READING 3 STATISTICAL MEASURES OF ASSET RETURNS

Question 1

The sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency.

A) Arithmetic mean
B) Median
C) Mode
D) Sample mean

Answer 1

A) Arithmetic mean

Question 2

Sum of all the values in a sample of a population (ΣX) divided by the number of observations (n). It is used to make inferences about the population mean.

A) Sample mean
B) Population mean
C) Arithmetic mean
D) Median

Answer 2

A) Sample mean

Question 3

Why is the median important?

Answer 3

Because the arithmetic mean can be affected by outliers, which are extremely large or small values. When this occurs, the median is a better measure of central tendency since it is not affected by extreme or erroneous values.

Question 4

When a distribution has one value that appears most frequently, it is said to be:

A) Unimodal
B) Bimodal
C) Trimodal

Answer 4

A) Unimodal

Question 5

When a distribution has two values that appears most frequently, it is said to be:

A) Unimodal
B) Bimodal
C) Trimodal

Answer 5

B) Bimodal

Question 6

When a distribution has three values that appears most frequently, it is said to be:

A) Unimodal
B) Bimodal
C) Trimodal

Answer 6

C) Trimodal

Question 7

For continuous data, such as investment returns, what do we do instead of identifying a single mode?

Answer 7

We divide the outcomes into intervals and identify the modal interval — the one with the highest number of observations.

Question 8

When a researcher decides to exclude outliers, what methods can be used?

A) Trimmed mean
B) Winsorized mean
C) Both
D) Neither

Answer 8

C) Both

Question 9

A mean that excludes a stated percentage of extreme observations. For example, A 1% would discard the lowest 0.5% and the highest 0.5% of the observations

A) Trimmed mean
B) Winsorized mean
C) Geometric mean
D) Harmonic mean

Answer 9

A) Trimmed mean

Question 10

A mean that substitutes values for extreme observations instead of discarding them.

A) Trimmed mean
B) Winsorized mean
C) Geometric mean
D) Harmonic mean

Answer 10

B) Winsorized mean

Question 11

What is the general term for a value at or below which a stated proportion of the data lies?

A) Quantile
B) Median
C) Percentile
D) Mode

Answer 11

A) Quantile

Question 12

How is a Quartile distribution divided?

Answer 12

Into 4 parts (25% each)

Question 13

How is a Quintile distribution divided?

Answer 13

Into 5 parts (20% each)

Question 14

How is a Decile distribution divided?

Answer 14

Into 10 parts (10% each)

Question 15

How is a Percentile distribution divided?

Answer 15

into 100 parts (1% each)

Question 16

Imagine 100 ranked exam scores. How are quartiles set?

Answer 16

1st Quartile: up to 25th score
2nd Quartile: up to 50th score (median)
3rd Quartile: up to 75th score

Note: The Interquartile Range (IQR) = Q3 - Q1.
Small IQR → Data tightly packed
Large IQR → Data spread out

Question 17

The variability around the central tendency is called:

A) Standard Deviation
B) Dispersion
C) MAD
D) Arithmetic Mean

Answer 17

B) Dispersion

Question 18

What is the common theme in finance and investments?

Answer 18

The tradeoff between reward and variability (risk).

Question 19

The average of the absolute deviations from the mean is:

A) Standard Deviation
B) MAD
C) Geometric mean
D) Harmonic mean

Answer 19

B) MAD

Question 20

A measure of dispersion for a sample:

A) Standard Deviation
B) Sample Variance
C) MAD
D) Harmonic mean

Answer 20

B) Sample Variance

Question 21

Why is the denominator for sample variance (n - 1) instead of n?

Answer 21

Using n would underestimate population variance. (n-1) improves the estimate, avoiding bias, especially with small samples.

Question 22

What is a major problem with variance, and how is it solved?

Answer 22

Variance is in squared units, making interpretation hard. Solution: Take the square root (Standard Deviation).

Question 23

When means differ between two data sets, what should be used to compare dispersion?

Answer 23

Use relative dispersion, measured by the Coefficient of Variation
(CV = Standard Deviation ÷ Mean).

Question 24

True or False:

A higher Coefficient of Variation (CV) is better.

Answer 24

False.

(Lower CV is better: less risk per unit of return.)

Question 25

What risk are we calculating with variance or standard deviation?

Answer 25

Risk based on outcomes above and below the mean.

Question 26

When measuring only outcomes below the mean, what risk measure is this?

Answer 26

Downside risk (e.g., target downside deviation).

Question 27

Skewness refers to: A) Symmetry B) Peakedness C) Asymmetry

Answer 27

C) Asymmetry

Question 28

A distribution that is characterized by outliers greater than the mean (in the upper region, or right tail). It is said to be skewed right because of its relatively long upper (right) tail. A) Positively skewed distribution B) Negatively skewed distribution C) Symmetrical distribution

Answer 28

A) Positively skewed distribution

Question 29

A distribution that is characterized by outliers greater less the mean that fall within its lower (left) tail. It is said to be skewed left because of its long lower tail A) Positively skewed distribution B) Negatively skewed distribution C) Symmetrical distribution

Answer 29

B) Negatively skewed distribution

Question 30

For a symmetrical distribution: A) Mean > Median > Mode B) Mean < Median < Mode C) Mean = Median = Mode

Answer 30

C) Mean = Median = Mode

Question 31

For a positively skewed, unimodal distribution: A) Mean > Median > Mode B) Mean < Median < Mode C) Mean = Median = Mode

Answer 31

A) Mean > Median > Mode (The mean is affected by outliers; in a positively skewed distribution, there are large, positive outliers, which will tend to pull the mean upward, or more positive.)

