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CFA Level I Exam · Statistical Characteristics of Asset Returns

Measures of Central Tendency for CFA Level I

Updated 6 October 2026 · Fact-checked

Measures of central tendency describe the centre of a data set. The arithmetic mean sums values and divides by n. The geometric mean gives compound growth over time. The harmonic mean suits averaging prices bought with equal money. Median is the middle value and mode is the most frequent. Pick the measure that fits the data.

Understand Measures of Central Tendency

A measure of central tendency gives one number that represents the typical value in a data set. Different data call for different measures, and the exam tests whether you can choose the right one.

The arithmetic mean adds all values and divides by the count. It is the best estimate of a single-period expected return. It is pulled by extreme values.

The geometric mean multiplies growth factors (1 + return), takes the nth root, then subtracts 1. It measures the compound growth rate over several periods. It is never larger than the arithmetic mean. The two are equal only when every value is identical. The more the returns vary, the wider the gap.

The harmonic mean is n divided by the sum of reciprocals. Use it when you invest a fixed amount each period at changing prices, such as cost averaging. The average price paid is the harmonic mean of the prices. For positive values that are not all equal: arithmetic mean > geometric mean > harmonic mean.

The median is the middle value after sorting. It is not distorted by outliers. The mode is the most frequent value. A data set can have no mode, one mode (unimodal), two (bimodal) or more. A weighted mean gives each value a weight, as in a portfolio return where the weights are portfolio proportions and sum to 1.

Key formulas to remember

Arithmetic mean
X̄ = (X₁ + X₂ + … + Xₙ) ÷ n
Use for the expected return in a single period. It is the average of the observations.
Geometric mean return
R_G = [(1 + R₁)(1 + R₂)…(1 + Rₙ)]^(1/n) − 1
Use for compound growth over multiple periods. Work with 1 + R, not R.
Harmonic mean
X_H = n ÷ Σ(1 ÷ Xᵢ)
Use for average price paid when the same money is invested each period. Values must be positive.
Weighted mean
X̄_w = Σ(wᵢ × Xᵢ), with Σwᵢ = 1
Portfolio return is a weighted mean of asset returns.
Median position
Position = (n + 1) ÷ 2 in sorted data
For an even n, average the two middle values.
Ordering of means
Arithmetic ≥ Geometric ≥ Harmonic
Holds for positive values. Equality only if all values are equal.

How to solve Measures of Central Tendency questions

Use this routine for any question on averages of returns or prices.

  1. 1Read what is being averaged: single-period returns, multi-period growth, prices, or portfolio components.
  2. 2Choose the measure: arithmetic for a typical one-period return, geometric for compound growth, harmonic for average price with equal money invested, weighted for portfolio parts, median or mode for the middle or most common value.
  3. 3Sort the data first if you need the median or mode.
  4. 4Convert returns to growth factors (1 + R) before applying the geometric mean.
  5. 5Compute, then take the nth root for geometric or the reciprocal of the average reciprocal for harmonic.
  6. 6Convert back: subtract 1 for a geometric return and express as a percentage.
  7. 7Sanity check: geometric must not exceed arithmetic, and harmonic must not exceed geometric.

Quickest way: Pick by keyword, then check the ordering

When to use it: Use it under time pressure when the three options differ and you must eliminate two.

  1. Spot the keyword: 'compound' or 'over the period' means geometric; 'average price' or 'equal amounts' means harmonic; 'expected' or 'one period' means arithmetic.
  2. Compute the arithmetic mean quickly as an upper bound for the geometric mean.
  3. Eliminate any option for geometric mean that exceeds the arithmetic mean.
  4. On a TI BA II Plus, find the geometric mean with the y^x key: product of growth factors, then y^x with 1/n entered as 1 ÷ n, then subtract 1.
  5. For harmonic mean, use the 1/x key on each value, sum, then divide n by the total.

Common mistakes in Measures of Central Tendency

  • Taking the geometric mean of returns instead of 1 + returns.

    Students treat returns like ordinary data. Raw returns can be zero or negative, so their product is meaningless or zero. The growth factor 1 + R is always positive for returns above −100%.

