Statistical concepts and market returns - Measures of centra...

Learning Outcomes

This article explains key statistical measures for analyzing market returns, including:

• Calculating and interpreting arithmetic mean, median, and mode for return series, and comparing how each summarizes typical investment outcomes
• Applying quantiles (quartiles and percentiles) and the interquartile range to describe the location and spread of returns within a distribution
• Computing and interpreting range, mean absolute deviation, variance, and standard deviation, and relating these measures to volatility and total risk
• Evaluating target downside deviation as a measure of downside risk and contrasting it with variance and standard deviation in portfolio analysis
• Using the coefficient of variation to compare risk per unit of return across assets, funds, or strategies
• Assessing the impact of skewness and outliers on different measures of central tendency and dispersion when analyzing return data
• Selecting the most appropriate statistic for specific CFA Level 1 exam-style questions on performance evaluation and risk assessment
• Critically interpreting numerical results to compare investment opportunities, explain risk–return trade-offs, and identify limitations of each measure in practice

CFA Level 1 Syllabus

For the CFA Level 1 exam, you are expected to understand the calculation, interpretation, and application of major statistical concepts for analyzing market returns, with a focus on the following syllabus points:

• Calculate, interpret, and evaluate measures of central tendency (mean, median, mode) and location (quantiles)
• Calculate, interpret, and evaluate measures of dispersion (range, mean absolute deviation, variance, standard deviation, target downside deviation, and coefficient of variation)
• Use these statistics to compare asset returns and risk in investment scenarios
• Assess the limitations and appropriate uses of these descriptive statistics for investment analysis

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. For a sample of annual returns: 5%, 8%, 15%, 18%, and 40%, what is the median? Is the mean higher or lower than the median in this dataset?
2. What measure best describes risk per unit of return for comparing two portfolios?
3. Why is the standard deviation preferred over the range as a risk measure in portfolio analysis?
4. What is the advantage of target downside deviation versus variance for measuring investment risk?

Introduction

Measures of central tendency and dispersion are fundamental tools for summarizing return data and evaluating investment performance and risk. Understanding these allows you to describe typical outcomes (using mean, median, or mode) and measure the risk or variability of returns (using variance, standard deviation, or related statistics). CFA exam questions frequently require knowing not just how to calculate these statistics, but also how to use them in interpreting investment data.

Key Term: central tendency
The value about which observations in a dataset tend to cluster, commonly measured by the mean, median, or mode.

Key Term: dispersion
The extent to which observations in a dataset vary or spread around a central value.

Measures of Central Tendency

There are three core measures of central tendency in investment statistics:

Arithmetic Mean (Average):
The sum of all observed returns divided by the number of observations. The mean is sensitive to extreme values or outliers.

Median:
The middle value in a sorted dataset. If the number of observations is even, the median is the average of the two central values. The median is robust to outliers.

Mode:
The most commonly occurring value in a dataset. The mode may be used for quantitative or qualitative data and a dataset may have multiple modes, or none at all.

Key Term: arithmetic mean
The sum of all values divided by their number; also called simply the mean.

Key Term: median
The midpoint value when observations are ordered from lowest to highest.

Key Term: mode
The value that occurs most frequently in a dataset.

Worked Example 1.1

A fund produced annual returns of 6%, 7%, 23%, 24%, and 40%. Find the mean, median, and mode.

Answer:
Ordered data: 6%, 7%, 23%, 24%, 40%.
Mean = (6 + 7 + 23 + 24 + 40) / 5 = 100 / 5 = 20%.
Median = 23% (middle value).
Mode: None (all values occur once).

Quantiles and Measures of Location

Quantiles describe position within a dataset.

• Quartiles: Divide data into quarters. The 1st quartile (Q1) marks the lowest 25% of values; the 2nd quartile is the median; the 3rd quartile separates the top 25%.
• Percentiles: Divide data into hundredths.

Key Term: quantile
A value at or below which a stated fraction of the data lies (e.g., quartile, decile, percentile).

Key Term: interquartile range (IQR)
The difference between the 75th percentile (Q3) and 25th percentile (Q1); measures spread of the middle half of the data.

Worked Example 1.2

Returns: 2%, 2%, 6%, 7%, 10%, 15%, 21%. What are Q1, Q2, Q3? What is the IQR?

