V1 Module 5 Errata: Statistical Characteristics of Asset Returns

1. The Mean-Median Comparison Does Not Prove Negative Outliers

Curriculum location: Measures of Central Tendency and Dispersion, German rate-spread case, printed p. 195.

That the median is below the mean suggests that there is a small number of large negative changes in the spread between the interest rates.

The displayed mean is −0.12 bp and the median is −0.20 bp, so the mean is actually above the median. A few large negative observations would tend to pull the mean downward. The histogram, reported skewness, and minimum may support a left-tail concern, but the mean-median ordering alone does not.

Correct reading: Do not infer negative outliers from this mean-median comparison. Assess the tail using the distribution’s shape, skewness, and extreme observations.

Following the printed rule could reverse the usual Level I mean-median skew heuristic.

2. Tail Percentages Mix Total and Per-Tail Conventions

Curriculum location: Measures of Central Tendency and Dispersion, trimmed-mean definition and Outliers and the Price of Oil case, printed pp. 196, 199–200.

A 1% trimmed mean would exclude the lowest 0.5% and the highest 0.5% of values, and the mean is computed using the remaining 99% of the data.

Trimming 1% removes two of the extreme observations, −79.82% and 33.51%, respectively, and the trimmed average changes to 0.83%. Trimming 10% removes 10 observations, and the trimmed average becomes 0.99%.

The outliers are replaced by the respective 1% and 10% percentile values.

Printed p. 196 defines the percentage as the total split across both tails. For 100 observations, removing or replacing one observation in each tail affects two observations, so it is 2% total—not 1%. The conflicting label is repeated in the case prose and exhibit labels on pp. 199–200. The winsorization paragraph also names the 5th and 95th percentiles for its 10% series, then calls them “10% percentile values.”

Correct reading: Apply the printed-page 196 total-percentage convention consistently. In 100 observations, one observation per tail is 2% total; 10% total is five observations per tail. Update the prose, exhibit labels, and comparisons together. For winsorization, refer to the lower and upper cutoff percentiles, and use “winsorizing,” not “trimming,” for Exhibit 8.

The printed wording can lead candidates to remove or replace the wrong number of observations and to enter the wrong percentage in spreadsheet functions.

3. Sample Standard Deviation Uses Population-Variance Notation

Curriculum location: Measures of Central Tendency and Dispersion, variance discussion, printed p. 208.

Note that, perhaps counterintuitively, the sample standard deviation, $s$, calculated as the square root of the variance, $\sqrt{{\sigma }^2}$, remains biased despite using Bessel’s correction!

The display crosses the sample statistic $s$ with the population variance ${\sigma }^2$.

Correct reading: Write $s=\sqrt{s^2}$. The statement that the usual sample standard deviation remains biased may remain.

The symbol mismatch can make candidates confuse sample and population notation.

4. The Gold Case Uses Inconsistent Indices for Ten Returns

Curriculum location: Measures of Central Tendency and Dispersion, gold case, printed pp. 210–211.

$\text{Average }\mu ={\sum }_{i=1}^{10}\frac{r_1}{n}=-2.85\%$

$r={\left({\prod }_{t=0}^{10}(1+r_t)\right)}^{1/10}-1$

The arithmetic-mean formula repeats $r_1$ rather than using the running index $r_i$. The geometric product’s inclusive limits contain eleven indices even though the table and exponent use ten returns.

Correct reading: Use $r_i$ in both arithmetic-mean sums, and index the ten geometric-return factors consistently as $t=1,\dots ,10$ or $t=0,\dots ,9$.

Following the printed indices could repeat the first return or multiply eleven holding-period relatives instead of ten.

5. Exhibit 16 Is a Dispersion Table

Curriculum location: Measures of Central Tendency and Dispersion, Exhibit 16, printed p. 211.

Exhibit 16: Excel and Google Sheets Functions for Measures of Central Tendency

Every row in the exhibit is a variance or standard-deviation function.

Correct reading: Retitle it “Excel and Google Sheets Functions for Measures of Dispersion.”

The current title places the entire table in the wrong statistical category.

6. The Gold Case Reverses the Frequency Comparison

Curriculum location: Measures of Central Tendency and Dispersion, root-of-time discussion, printed p. 212.

