You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

Learning Module 3 of Quantitative Methods 6–9% of the exam

Statistical Measures of Asset Returns

6 lessons. About 34 minutes in cram mode, 140 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
Mean, median, mode and what outliers do to them
calculate 6 min cram, 24 min full
2
Quartiles, percentiles and box plots
calculate 6 min cram, 24 min full
3
Range, mean absolute deviation, variance and standard deviation
calculate 6 min cram, 24 min full
4
Downside deviation and coefficient of variation
calculate 6 min cram, 24 min full
5
Skewness and kurtosis
interpret 4 min cram, 20 min full
6
Covariance and correlation
interpret 6 min cram, 24 min full

What the exam expects you to do

Calculate, interpret, and evaluate measures of central tendency and location to address an investment problem.Mean, median, mode and what outliers do to them
Calculate, interpret, and evaluate measures of dispersion to address an investment problem.Range, mean absolute deviation, variance and standard deviation
Interpret and evaluate measures of skewness and kurtosis to address an investment problem.Skewness and kurtosis
Interpret correlation between two variables to address an investment problem.Covariance and correlation

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

Hide what I know

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Mean, median, mode and what outliers do to them

Known

MemorizeDropping versus capping extremes

Mean	Extremes	n in the divisor
Arithmetic	kept as they are	n
Trimmed	dropped	n − (values dropped)
Winsorized	replaced by the nearest kept values	n

Memory hook Trim = throw away (fewer numbers). Winsorize = clip the ends (same count). Winsorized keeps all n.

Know it cold Even n: median = average of the two middle values. A "5% trimmed mean" drops 2.5% from each end.

MemorizeWhich center fits

Memory hook The mean is dragged toward the tail; the median stays put. Skewed or outlier-heavy data: report the median.

Know it cold Mode is the only center that works for categories such as sector or rating.

Key terms

measure of central tendency Know the idea
arithmetic mean Memorize
sample mean Use it
median Memorize
mode Use it
unimodal Know the idea
bimodal Know the idea
outlier Use it
trimmed mean Memorize
winsorized mean Memorize

Measures of the middle

Measure of central tendencywhere a data set is centered

Mean Memorize sum ÷ n, uses every value
- Pulled by outliers drifts toward the tail
Median Memorize the middle value after sorting
- Resists outliers best center for skewed data
Mode Use it the most frequent value
- Unimodal, bimodal one, two (or more) peaks; may have none
- Only measure for categories works with nominal data
Outlier-resistant means Memorize still average, with the extremes handled
- Trimmed mean drop the extremes, n falls
- Winsorized mean replace extremes with nearest kept values, n stays

What to do with an outlier

Extreme value in the datafirst ask whether it is real

Recording error fix or delete it
Real and important Use it keep it, report the mean and the median
Real but mostly noise Use it use a trimmed or winsorized mean

Formulas

Sample mean Memorize

\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}

Quick example

Four quarterly returns of 3%, −1%, 4% and 2%. (3 − 1 + 4 + 2) ÷ 4 = 2.00%.

On your calculator

3+1+|-+4+2=÷4= 2

+|- change sign

3ENTER1CHS+4+2+4÷ 2

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Median position Memorize

n \text{ odd: position } \frac{n+1}{2}; \quad n \text{ even: average positions } \frac{n}{2} \text{ and } \frac{n}{2}+1

Quick example

Six sorted returns 1.1, 1.9, 2.4, 2.8, 3.5, 6.0. Average the 3rd and 4th: (2.4 + 2.8) ÷ 2 = 2.60%.

On your calculator

2.4+2.8=÷2= 2.6

2.4ENTER2.8+2÷ 2.6

ENTER enter (on the HP, puts the number on the stack)

Quartiles, percentiles and box plots

Known

MemorizeQuantile names

Name	Parts	Cut points
Median	2	1
Quartile	4	3
Quintile	5	4
Decile	10	9
Percentile	100	99

Memory hook Cut points are always one fewer than the parts: 4 parts, 3 cuts.

Know it cold $L_y = (n+1)\,y/100$ , then interpolate. $IQR = Q_3 - Q_1$ . Fences sit 1.5 IQR outside the box.