Answer:
Ordered data: 2, 2, 6, 7, 10, 15, 21.
Q2 (median) = 7%
Q1 = median of lower half (2, 2, 6): 2%
Q3 = median of upper half (10, 15, 21): 15%
IQR = Q3 − Q1 = 15% − 2% = 13%

Measures of Dispersion

Dispersion measures the degree of variability in investment returns:

Range:
Difference between the highest and lowest values.

Mean Absolute Deviation (MAD):
The average of the absolute differences between each value and the mean.

Variance:
The average squared deviation from the mean. For a sample, this is:
Variance =

Standard Deviation (SD):
The square root of variance; interpretable in the same units as the data and widely used as a risk measure in finance.

Key Term: range
The difference between the maximum and minimum values.

Key Term: mean absolute deviation (MAD)
The mean of the absolute differences between each value and the mean.

Key Term: variance
The mean squared difference between each value and the mean; for a sample, divided by n − 1.

Key Term: standard deviation
The positive square root of variance.

Worked Example 1.3

For monthly returns of 4%, 5%, 7%, 9%, and 15%, calculate the range, MAD, variance, and standard deviation.

Answer:
Range = 15% − 4% = 11%
Mean = (4 + 5 + 7 + 9 + 15) / 5 = 8%
MAD = (|4−8| + |5−8| + |7−8| + |9−8| + |15−8|) / 5 = (4 + 3 + 1 + 1 + 7) / 5 = 3.2%
Variance = [(4−8)² + (5−8)² + (7−8)² + (9−8)² + (15−8)²]/(5−1) = (16 + 9 + 1 + 1 + 49)/4 = 76/4 = 19%
Standard deviation = √19 ≈ 4.36%

Target Downside Deviation and Coefficient of Variation

For investors focused on safety, downside risk is more relevant than total risk.

Target Downside Deviation (Semideviation):
Only considers deviations below a chosen target (often zero or the mean). Useful for investors focused on minimum acceptable returns.

Coefficient of Variation (CV):
CV = standard deviation / mean
Compares relative risk (per unit of return); higher CV signals greater risk for each unit of expected return.

Key Term: downside deviation
The standard deviation of observations falling below a chosen target (such as a minimum return).

Key Term: coefficient of variation (CV)
Standard deviation divided by the mean. Measures risk per unit of expected return.

Worked Example 1.4

A fund had monthly returns: −2%, −1%, 1%, 2%, 5%, 10%. Calculate downside deviation for a 0% target, and CV.

Answer:
Returns below 0%: −2%, −1%.
Squared deviations: (−2)² = 4, (−1)² = 1.
Average squared deviation = (4 + 1)/6 = 0.833%.
Downside deviation = √0.833 ≈ 0.91%
Mean = (−2 + (−1) + 1 + 2 + 5 + 10)/6 = 15/6 = 2.5%
SD: Calculate as above (≈4.58%)
CV = 4.58 / 2.5 ≈ 1.83

Exam Warning

The mean is more affected by outliers than the median. In skewed or non-normal datasets with extreme values, always consider using the median for a better measure of typical return.

Revision Tip

For the CFA exam, practice hand-calculating the mean, median, mode, range, variance, SD, and CV for short lists of returns. CFAs frequently test not just calculation but also conceptual understanding of which statistic is appropriate in a context.

Summary

Measures of central tendency tell us about expected or typical outcomes—key for communicating investment results. Measures of dispersion quantify risk and allow comparison of volatility across assets or portfolios. Downside measures and the coefficient of variation provide further clarity into the nature of risk for practical investment analysis. For the CFA exam, you must not only perform these calculations but also critically interpret and compare assets using them.

Key Point Checklist

This article has covered the following key knowledge points:

• Calculate and interpret arithmetic mean, median, and mode for investment returns
• Compute quantiles (quartiles, percentiles) and interquartile range, and interpret their meaning as measures of location
• Calculate and interpret key measures of dispersion: range, mean absolute deviation, variance, and standard deviation, for return datasets
• Interpret downside deviation and coefficient of variation, and compare their use in risk analysis
• Assess which measure of central tendency or dispersion is most appropriate in different investment scenarios, especially with skewed or outlier-prone data

Key Terms and Concepts

• central tendency
• dispersion
• arithmetic mean
• median
• mode
• quantile
• interquartile range (IQR)
• range
• mean absolute deviation (MAD)
• variance
• standard deviation
• downside deviation
• coefficient of variation (CV)