We show that the observed variance and standard deviation in lower-frequency data can be higher than expected from scaling longer-horizon variance or standard deviation; this is a typical occurrence.

The following case instead shows observed daily, higher-frequency volatility exceeding a daily estimate scaled down from annual, longer-horizon data.

Correct reading: Observed higher-frequency or shorter-horizon dispersion can exceed the value implied by scaling from lower-frequency or longer-horizon data.

The printed direction can invert a candidate’s interpretation of the case.

7. Skewness Does Not Determine Counts, Frequencies, or Average Magnitudes

Curriculum location: Measures of Shape of a Distribution, skewness discussion, printed pp. 217–218.

Negative skewness indicates that the left tail, or lower values or negative returns, are longer or fatter than the right tail and means there are more low-value or negative outliers.

Positive skewness indicates that the right tail, or higher values or positive returns, are longer or fatter than the left tail and means there are more high-value or positive outliers.

A positively skewed return distribution of an investment suggests that there is a higher probability of achieving high returns compared to low returns.

If skewness is negative, the average magnitude of negative deviations from the mean is larger than the average magnitude of positive deviations.

The same paragraph also states:

In financial asset return terms, on average, the negative returns must be of greater magnitude than the positive returns; there are frequent small gains and a few extreme losses.

Skewness is based on standardized cubed deviations. Its sign describes which tail contributes more strongly to the third moment, but it does not identify outlier counts or make the ordinary average-magnitude statement necessary. “Frequent small gains and a few extreme losses” is a familiar negative-skew return pattern at Level I, but it is an illustration—not a theorem or definition.

Correct reading: Positive skew indicates greater right-tail contribution; negative skew indicates greater left-tail contribution. Skewness does not by itself determine outlier counts or ordinary conditional average magnitudes. A familiar frequency pattern may be used as an example, but not as a necessary implication of the statistic.

A single extreme observation can dominate skewness, so count-based rules are unsafe.

8. Exhibit 25 Omits the Normalization in Skewness and Kurtosis

Curriculum location: Measures of Shape of a Distribution, Exhibit 25, printed p. 222.

$\text{Skew}={\sum }_{i=1}^n{\left(\frac{X_i-\mu }{\sigma }\right)}^3$

$\text{Kurt}={\sum }_{i=1}^n{\left(\frac{X_i-\mu }{\sigma }\right)}^4$

Both moment formulas are raw sums rather than averages, and their population bounds use $n$ rather than $N$.

Correct reading: Use $N^{-1}{\sum }_{i=1}^N((X_i-\mu )/\sigma {)}^3$ for population skewness and $N^{-1}{\sum }_{i=1}^N((X_i-\mu )/\sigma {)}^4$ for population raw kurtosis, with the corresponding $1/n$ sample approximations.

Omitting the averaging factor multiplies the reported moment by the population or sample size.

9. Question Set 2 Gives Three Incompatible Frequency Rules for Negative Skew

Curriculum location: Measures of Shape of a Distribution, Question Set 2, printed p. 223.

Hence, daily returns of the front-month Brent crude oil futures returns are not symmetric and are characterized by infrequent, large daily negative returns and frequent, small daily positive returns.

The distribution is skewed to the left, suggesting that there may be more negative daily returns than small, positive returns.

The negative skew, −0.9393, specifically indicates that there are either frequent, small negative returns; infrequent, large negative returns; or both, rather than infrequent, large positive returns.

The keyed tail-direction explanation is sufficient, but the additional frequency claims disagree with one another and infer counts from a statistic that does not contain them.

Correct reading: Keep option B because the negative skewness value indicates a longer left tail. Remove the option A inference that negative skew means more negative returns. In the option C discussion, do not present frequent small negative returns with a few extreme positive returns as a negative-skew signature; that is a common positive-skew pattern.

The keyed answer survives, but the explanation teaches mutually incompatible rules.

10. Positive Correlation Does Not Automatically Mean a Close Link

Curriculum location: Covariance and Correlation between Variables, excess-return discussion, printed p. 228.

A positive correlation suggests that the asset’s performance, after adjusting for the risk-free rate, is closely linked to the market.

The sign gives direction. Strength depends on the magnitude of the correlation.