    Fix: Always convert to 1 + R, multiply, take the root, then subtract 1.

  • Using the arithmetic mean to describe multi-year growth.

    It is the familiar average and gives a higher, more flattering number.

    Fix: Use the geometric mean for compound growth. The arithmetic mean overstates it when returns vary.

  • Averaging prices with the arithmetic mean in cost averaging.

    Students forget the same money buys more shares when the price is low.

    Fix: When equal amounts are invested each period, use the harmonic mean of the prices.

  • Forgetting to average the two middle values for the median when n is even.

    Students pick one middle value or compute the position wrongly.

    Fix: Sort, find positions n/2 and n/2 + 1, and average them.

  • Wrong weights in a weighted mean.

    Students use cost weights instead of market-value weights, or weights that do not sum to 1.

    Fix: Use the weights the question states, check they sum to 1, and then compute Σ(w × X).

Worked examples

Example 1

A fund returned +20%, −10% and +10% over three years. Which is closest to the geometric mean annual return? A) 5.4% B) 5.9% C) 6.7%

Show the solution
  1. Convert to growth factors: 1.20, 0.90, 1.10.
  2. Multiply: 1.20 × 0.90 = 1.08; 1.08 × 1.10 = 1.188.
  3. Take the cube root: 1.188^(1/3) ≈ 1.0590.
  4. Subtract 1: the geometric mean is about 5.90%.
  5. The arithmetic mean is (20 − 10 + 10) ÷ 3 = 6.67%. The geometric mean must be below this, so C (6.7%) is eliminated.
  6. Check A: 1.054³ ≈ 1.171, which does not match 1.188, so 5.4% is also eliminated. Check B: 1.059³ ≈ 1.188, which matches.

Answer: B. The geometric mean is about 5.9%, below the arithmetic mean of 6.67%.

Example 2

An investor buys €1,000 of a stock each month for three months at prices of €10, €8 and €5. What is the average price paid per share? A) €6.67 B) €7.06 C) €7.67

Show the solution
  1. Equal money each month, so use the harmonic mean.
  2. Reciprocals: 1/10 = 0.1; 1/8 = 0.125; 1/5 = 0.2.
  3. Sum = 0.425.
  4. Harmonic mean = 3 ÷ 0.425 = 7.0588.
  5. Check by totals: shares = 100 + 125 + 200 = 425; cost = €3,000; €3,000 ÷ 425 = €7.06.
  6. The arithmetic mean is (10 + 8 + 5) ÷ 3 = €7.67, which is the trap option.

Answer: B. The average price paid is about €7.06 per share. This is below the arithmetic mean of €7.67, as the harmonic mean always is for positive values that are not all equal.

Exam tips

  • Options are listed smallest to largest, so use the ordering arithmetic ≥ geometric ≥ harmonic to discard the wrong ones fast.
  • Words like 'compound', 'cumulative' and 'over the period' point to geometric. 'Average cost per share' with equal money points to harmonic.
  • When you see a trap option equal to the arithmetic mean in a geometric or harmonic question, it is almost always wrong unless all values are equal.
  • For median, always sort first and check whether n is odd or even.
  • Check your calculator is in the right mode and clear memory registers before multi-step root calculations.

Practice questions from Statistical Characteristics of Asset Returns

Measures of Central Tendency: frequently asked questions

What is the difference between arithmetic mean and geometric mean returns?

The arithmetic mean is the simple average of returns and estimates a typical single-period return. The geometric mean is the compound growth rate over several periods. The geometric mean is lower whenever returns vary.

How do I calculate the harmonic mean on the CFA Level I exam?

Take the reciprocal of each value, add them, and divide n by that sum. On the BA II Plus, use the 1/x key for each value and add the results. Then compute n ÷ total.

When should I use the median instead of the mean?

Use the median when outliers distort the mean, such as a data set with a few extreme returns. The median is the middle value and ignores how large the extremes are. It is a more robust centre for skewed data.

Can a data set have more than one mode?

Yes. A data set can have no mode, one mode, two modes (bimodal) or more. The mode is the value that occurs most often.