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Investment-Measures of Dispersion
Whereas measures of central tendency are used to estimate representative or center values of a data collection, measurements of dispersion are crucial for describing the spread of the data or its variation around a central value. Two data sets may have the same mean or median but radically different amounts of variability or vice versa. A description of a data collection should include both a measure of central tendency, such as the mean, and a measure of dispersion.
Suppose two companies each pay an average annual salary of USD50,000. In one company, most incomes are clustered close to the average, whereas in the second, they are spread out with many people earning very little and some earning a lot. It would be useful to have a measure of dispersion that can help discover such disparities between data sets
Another reason why measurements of dispersion are significant in finance is because investment risk is generally measured using some measure of variability. When investors are considering investing in a security, they are interested in the likely (anticipated) return on that investment as well as in the risk that the return could differ from the expected return (its variability). A risk-averse investor assessing two investments that have equal expected returns but substantially different amounts of fluctuation (risk) around those expected returns often favors the asset with the lowest variability.
Two popular metrics of dispersion of a data collection are the range and the standard deviation.
Range
The range is the difference between the highest and lowest values in a data set. It is the easiest measure of dispersion to compute and understand, but it is particularly sensitive to outliers. To compute the range in a data set:
Find the lowest value (Min) and the maximum value (Max)
Calculate the difference between the Max and the Min
Range = Max – Min
Clearly, the range is affected by extreme values and, if there are outliers, it indicates little about the distribution of the data between those extremes.
If there are a significant number of observations listed in order of size, the range can be partitioned into 100 equal-sized intervals. Dividing points between intervals are termed percentiles. The 50th percentile is the median and divides the observations so that 50% are higher and 50% are lower than the median. The 20th percentile is the value below which 20% of observations in the series fall. Accordingly, the dispersion of the observations can be characterized in terms of percentiles. Observations can also be separated into various equal-sized intervals. Commonly used intervals include quartiles (the observations are divided into four equal-sized intervals) and deciles (the observations are divided into 10 equal-sized intervals).
Standard Deviation
A commonly used metric of dispersion is standard deviation. It quantifies the variability or volatility of a data set around the average value (the arithmetic mean) of that data collection. Although, as said before, you are not responsible for any computations, you may find it beneficial to look at the formula for how standard deviation is computed.
Whereas measures of central tendency are used to estimate representative or center values of a data collection, measurements of dispersion are crucial for describing the spread of the data or its variation around a central value. Two data sets may have the same mean or median but radically different amounts of variability or vice versa. A description of a data collection should include both a measure of central tendency, such as the mean, and a measure of dispersion.
Suppose two companies each pay an average annual salary of USD50,000. In one company, most incomes are clustered close to the average, whereas in the second, they are spread out with many people earning very little and some earning a lot. It would be useful to have a measure of dispersion that can help discover such disparities between data sets
Another reason why measurements of dispersion are significant in finance is because investment risk is generally measured using some measure of variability. When investors are considering investing in a security, they are interested in the likely (anticipated) return on that investment as well as in the risk that the return could differ from the expected return (its variability). A risk-averse investor assessing two investments that have equal expected returns but substantially different amounts of fluctuation (risk) around those expected returns often favors the asset with the lowest variability.
Two popular metrics of dispersion of a data collection are the range and the standard deviation.
Range
The range is the difference between the highest and lowest values in a data set. It is the easiest measure of dispersion to compute and understand, but it is particularly sensitive to outliers. To compute the range in a data set:
Find the lowest value (Min) and the maximum value (Max)
Calculate the difference between the Max and the Min
Range = Max – Min
Clearly, the range is affected by extreme values and, if there are outliers, it indicates little about the distribution of the data between those extremes.