Correct reading: A positive correlation indicates a same-direction linear association. Describe the link as close or strong only when the magnitude supports that description.

Without the magnitude distinction, even a correlation just above zero could be called strong.

11. The Benchmark Correlations Have Two Different Printed Value Pairs

Curriculum location: Covariance and Correlation between Variables, Exhibits 32–33 and following discussion, printed p. 229.

The Correlation of the Monthly Excess Returns between LVMH Moet Hennessy Louis Vuitton and the CAC 40 Index is 0.7581

The Ccorrelation of the Monthly Excess Returns between Polo Ralph Lauren and the S&P 500 Index is 0.6188

Specifically, LVMH Moet Hennessy Louis Vuitton has a higher correlation with its benchmark (0.7546) than does Polo Ralph Lauren (0.6261).

The charts and prose give different exact values for both relationships, with no stated change in period, observations, or method.

Correct reading: The ordering—LVMH’s correlation is higher—is consistent. Do not quote either exact pair as authoritative until the coefficients can be recomputed from the underlying monthly excess-return data or confirmed by an authoritative author-side source.

The page establishes a contradiction but does not contain enough data to choose the correct exact pair.

12. Question Set 3 Switches from Growth to Levels

Curriculum location: Covariance and Correlation between Variables, Question Set 3, printed p. 233.

The solution says:

there is a positive association between current earnings yield and expected dividend yield.

The stem and option B concern growth or changes, but the explanation switches to current and expected levels.

Correct reading: Edit only the explanation so it remains on growth in earnings yield and growth in dividend yield, as named in the stem. Option B is already correct and should remain unchanged.

Correlations of levels and growth are not interchangeable, even when the keyed direction is unchanged.

13. Equation 15 Mixes Population Notation with a Sample Convention

Curriculum location: Other Dispersion Measures, Equation 15 and following discussion, printed pp. 235–236.

${\sigma }_{\text{Semi}}^2={\sum }_{i:x_i<\bar{X}}^n\frac{(X_i-\bar{X}{)}^2}{n}$

Because these are samples drawn from the population of all returns, both divide through by $n-1$, even though semi-deviation uses fewer observations.

The equation and prose disagree on the denominator. The equation also uses $\bar{X}$, a sample center, with the population symbol ${\sigma }^2$.

Correct reading: Under the sample convention stated in the text, write $s_{\text{Semi}}^2={\sum }_{i:X_i<\bar{X}}(X_i-\bar{X}{)}^2/(n-1)$.

A candidate otherwise has to choose between two printed denominators and two incompatible notation systems.

14. Semi-Deviation Is Not Variance

Curriculum location: Other Dispersion Measures, Question Set 4, printed p. 238.

The correct response is B. Semi-deviation is the variance for observations below the mean.

Semi-variance is the squared measure. Semi-deviation is its positive square root.

Correct reading: Semi-deviation is the square root of semi-variance and measures downside dispersion below the mean or a target.

The distinction matters because variance is in squared units while deviation is in the original return units.

Complete Module 5 Body-Text Errata Index

Errata scope

• Curriculum: CFA Level I 2027 Curriculum
• Volume: Volume 1, Quantitative Methods
• Module: Module 5, Statistical Characteristics of Asset Returns
• Topics covered: Measures of Central Tendency and Dispersion; Measures of Shape of a Distribution; Covariance and Correlation between Variables; Other Dispersion Measures
• Scope: Module 5 printed pp. 187–240, body text and embedded support material

The table lists every confirmed source-authored error found in the reviewed topic body and embedded support material. It excludes extraction or transcription artifacts that do not appear in the printed curriculum, and end-of-module practice material outside the stated scope.

Page references use printed curriculum page numbers.

References

• NIST/SEMATECH: Measures of location
• NIST/SEMATECH: Skewness and kurtosis
• NIST/SEMATECH: Standard deviation
• Microsoft Support: TRIMMEAN function
• Penn State STAT 500: Correlation

This article is an independent candidate-focused analysis of confirmed errors in the CFA Level I 2027 Curriculum, Volume 1 Quantitative Methods, Module 5 Statistical Characteristics of Asset Returns. It is not an official curriculum errata notice.