Key terms

measures of location Know the idea
quantile Use it
quartiles Memorize
quintiles Use it
deciles Use it
percentiles Memorize
interquartile range Memorize
box and whisker plot Use it

Quantiles: cutting sorted data into equal parts

Quantilea measure of location; a cut point in sorted data

Median Memorize 2 equal parts, 1 cut point (the 50th percentile)
Quartiles Memorize 4 parts, 3 cut points: Q1, Q2 (median), Q3
- Interquartile range Q3 − Q1, the spread of the middle half
Quintiles Use it 5 parts, 4 cut points (20th, 40th, 60th, 80th percentiles)
Deciles Use it 10 parts, 9 cut points
Percentiles Memorize 100 parts, 99 cut points

Parts of a box and whisker plot

Box and whisker plota picture of the quartiles

Box Memorize from Q1 to Q3; its height is the IQR
Line inside the box the median (Q2)
Whiskers run out to the fences or to the extreme values
Fences Use it Q1 − 1.5 IQR and Q3 + 1.5 IQR in the common version
- Points beyond a fence flagged as outliers and drawn as dots

Formulas

Percentile position Memorize

L_y = (n + 1)\,\frac{y}{100}

Quick example

19 sorted returns, 80th percentile. L = (19 + 1) × 0.80 = 16, so the 16th value exactly.

On your calculator

19+1=×0.8= 16

19ENTER1+0.8× 16

ENTER enter (on the HP, puts the number on the stack)

Interpolated percentile Memorize

X_y = X_{\text{lower}} + (L - \text{lower})\,(X_{\text{upper}} - X_{\text{lower}})

Quick example

L = 6.4 sits between the 6th value, 1.2, and the 7th, 1.7. 1.2 + 0.4 × (1.7 − 1.2) = 1.40%.

On your calculator

1.7−1.2=×0.4+1.2= 1.4

1.7ENTER1.2−0.4×1.2+ 1.4

ENTER enter (on the HP, puts the number on the stack)

Interquartile range Memorize

IQR = Q_3 - Q_1

Quick example

Q1 = −0.8% and Q3 = 2.2%. IQR = 2.2 − (−0.8) = 3.0 percentage points.

On your calculator

2.2−0.8+|-= 3

+|- change sign

2.2ENTER0.8CHS− 3

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Outlier fences Memorize

F_U = Q_3 + 1.5\,IQR, \qquad F_L = Q_1 - 1.5\,IQR

Quick example

Q1 = −0.6% and Q3 = 2.4%, so IQR = 3.0. Upper fence 2.4 + 1.5 × 3.0 = 6.90%; lower fence −0.6 − 4.5 = −5.10%.

On your calculator

2.4−0.6+|-=×1.5+2.4= 6.9

+|- change sign

2.4ENTER0.6CHS−1.5×2.4+ 6.9

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Range, mean absolute deviation, variance and standard deviation

Known

MemorizeSample dispersion formulas

Measure	Formula	Units
Range	max − min	%
MAD	Σ\|X − X̄\| ÷ n	%
Variance	Σ(X − X̄)² ÷ (n − 1)	%²
Std. deviation	√variance	%

Memory hook Only the variance and standard deviation divide by n − 1. MAD ≤ standard deviation always.

Know it cold Sample: divide by n − 1 (one degree of freedom is used up estimating the mean).

Key terms

dispersion Know the idea
absolute dispersion Know the idea
range Use it
mean absolute deviation Use it
sample variance Memorize
sample standard deviation Memorize
degrees of freedom Know the idea

Measures of absolute dispersion

Dispersionhow far returns spread around the center

Range Use it max − min, units: %
- Weakness uses only two values
Mean absolute deviation Use it average of |X − mean|, units: %
- Weakness absolute values are awkward in later maths
Variance Memorize Σ(X − mean)² ÷ (n − 1), units: %²
- Weakness squared units, hard to read
Standard deviation Memorize √variance, units: %
- Weakness treats up and down moves alike

Formulas

Range Use it

\text{Range} = X_{\max} - X_{\min}

Quick example

Returns of 4%, −2%, 9% and 1%. Range = 9 − (−2) = 11 percentage points.