Question 32

For a negatively skewed, unimodal distribution: A) Mean > Median > Mode B) Mean < Median < Mode C) Mean = Median = Mode

Answer 32

B) Mean < Median < Mode (Large, negative outliers that tend to pull the mean downward (to the left)).

Question 33

True or False: Skew affects the mean more than median and mode.

Answer 33

True

Question 34

Skewness is measured by:

Answer 34

Sum of cubed deviations from mean ÷ cubed standard deviation ÷ number of observations. Positive skewness → Right skewed Negative skewness → Left skewed 0 skewness → Symmetrical (>0.5 absolute value = Significant skew)

Question 35

Measure of the degree to which a distribution is more or less peaked than a normal distribution. A) Sample skewness B) Kurtosis C) Correlation D) Covariance

Answer 35

2. Kurtosis

Question 36

Describes a distribution that is more peaked than a normal distribution: A) Leptokurtic B) Platykurtic C) Mesokurtic

Answer 36

A) Leptokurtic

Question 37

Refers to a distribution that is less peaked, or flatter than a normal one: A) Leptokurtic B) Platykurtic C) Mesokurtic

Answer 37

B) Platykurtic

Question 38

A distribution that has the same kurtosis as a normal distribution: A) Leptokurtic B) Platykurtic C) Mesokurtic

Answer 38

C) Mesokurtic

Question 39

True or False: A greater likelihood of large deviations from the mean increases perceived risk.

Answer 39

True

Question 40

What is the computed kurtosis for all normal distributions?

Answer 40

3 Excess kurtosis = Kurtosis - 3

Question 41

A normal distribution has excess kurtosis equal to: A) Zero B) Greater than zero C) Lower than zero

Answer 41

A) Zero

Question 42

A leptokurtic distribution has excess kurtosis: A) Zero B) Greater than zero C) Lower than zero

Answer 42

B) Greater than zero

Question 43

A platykurtic distribution has excess kurtosis: A) Zero B) Greater than zero C) Lower than zero

Answer 43

C) Lower than zero

Question 44

A measure that is approximated using deviations raised to the fourth power?

Answer 44

Sample kurtosis

Question 45

What is a measure that measures how two variables move together. A) Covariance B) Correlation C) Coefficient of variation

Answer 45

A) Covariance However, we cannot interpret the relative strength of the relationship between two variables. Knowing that the covariance of X and Y is 0.8756 for example tells us only that they tend to move together because the covariance is positive.

Question 46

A measure that mainly tells you if there’s a relationship. But it doesn’t tell you how strong that relationship is or give you an easy number to compare across different data: A) Correlation B) Coefficient of variation C) Covariance

Answer 46

C) Covariance Positive → they move together. Negative → they move opposite.

Question 47

Is there a connection? A) Correlation B) Covariance

Answer 47

B) Covariance

Question 48

How strong is the connection? A) Correlation B) Covariance

Answer 48

A) Correlation

Question 49

A standardized measure of the linear relationship between two variables. It measures the strength of the linear relationship between two random variables. A) Correlation B) Coefficient of variation C) Covariance

Answer 49

A) Correlation

Question 50

For correlation, this means that a movement in one random variable result in a proportional positive movement in the other relative to its mean: A) If ρXY = 1.0 B) If ρXY = −1.0 C) If ρXY = 0

Answer 50

A) If ρXY = 1.0

Question 51

For correlation, this means that a movement in one random variable result in an exact opposite proportional movement in the other relative to its mean: A) If ρXY = 1.0 B) If ρXY = −1.0 C) If ρXY = 0

Answer 51

B) If ρXY = −1.0

Question 52

For correlation, there is no linear relationship between the variables, indicating that prediction of Y cannot be made on the basis of X using linear methods: A) If ρXY = 1.0 B) If ρXY = −1.0 C) If ρXY = 0

Answer 52

C) If ρXY = 0

Question 53

What does a significant correlation between two variables imply? A) Causation is confirmed B) A strong causal link C) Only association, not causation D) That one variable causes the other

Answer 53

C) Only association, not causation

Question 54

Q: What is spurious correlation? A) A true relationship between two variables B) A correlation caused by chance or a third variable C) A perfect positive relationship D) A correlation caused by direct causation

Answer 54

B) A correlation caused by chance or a third variable

Question 55

Q: Why is it important to investigate outliers when analyzing correlation? A) Outliers always prove causation B) Outliers can artificially inflate or distort correlation C) Outliers always decrease correlation strength D) Outliers are irrelevant for correlation analysis

Answer 55

B) Outliers can artificially inflate or distort correlation

Question 56

Q: If removing outliers drastically reduces the correlation between two variables, what should a researcher do? A) Conclude no relationship exists B) Assume causation C) Investigate further if outliers were informative or random noise D) Ignore the change

Answer 56

C) Investigate further if outliers were informative or random noise

Question 57

Q: Two variables both change over time because they are influenced by inflation. What is this an example of? A) Spurious correlation B) Perfect correlation C) Negative correlation D) Nonlinear correlation

Answer 57

A) Spurious correlation