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6. Measures of Dispersion

g. calculate and interpret 1) a range and a mean absolute deviation, and 2) the variance and standard deviation of a population and of a sample;

What is the definition of dispersion? Dispersion is defined as “variability around the central tendency”.

Why is understanding dispersion important for investment? Investment is all about reward vs variability (risk). A central tendency is a measure of the reward of an investment and dispersion is a measure of investment risk.

What is the difference between absolute and relative dispersion? 1. Absolute dispersion - dispersion without any comparison to benchmarks 2. Relative dispersion - amount of dispersion in comparison to a benchmark

What is “range”? Range is the simplest measure of dispersion. To get range subtract the biggest value in a set from the smallest.

When can range be a good measure of dispersion and when is it bad? Range can be good because it is easily understood. It can be bad because it is only based on two values and is very sensitive to outliers. If you have one very low or very high value, the range is significantly impacted.

In addition to Range, what are some other good measure of spread that should be used alongside it? It’s a good idea to use standard deviation or semi-interquartile range.

What is the Mean Absolute Deviation and how is it calculated? - Mean absolute deviation (MAD) is the arithmetic average of all the mean deviations in the data set. (Deviation = an observation’s distance from the mean) - \(MAD = \frac{\sum |X_i - \bar{X}|}{n}\)

fWhy is MAD superior to range as a measure of dispersion? Mean absolute deviation uses every observation in the sample whereas range only uses the smallest and biggest (and is thus sensitive to outliers)

What is Variance and what is the math formula to get Variance for the entire population? - Variance is a measure of how spread out a distribution is. - Variance is computed as the average squared deviation of each number from its mean - \(\sigma^2 = \frac{\sum (X_i - \mu)^2}{N}\)

How does the formula for a population sample differ from the entire population formula? - For a sample variance, just subtract one from the number of observations, like this: - \(s^2 = \frac{\sum (X_i - m)^2}{N - 1}\) - We do this for samples because it gives a better approximation and avoids underestimating the population variance.

How do you derive the standard deviation of a set? 1. Calculate the variance 2. Take the variance’s square root

In simple terms, what is the standard deviation? Standard deviation is a measure of the average deviation from the mean.

What is the empirical rule? In a normal distribution, 68% of the values of a set are within one standard deviation, 95% are within two standard deviations, and 99.7% are within three standard deviations.

What is relative dispersion? Relative dispersion is the amount of variability present in comparison to a reference point or benchmark.

What is coefficient of variation (CV) and what is the formula to calculate it? - CV gives a measure of the risk per unit of return and an idea of the magnitude of variation in percentage terms - \(CV = \frac{s}{\bar{X}}\)

What is another name for coefficient of variation (CV)? Relative standard deviation