and the y-axis shows frequency or relative frequency. Its shape shows skewness and fat tails.

A box plot (box and whisker plot) shows the median as a line inside a box that runs from Q1 to Q3. Whiskers extend toward the minimum and maximum, or to a set limit, and points beyond are shown as outliers. If the median sits off-centre in the box or one whisker is longer, the data is skewed that way.

Key formulas to remember

Position of the y-th percentile
Ly = (n + 1) × y ÷ 100
n is the number of observations in the sorted data. Use y = 25, 50, 75 for quartiles. If Ly is not a whole number, interpolate.
Interpolation
Value = X(lower) + fraction × [X(upper) − X(lower)]
fraction is the decimal part of Ly. X(lower) and X(upper) are the sorted values on either side of the position.
Interquartile range
IQR = Q3 − Q1
Measures spread of the middle 50% of observations. Not affected by extreme values.
Relative frequency
Relative frequency = interval frequency ÷ total observations
Relative frequencies across all intervals sum to 1 (100%).
Quantile equivalents
Median = Q2 = D5 = P50
Q1 = P25, Q3 = P75. Each decile is 10 percentiles; each quintile is 20 percentiles.

How to solve Quantiles, Percentiles and Data Visualization questions

Use this method for any question on quantiles, frequency tables or plots.

  1. 1Identify what is asked: a quartile, decile or percentile value, a position, the IQR, or a reading from a chart.
  2. 2Sort the data from smallest to largest. Count n.
  3. 3Apply Ly = (n + 1) × y ÷ 100 to get the position.
  4. 4If the position is a whole number, take that sorted value. If not, interpolate between the two neighbouring values.
  5. 5For IQR, find Q1 and Q3 separately, then subtract.
  6. 6For histograms, check whether the axis shows counts or relative frequency, and add bars as needed. Relative frequencies must total 1.
  7. 7For box plots, read median, Q1, Q3 and whisker lengths. Judge skew from the median's position in the box and whisker lengths.
  8. 8Check that your answer lies within the data range and matches one of the three options.

Quickest way: Position first, then interpolate

When to use it: Use when you must find a percentile or quartile from a short sorted list in about 90 seconds.

  1. Write n and compute (n + 1) × y ÷ 100 straight away.
  2. Read off the whole-number part as the lower rank and the decimal as the fraction.
  3. Compute lower + fraction × gap. Do it mentally or on the calculator.
  4. Eliminate options outside the range of the two neighbouring values.
  5. For chart questions, look at the shape first: a long right tail means positive skew, with the mean above the median.

Common mistakes in Quantiles, Percentiles and Data Visualization

  • Using n × y ÷ 100 instead of (n + 1) × y ÷ 100.

    Other textbooks and software use different position rules.

    Fix: Use (n + 1) × y ÷ 100 as in the CFA curriculum.

  • Not sorting the data before finding the position.

    The data is given in time order, such as monthly returns.

    Fix: Always rewrite the data from smallest to largest first.

  • Rounding the position to a whole number instead of interpolating.

    Students want a value that appears in the data set.

    Fix: Take the lower value plus the fraction times the gap to the next value.

  • Confusing IQR with the range.

    Both measure spread and both are differences.

    Fix: Range is maximum − minimum. IQR is Q3 − Q1.

  • Reading histogram bar height as the exact value of observations.

    Students forget bars show counts for an interval, not individual data points.

    Fix: Bar height gives frequency or relative frequency for that whole interval. Check the y-axis label.

  • Treating whiskers as always showing the minimum and maximum.

    Box plot definitions vary, and outliers may be plotted as separate points.

    Fix: Read the question's description. Points beyond the whiskers are outliers, not part of the whisker.

Worked examples

Example 1

An analyst has 11 sorted monthly returns (%): -4, -2, -1, 0, 1, 2, 3, 5, 6, 8, 12. What is the third quartile (Q3)? Options: A) 5, B) 6, C) 7.

Show the solution
  1. n = 11 and the data is already sorted.
  2. Position of Q3 = (11 + 1) × 75 ÷ 100 = 12 × 0.75 = 9.
  3. The position is a whole number, so no interpolation is needed.
  4. The 9th value in the sorted list is 6.

Answer: B) 6%

Example 2

Eight sorted annual returns (%) are: 2, 4, 5, 7, 9, 10, 14, 20. What is the interquartile range? Options: A) 5.5, B) 8.75, C) 12.

Show the solution
  1. n = 8. Position of Q1 = 9 × 25 ÷ 100 = 2.25.
  2. Q1 = 4 + 0.25 × (5 − 4) = 4.25.
  3. Position of Q3 = 9 × 75 ÷ 100 = 6.75.
  4. Q3 = 10 + 0.75 × (14 − 10) = 10 + 3 = 13.
  5. IQR = 13 − 4.25 = 8.75.

Answer: B) 8.75 percentage points (Q1 = 4.25, Q3 = 13)

Exam tips

  • Memorize Ly = (n + 1) × y ÷ 100. Most calculation questions on this topic need only this formula and one interpolation.
  • Options are listed smallest to largest, so a quick range check usually removes one or two choices.
  • For box plots, look at where the median sits between Q1 and Q3 and which whisker is longer to judge skew.
  • Know the equivalences: median = Q2 = D5 = P50, and a quintile spans 20 percentiles.
  • For frequency tables, check that relative frequencies sum to 1 before trusting a derived value.

Practice questions from Statistical Characteristics of Asset Returns

Quantiles, Percentiles and Data Visualization: frequently asked questions

What is the difference between quartiles, quintiles, deciles and percentiles?

They differ only in how many equal parts the sorted data is split into. Quartiles give 4 parts, quintiles 5, deciles 10 and percentiles 100. All are types of quantiles.

How do I find the position of a percentile in CFA Level I?

Sort the data, then compute (n + 1) × y ÷ 100. If the result has a decimal, interpolate between the two nearest sorted values. For example, with n = 8, the 25th percentile sits at position 2.25.

What is the interquartile range used for?

The IQR is Q3 − Q1 and shows the spread of the middle half of the data. Because it ignores the tails, extreme returns do not affect it much.

How do I read a box and whisker plot?

The line in the box is the median and the box edges are Q1 and Q3. Whiskers show the spread beyond the box, and separate points are outliers. A median closer to one edge or an uneven whisker suggests skew.