Reading 3

Measures of Central Tendency

Measures of central tendency identify the center, or average, of a dataset. This central point can then be used to represent the typical, or expected, value in the dataset.

The arithmetic mean is the sum of the observation values divided by the number of observations. It is the most widely used measure of central tendency. The sample mean is the sum of all values in a sample, $\sum X$, divided by the number of observations, $n$. It is used to make inferences about the population mean:

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

The median is the midpoint of a dataset, where the data are arranged in ascending or descending order. Half of the observations lie above the median and half below. The median is important because the arithmetic mean can be affected by outliers (extremely large or small values). When this occurs, the median is a better measure of central tendency than the mean.

The mode is the value that occurs most frequently in a dataset. A dataset may have more than one mode, or no mode. A distribution with one most-frequent value is unimodal; with two or three, bimodal or trimodal.

For continuous data such as investment returns, we typically do not identify a single outcome as the mode. Instead, we divide the relevant range of outcomes into intervals and identify the modal interval as the one into which the largest number of observations fall.

Methods for Dealing With Outliers

In some cases, a researcher may decide that outliers should be excluded from a measure of central tendency.

• A trimmed mean excludes a stated percentage of the most extreme observations. A 1% trimmed mean discards the lowest 0.5% and the highest 0.5% of observations.
• A winsorized mean substitutes a value for the most extreme observations rather than discarding them. To compute a 90% winsorized mean: determine the 5th and 95th percentiles, replace any value below the 5th percentile with the 5th percentile and any value above the 95th percentile with the 95th percentile, then average the revised dataset.

Measures of Location

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies. Examples include:

• Quartile — distribution divided into quarters (4)
• Quintile — divided into fifths (5)
• Decile — divided into tenths (10)
• Percentile — divided into hundredths (100)

Any quantile can be expressed as a percentile. The third quartile is the 75th percentile. The difference between the third quartile and the first quartile (25th percentile) is known as the interquartile range (IQR).

To visualize a dataset based on quantiles, we can create a box and whisker plot. The box represents the central portion of the data (interquartile range), while the vertical line ("whisker") represents the entire range. If the largest observation is farther from the center than the smallest observation, the data may include one or more outliers on the high side.

Range and Mean Absolute Deviation

Dispersion is the variability around the central tendency. The common theme in finance is the tradeoff between reward (central tendency) and variability (risk).

The range is the distance between the largest and smallest values in the dataset:

range = maximum value − minimum value

The mean absolute deviation (MAD) is the average of the absolute values of the deviations of individual observations from the arithmetic mean:

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

The MAD uses absolute values because the sum of actual deviations from the arithmetic mean is zero.

Sample Variance and Standard Deviation

The sample variance, $s^2$, is the measure of dispersion that applies when we are evaluating a sample of $n$ observations from a population:

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

The denominator is $n - 1$ (one less than $n$). Using $n$ would systematically underestimate the population variance — a biased estimator. Using $n - 1$ improves the statistical properties of $s^2$ as an unbiased estimator of the population variance.

A major problem with variance is interpreting its squared units (squared percentages, squared dollars). The sample standard deviation, $s$, is the positive square root of $s^2$ — and is in the original units of the data:

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

Coefficient of Variation

A direct comparison between two or more measures of dispersion may be difficult when the means differ greatly. To make a meaningful comparison, a relative measure of dispersion must be used. Relative dispersion is commonly measured with the coefficient of variation (CV):

$\displaystyle CV = \frac{s}{\bar{X}} = \frac{\text{standard deviation of }x}{\text{average value of }x}$

CV measures the amount of dispersion in a distribution relative to its mean. In an investments setting, CV measures the risk (variability) per unit of expected return — lower CV is better.

Target Downside Deviation

Variance and standard deviation calculate risk based on outcomes both above and below the mean. In some situations it is more appropriate to consider only outcomes less than the mean (or some other specific value). In this case, we are measuring downside risk.

One measure of downside risk is target downside deviation (also known as target semideviation). We choose a target value $B$ and only include deviations from the target in the numerator if the outcomes are below $B$:

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

Note that the denominator remains $n - 1$ (the full sample size minus one), even though we are not using all observations in the numerator.

Skewness

A distribution is symmetrical if it is shaped identically on both sides of its mean. Symmetry implies that intervals of losses and gains will exhibit the same frequency. The extent to which a returns distribution is symmetrical is important because it tells analysts if deviations from the mean are more likely to be positive or negative.

Skewness (or skew) refers to the extent to which a distribution is not symmetrical. Nonsymmetrical distributions result from outliers — observations extraordinarily far from the mean:

• A positively skewed distribution has outliers greater than the mean (in the upper, or right tail) — said to be skewed right.
• A negatively skewed distribution has a disproportionately large amount of outliers less than the mean (in the lower, or left tail) — said to be skewed left.