If there are a significant number of observations listed in order of size, the range can be partitioned into 100 equal-sized intervals. Dividing points between intervals are termed percentiles. The 50th percentile is the median and divides the observations so that 50% are higher and 50% are lower than the median. The 20th percentile is the value below which 20% of observations in the series fall. Accordingly, the dispersion of the observations can be characterized in terms of percentiles. Observations can also be separated into various equal-sized intervals. Commonly used intervals include quartiles (the observations are divided into four equal-sized intervals) and deciles (the observations are divided into 10 equal-sized intervals).
Standard Deviation
A commonly used metric of dispersion is standard deviation. It quantifies the variability or volatility of a data set around the average value (the arithmetic mean) of that data collection. Although, as said before, you are not responsible for any computations, you may find it beneficial to look at the formula for how standard deviation is computed.
The disparities between the observed values of X and the mean value of X capture the variability of X. These discrepancies are squared and totaled. Note that, because the differences are squared, what matters is the size of the difference, not the sign of the difference. The sum is then divided by the number of observations. Finally, the square root of this value is calculated to determine the standard deviation.
Variance
The value before the square root is used when calculating standard deviation is known as the variance, which is another measure of dispersion. The standard deviation is the square root of the variance. The standard deviation and the variance capture the same thing – how far out from the mean the data are. The advantage of the standard deviation is that it is expressed in the same unit as the mean. For example, if the mean is stated as minutes of journey time, the standard deviation will also be expressed as minutes, however the variance will be expressed as minutes squared, making the standard deviation an easier measure than range to use and compare with the mean.
Calculating Standard Deviation
To explain the calculation of the standard deviation, let us return to the example presented in Lesson 5 of a three-year investment that yields 8% or 0.08 the first year, 3% or 0.03 the second year, and 7% or 0.07 the third year. The arithmetic mean return is 6% or 0.06. The standard deviation is roughly 2.16%.
Variance
The value before the square root is used when calculating standard deviation is known as the variance, which is another measure of dispersion. The standard deviation is the square root of the variance. The standard deviation and the variance capture the same thing – how far out from the mean the data are. The advantage of the standard deviation is that it is expressed in the same unit as the mean. For example, if the mean is stated as minutes of journey time, the standard deviation will also be expressed as minutes, however the variance will be expressed as minutes squared, making the standard deviation an easier measure than range to use and compare with the mean.
Calculating Standard Deviation
To explain the calculation of the standard deviation, let us return to the example presented in Lesson 5 of a three-year investment that yields 8% or 0.08 the first year, 3% or 0.03 the second year, and 7% or 0.07 the third year. The arithmetic mean return is 6% or 0.06. The standard deviation is roughly 2.16%.
Larger levels of standard deviation compared to the mean imply greater variety in a data set. Also, by utilizing standard deviation, you may predict how likely it is that any given observation will occur depending on its distance from the mean.
Example: Comparison of Investments
The next example compares the returns on the investment sample we have been using to the returns on another investment over the same period using mean and standard deviation.
An investment receives returns over a 10-year period with the following characteristics:
Number of observations = 10
Mean = 6.3%
Standard deviation = 7.1%
Another investment over the same time period has the following characteristics:
Number of observations = 10
Mean = 6.5%
Standard deviation = 2.6%
Based on mean and standard deviation, the second investment is better than the first investment. It has a higher mean return and lower standard deviation, hence less variability (which suggests less risk) in its returns.
Larger levels of standard deviation compared to the mean imply greater variety in a data set. Also, by utilizing standard deviation, you may predict how likely it is that any given observation will occur depending on its distance from the mean.
Example: Comparison of Investments
The next example compares the returns on the investment sample we have been using to the returns on another investment over the same period using mean and standard deviation.
An investment receives returns over a 10-year period with the following characteristics:
Number of observations = 10
Mean = 6.3%
Standard deviation = 7.1%
Another investment over the same time period has the following characteristics:
Number of observations = 10
Mean = 6.5%
Standard deviation = 2.6%
Based on mean and standard deviation, the second investment is better than the first investment. It has a higher mean return and lower standard deviation, hence less variability (which suggests less risk) in its returns.
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