Updated for the 2026-2027 CFA® Level I curriculum.
Averages can tell different stories about the same investment data. The arithmetic mean may summarize a typical period, the geometric mean may show compounded growth, and the median may give a more useful picture when extreme observations are present.
For CFA Level I, you should know how to calculate, interpret, and evaluate each measure based on the investment problem being presented.
Quick Answer
Measures of central tendency summarize a dataset with a representative value, such as the mean, median, or mode. Measures of location, including quartiles and percentiles, show where an observation sits within ordered data. The appropriate measure depends on the question, the structure of the data, and whether weighting, compounding, ratios, or extreme values are involved.
Key Takeaways About Measures of Central Tendency and Location
The arithmetic mean gives the simple average of a group of observations.
The weighted mean accounts for differences in the importance or size of each observation.
The geometric mean measures compounded growth across multiple periods.
The harmonic mean is useful for ratios with a fixed numerator, such as investing the same amount at different prices.
The median identifies the middle observation and is less affected by extreme values.
The mode identifies the value or category that appears most often.
Quantiles describe the relative position of an observation within an ordered dataset.
What You Need to Know for CFA Level I
For CFA Level I, focus on the relationship between the investment problem and the measure you choose.
You should be able to:
Calculate the arithmetic, weighted, geometric, and harmonic means.
Find the median and mode of a dataset.
Explain how outliers affect the mean and median differently.
Choose an appropriate average for one-period returns, compounded returns, portfolio weights, or fixed-investment amounts.
Identify quartiles, quintiles, deciles, and percentiles.
Interpret what each result tells an analyst about the dataset.
Recognize wording that signals which measure a question requires.
What Do Measures of Central Tendency and Location Tell You?
A measure of central tendency gives you one value that represents a broader dataset. Analysts use these measures to summarize information such as returns, valuation multiples, prices, fees, and company financial results.
A measure of location answers a different question. It shows where an observation ranks within an ordered dataset. For example, a fund in the 75th percentile has a different relative position from a fund with an average return, even though both statements may describe the same dataset.
Exam questions often signal the required measure through their wording:
“What was the average one-period return?” generally points to the arithmetic mean.
“What compounded annual return did the investor earn?” points to the geometric mean.
“Which quartile contains the manager’s result?” asks for a measure of location.
“What was the typical value in this skewed dataset?” may make the median more informative.
The measure should match the investment question you are trying to answer.
Main Measures of Central Tendency
Arithmetic Mean
The arithmetic mean adds all observations and divides the total by the number of observations. It is the familiar form of an average and works well when each observation carries equal weight.
For investment analysis, it is commonly used to summarize comparable one-period returns. Extreme values can pull it noticeably higher or lower, so the arithmetic mean should be interpreted carefully when the dataset is skewed.
Weighted Mean
The weighted mean gives each observation a specified level of importance. You multiply every value by its weight and then add the results.
Portfolio return is a common application. An asset representing 60% of a portfolio has a larger effect on the portfolio return than an asset representing 10%.
The weights should normally sum to 1, or 100%.
Geometric Mean
The geometric mean measures the constant rate of growth that would produce the same compounded result across several periods.
It is useful for multi-period investment returns because each period builds on the value produced by the previous period. Greater variation between period returns creates a larger difference between the arithmetic and geometric means.
The standard calculation requires valid positive growth factors. A return of reduces the investment value to zero and prevents continued compounding from that point.
Harmonic Mean
The harmonic mean is used when you are averaging ratios and the numerator remains fixed.
A common investment example is dollar-cost averaging. When an investor contributes the same amount of money at several different prices, more shares are purchased at lower prices and fewer shares are purchased at higher prices. The harmonic mean captures that relationship.
The observations must be positive and nonzero because the calculation uses their reciprocals.
Median
The median is the middle value after the observations have been arranged in order.
For an odd number of observations, the median is the single middle value. For an even number, it is the arithmetic mean of the two middle values.
The median works well for skewed data because its position changes less than the arithmetic mean when a dataset contains a very large or very small observation. It uses the order of the data rather than the distance between every value.
Mode
The mode is the observation or category that occurs most frequently.
It can be useful for identifying the most common credit rating, price range, survey response, or other recurring result. A dataset may have one mode, several modes, or no mode when every value appears equally often.
Trimmed and Winsorized Means
A trimmed mean removes a specified percentage of observations from the upper and lower ends of an ordered dataset before calculating the arithmetic mean.
A winsorized mean keeps the same number of observations but replaces extreme values with the nearest values that remain within the selected limits.
Both methods reduce the effect of outliers. The trimmed mean discards observations, while the winsorized mean limits their magnitude.
Measures of Location
Measures of location describe the relative position of an observation within an ordered dataset.
Quantiles divide the data into groups of equal size:
Quantile | Number of Groups | Example Interpretation |
|---|---|---|
Quartiles | 4 | The third quartile separates the highest 25% from the lower 75% |
Quintiles | 5 | The fifth quintile contains the highest 20% |
Deciles | 10 | The ninth decile marks the beginning of the highest 10% |
Percentiles | 100 | The 80th percentile is above approximately 80% of observations |
The observations must be arranged from lowest to highest before you calculate a quantile.
Different software packages may use slightly different quantile conventions. In CFA questions, apply the position method provided by the curriculum or question and use it consistently throughout the calculation.
How to Choose the Right Measure
Choosing the correct measure begins with identifying what the question is trying to summarize.