On your calculator

9−2+|-= 11

+|- change sign

9ENTER2CHS− 11

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Mean absolute deviation Use it

MAD = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n}

Quick example

Returns of 5%, −3%, 9% and 1% have mean 3. Distances 2, 6, 6, 2 add to 16, and 16 ÷ 4 = 4.0%.

On your calculator

2+6+6+2=÷4= 4

2ENTER6+6+2+4÷ 4

ENTER enter (on the HP, puts the number on the stack)

Sample variance Memorize

s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1}

Quick example

Squared deviations of 4 returns add to 80. 80 ÷ (4 − 1) = 26.67 (%²).

On your calculator

80÷(4−1)= 26.67

80ENTER4ENTER1−÷ 26.67

ENTER enter (on the HP, puts the number on the stack)

Sample standard deviation Memorize

s = \sqrt{s^2}

Quick example

Variance 26.67 (%²). √26.67 = 5.16%.

On your calculator

26.67√x 5.16

√x square root

26.67g√x 5.16

g blue shift key · √x square root

Downside deviation and coefficient of variation

Known

MemorizeDownside deviation and CV

Measure	Formula	Better when
Target downside deviation	√[ Σ(X − B)² for X < B ÷ (n − 1) ]	lower
Coefficient of variation	s ÷ X̄	lower

Memory hook Downside deviation: keep only the bad months, square the misses, but divide by all n − 1, not by how many were bad.

Know it cold Raise the target B and the downside deviation can only rise. CV is unit-free, so it compares assets with different returns.

Key terms

downside risk Know the idea
target downside deviation Memorize
target semideviation Use it
coefficient of variation Memorize
relative dispersion Know the idea

Three ways to put a number on risk

Dispersion of returnsfrom the previous lesson to this one

Standard deviation Memorize all spread around the mean, up and down alike
Target downside deviation Memorize only shortfalls below a target B
- Divisor n − 1, with n counting every observation
- Higher target more observations count, bigger shortfalls, larger result
Coefficient of variation Memorize standard deviation ÷ mean, a relative measure
- Reading it risk per unit of return, lower is better
- Caution only meaningful with a positive mean not near zero

Formulas

Target downside deviation Memorize

s_{target} = \sqrt{\frac{\sum_{X_i < B} (X_i - B)^2}{n - 1}}

Quick example

Returns 3%, −1%, 2%, −4%, 5% and a 0% target. Shortfalls −1 and −4: (1 + 16) ÷ (5 − 1) = 4.25, and √4.25 = 2.06%.

On your calculator

1x²+4x²= 17
÷4=√x 2.06

x² square · √x square root

1ENTER×4ENTER×+ 17
4÷g√x 2.06

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Coefficient of variation Memorize

CV = \frac{s}{\bar{X}}

Quick example

Mean 10%, standard deviation 12%. CV = 12 ÷ 10 = 1.20 units of risk per unit of return.

On your calculator

12÷10= 1.2

12ENTER10÷ 1.2

ENTER enter (on the HP, puts the number on the stack)

Skewness and kurtosis

Known

MemorizeSkewness and the order of the averages

Shape	Long tail	Order
Positive skew	right	mode < median < mean
Symmetric	none	mean = median = mode
Negative skew	left	mean < median < mode

Memory hook The mean is dragged toward the tail. The median sits between the mean and the mode.

Know it cold Skew has the sign of the tail: left tail, negative skew.

MemorizeKurtosis names

Name	Kurtosis	Excess	Tails
Leptokurtic	> 3	> 0	fat
Mesokurtic	= 3	= 0	normal
Platykurtic	< 3	< 0	thin

Memory hook "Lepto" = leaping peak and fat tails (extreme events leap out). "Platy" = flat, like a plateau.

Know it cold Excess kurtosis = kurtosis − 3. The normal distribution has kurtosis 3 and excess kurtosis 0.

Key terms

skewed Know the idea
skewness Memorize
sample skewness Know the idea
kurtosis Memorize
excess kurtosis Memorize
sample excess kurtosis Know the idea
leptokurtic Memorize
platykurtic Memorize
mesokurtic Memorize
fat-tailed Use it
thin-tailed Use it

Skewness: which way is the tail?