Effect of skewness on mean, median, mode:

• Symmetrical: mean = median = mode.
• Positively skewed (unimodal): mode < median < mean. Large positive outliers pull the mean upward. Example: a neighborhood of 99 homes selling for $100,000 plus one selling for $1,000,000 — median & mode = $100,000; mean = $109,000.
• Negatively skewed (unimodal): mean < median < mode. Large negative outliers pull the mean downward.

Sample skewness is approximated for large samples as:

$\displaystyle \text{sample skewness} \approx \left(\frac{1}{n}\right) \frac{\sum_{i=1}^{n}(X_i - \bar{X})^3}{s^3}$

The denominator $s^3$ is always positive; the numerator's sign depends on whether observations above or below the mean tend to be farther from the mean. Right-skewed → positive skewness; left-skewed → negative skewness. Values of sample skewness in excess of 0.5 in absolute value are considered significant.

Kurtosis

Kurtosis measures the degree to which a distribution is more or less peaked than a normal distribution.

• Leptokurtic — more peaked than normal (fat tails). Greater % of small deviations from the mean AND greater % of extremely large deviations from the mean.
• Platykurtic — less peaked, flatter than normal (thin tails).
• Mesokurtic — same kurtosis as a normal distribution.

A leptokurtic distribution implies a relatively greater probability of observing values either close to the mean or far from the mean. For investment returns, a greater likelihood of large deviations from the mean is often perceived as an increase in risk.

A distribution exhibits excess kurtosis if it has either more or less kurtosis than the normal distribution. Computed kurtosis for a normal distribution is 3. Statisticians sometimes report excess kurtosis = kurtosis − 3:

• Normal distribution: excess kurtosis = 0.
• Leptokurtic: excess kurtosis > 0 (positive).
• Platykurtic: excess kurtosis < 0 (negative).

Kurtosis is critical in risk management. Most research shows actual securities returns exhibit both skewness and kurtosis. When returns are modeled assuming a normal distribution, the predictions will not account for the potential for extremely large, negative outcomes. Risk managers focus on the tails of the distribution — that is where the risk is. Greater excess kurtosis and more negative skew indicate increased risk.

Sample kurtosis (large sample approximation):

$\displaystyle \text{sample kurtosis} \approx \left(\frac{1}{n}\right) \frac{\sum_{i=1}^{n}(X_i - \bar{X})^4}{s^4}$

Scatter Plots and Covariance

Scatter plots display the relationship between two variables. With one variable on each axis, paired observations are plotted as single points. Common patterns include:

• (a) No relationship — points scattered randomly with no clear trend.
• (b) Strong linear relationship — points cluster around a line (high correlation coefficient).
• (c) Non-linear relationship — clear pattern but not linear; correlation coefficient close to zero. This is a key advantage of scatter plots: they reveal nonlinear relationships that are not described by the correlation coefficient.

Covariance is a measure of how two variables move together:

$\displaystyle s_{XY} = \frac{\sum_{i=1}^{n}\left[(X_i - \bar{X})(Y_i - \bar{Y})\right]}{n - 1}$

Covariance is difficult to interpret because:

• Its value depends on the units of the variables (e.g., yen vs. dollars).
• Its units are the square of the units used for the data.
• We cannot interpret the relative strength of the relationship — a covariance of 0.8756 only tells us X and Y tend to move together (positive sign).

Correlation Coefficient

A standardized measure of the linear relationship between two variables is the correlation coefficient, or simply correlation:

$\displaystyle \rho_{XY} = \frac{s_{XY}}{s_X \, s_Y}$   which implies   $s_{XY} = \rho_{XY}\, s_X\, s_Y$

Properties of correlation:

• Measures the strength of the linear relationship between two random variables.
• Has no units.
• Ranges from −1 to +1: $-1 \le \rho_{XY} \le +1$.
• $\rho_{XY} = +1$ → perfect positive correlation (proportional positive movement).
• $\rho_{XY} = -1$ → perfect negative correlation (exact opposite proportional movement).
• $\rho_{XY} = 0$ → no linear relationship; prediction of $Y$ from $X$ via linear methods is not possible.

Care should be taken when drawing conclusions based on correlation. Causation is not implied just from significant correlation. Even if causation were present, correlation does not reveal which variable is causing change in the other. It is more prudent to say two variables exhibit positive (or negative) association — the nature of any causal relationship must be separately investigated or grounded in theory subject to additional tests.

Outliers and correlation: If removing outliers significantly reduces the calculated correlation, further inquiry is necessary into whether the outliers provide information or are caused by noise (randomness) in the data.

Spurious correlation refers to correlation that is either:

• The result of chance; or
• Present due to changes in both variables over time caused by their association with a third variable.

Examples (Tyler Vigen, Spurious Correlations):

• The correlation between the age of each year's Miss America and the number of films Nicolas Cage appeared in that year is 87% — random.
• U.S. spending on science, space, and technology vs. suicides by hanging, strangulation, and suffocation (1999–2009): 99.87% — both variables increased approximately linearly over the period.