Investment Problem | Preferred Measure | Why It Fits | Common Wrong Choice |
|---|---|---|---|
Average return across comparable single periods | Arithmetic mean | Gives each period equal importance | Geometric mean |
Compounded return over several periods | Geometric mean | Accounts for the sequence of growth and losses | Arithmetic mean |
Portfolio return with unequal allocations | Weighted mean | Reflects each holding’s portfolio weight | Simple arithmetic mean |
Equal amount invested at different prices | Harmonic mean | Accounts for buying more units at lower prices | Arithmetic mean |
Typical value in a skewed dataset | Median | Limits the influence of extreme observations | Arithmetic mean |
Most frequent value or category | Mode | Identifies the most common outcome | Mean or median |
Relative position within ordered data | Quantile | Shows rank within the distribution | A central-tendency measure |
The arithmetic mean often appears to be the easiest choice because it is familiar. Exam questions reward the measure that fits the data-generating process, rather than the calculation that is easiest to perform.
Comparison of the Main Measures
Measure | Core Idea | Best Used For | Main Limitation |
|---|---|---|---|
Arithmetic mean | Sum divided by count | Comparable one-period observations | Sensitive to extreme values |
Weighted mean | Values multiplied by assigned weights | Portfolios, indexes, or unequal importance | Depends on appropriate weights |
Geometric mean | Constant compounded growth rate | Multi-period returns | Requires valid growth factors |
Harmonic mean | Reciprocal-based average | Fixed-numerator ratios | Highly affected by values near zero |
Median | Middle ordered observation | Skewed data | Does not reflect distances between most values |
Mode | Most frequent observation | Common values or categories | May be absent or non-unique |
Quantile | Relative position in ordered data | Ranking and distribution analysis | Results may depend on the position convention |
Key Formulas for Measures of Central Tendency and Location
Arithmetic Mean Formula
Where:
= arithmetic mean
= observation
= number of observations
Weighted Mean Formula
Where:
= weighted mean
= weight assigned to observation
= value of observation
= number of observations
The weights normally satisfy:
Geometric Mean Return Formula
Where:
= geometric mean return
= return in period , expressed as a decimal
= number of periods
Harmonic Mean Formula
Where:
= harmonic mean
= each positive, nonzero observation
= number of observations
Quantile Position Formula
Using the position convention applied in this example:
Where:
= location of percentile
= number of observations
= required percentile
Confirm the required position convention before applying this formula, especially when working with software output.
Worked Investment Example
A fund reports the following annual returns over five years:
The 34% return is much higher than the other observations, so this dataset shows how the choice of measure can affect the interpretation.
Arithmetic Mean
The arithmetic mean reports an average one-year return of 13%. The unusually strong fifth year pulls this result above most of the individual observations.
Median
The observations are already ordered:
The third observation is the median:
The median gives a clearer view of the middle annual result because the 34% observation does not change its position.
Geometric Mean
The geometric mean shows that the compounded annual growth rate was approximately 12.53%.
The three measures answer related but separate questions:
The arithmetic mean of 13% summarizes the average single-period return.
The median of 9% identifies the middle annual observation.
The geometric mean of 12.53% gives the annualized compounded return.
The geometric mean sits below the arithmetic mean because the returns vary from year to year.
75th Percentile
Using the position convention:
Position lies halfway between the fourth observation, 11%, and the fifth observation, 34%.
Under this convention, the 75th percentile is approximately 22.5%. A return above this value would fall within the highest quarter of the five observations.
Common Exam Traps
Using the arithmetic mean when the question asks for compounded multi-period growth.
Using the geometric mean for a simple average of independent one-period observations.
Treating the median as though it incorporates the magnitude of every observation.
Confusing the weighted mean with the geometric mean because both differ from a simple average.
Applying the harmonic mean when the numerator of the ratios changes.
Forgetting to order observations before finding the median or a quantile.
Assuming the mean, median, and mode are always equal.
Applying a quantile formula without checking the position convention used in the question.
Practice Question
An investor purchases a fixed $2,000 of the same stock at the end of each quarter. The share prices during the three purchases are $40, $50, and $80.
What is the investor’s average price paid per share?
$52.17
$54.29
$56.67
Correct Answer: A
Because the investor contributes the same dollar amount at each price, the harmonic mean gives the appropriate average price.
The investor buys more shares when the price is $40 and fewer shares when the price is $80. The harmonic mean reflects this difference in the number of shares purchased.
Option B is the geometric mean of the three prices. The prices do not represent compounded period returns.
Option C is the arithmetic mean. It gives every price equal weight without accounting for the larger number of shares purchased at lower prices.
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FAQs About Measures of Central Tendency and Location
What Is the Difference Between Central Tendency and Location?
Central tendency summarizes a dataset using a representative value, such as the arithmetic mean, median, or mode.
A measure of location shows the relative position of an observation within ordered data. Quartiles, deciles, and percentiles are common measures of location.
When Should You Use the Geometric Mean Instead of the Arithmetic Mean?
Use the geometric mean when returns compound across several periods. It accounts for the way gains and losses build on the investment value produced in earlier periods.
Use the arithmetic mean when you need the simple average of comparable one-period observations.
Why Is the Median Useful for Skewed Investment Data?
The median depends on the position of the middle observation. A small number of unusually high or low values therefore has less influence on it than on the arithmetic mean.
This makes the median useful for skewed datasets such as compensation figures, valuation multiples, property prices, or investment returns with extreme observations.
What Is the Best Measure of Center?
The appropriate measure of center depends on the dataset and the investment question.
The arithmetic mean works well for comparable observations without severe outliers. The median may be more informative for skewed data, while the geometric or harmonic mean is appropriate for specific return and ratio problems.
Why Is the Harmonic Mean Used for Equal Investment Amounts?
Equal investment amounts purchase different numbers of shares as the price changes. Lower prices result in more shares, while higher prices result in fewer shares.
The harmonic mean accounts for this inverse relationship and gives the effective average price paid per share.