Skewnessasymmetry of returns around the mean

Symmetric (skew 0) Memorize mean = median = mode
Positive skew Memorize long right tail: frequent small losses, a few large gains
- Order mode < median < mean
- Examples lottery-like payoffs, bought options
Negative skew Memorize long left tail: frequent small gains, a few large losses
- Order mean < median < mode
- Examples selling insurance, equity index returns

Kurtosis: how heavy are the tails?

Kurtosisweight of the tails relative to the normal (kurtosis 3)

Mesokurtic Memorize kurtosis = 3, excess = 0, like the normal
Leptokurtic Memorize kurtosis > 3, excess > 0, fat-tailed
- Consequence extreme returns more often than the normal predicts
Platykurtic Memorize kurtosis < 3, excess < 0, thin-tailed

Formulas

Excess kurtosis Memorize

K_E = K - 3

Quick example

Kurtosis of 2.4. Excess kurtosis = 2.4 − 3 = −0.6, so thin-tailed (platykurtic).

On your calculator

2.4−3= -0.6

2.4ENTER3− -0.6

ENTER enter (on the HP, puts the number on the stack)

Sample skewness (large n) Know the idea

\text{Skewness} \approx \frac{\frac{1}{n}\sum_{i=1}^{n} (X_i - \bar{X})^3}{s^3}

Quick example

Average cubed deviation −0.9 (in %³) and s = 1.2%. −0.9 ÷ 1.2³ = −0.52: negative skew.

On your calculator

1.2y^x3=1/x×0.9+|-= -0.5208

y^x power · 1/x reciprocal · +|- change sign

0.9CHSENTER1.2ENTER3y^x÷ -0.5208

CHS change sign · ENTER enter (on the HP, puts the number on the stack) · y^x power

Sample excess kurtosis (large n) Know the idea

K_E \approx \frac{\frac{1}{n}\sum_{i=1}^{n} (X_i - \bar{X})^4}{s^4} - 3

Quick example

Average fourth-power deviation 9.9 (in %⁴) and s = 1.2%. 9.9 ÷ 1.2⁴ − 3 = 4.77 − 3 = 1.77: fat-tailed.

On your calculator

1.2y^x4=1/x×9.9−3= 1.7743

y^x power · 1/x reciprocal

9.9ENTER1.2ENTER4y^x÷3− 1.7743

ENTER enter (on the HP, puts the number on the stack) · y^x power

Covariance and correlation

Known

MemorizeCorrelation at a glance

r	Scatter plot	Meaning
+1	points on an upward line	perfect positive linear link
0	no straight-line pattern	no linear link (curves possible)
−1	points on a downward line	perfect negative linear link

Memory hook Correlation is covariance with the units divided out: that is why it is stuck between −1 and +1.

Know it cold $r = s_{xy} \div (s_x s_y)$ , so $s_{xy} = r\,s_x s_y$ . Sample covariance divides by $n - 1$ .

Key terms

scatter plot Use it
sample covariance Memorize
sample correlation coefficient Memorize
correlation Use it
spurious correlation Use it
outlier Use it

What a correlation cannot tell you

Correlation coefficient rbetween −1 and +1, no units

Reading r Memorize +1 upward line, −1 downward line, 0 no linear pattern
Limitations Use it check these before trusting r
- Linear only a strong curve can give r near 0
- Outliers one point can create or hide correlation
- Spurious correlation chance, or a hidden third variable
- Not causation association says nothing about cause

Formulas

Sample covariance Memorize

s_{xy} = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1}

Quick example

Cross-products of the deviations add to 10 over n = 4 pairs. 10 ÷ (4 − 1) = 3.33 (%²).

On your calculator

10÷(4−1)= 3.33

10ENTER4ENTER1−÷ 3.33

ENTER enter (on the HP, puts the number on the stack)

Sample correlation coefficient Memorize

r = \frac{s_{xy}}{s_x\, s_y}

Quick example

Covariance 0.0045, standard deviations 15% and 10%. r = 0.0045 ÷ (0.15 × 0.10) = 0.30.

On your calculator

0.0045÷(0.15×0.1)= 0.3

0.0045ENTER0.15ENTER0.1×÷ 0.3

ENTER enter (on the HP, puts the number on the stack)

Cheat sheet · PDF · 4 A4 pages

Statistical Measures of Asset Returns on 4 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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