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Investment - Descriptive Statistics
As the name suggests, descriptive statistics are used to characterize data. Often, you are confronted by data that you need to structure in order to understand.
Let's take everyday commute time as an example. You have the sensation that the commute home from work is growing slower, and you are thinking of altering your route. How could you judge whether the travel actually is growing slower? Suppose you calculated and compared your average daily travel time each month over a year.
The first thing you need to address is: What is meant by average?
There are a lot of alternative approaches to calculate averages, as you shall see, each of which has advantages and downsides.
Types of Descriptive Statistics
In general, descriptive statistics are figures that summarise fundamental properties of a data set. A data set pertains to a certain variable — for example, the time it takes to travel home from work. The data set includes several observations – that is, observable values for the variable. For example, if you keep track of your everyday commute time for a year, you will end up with around 250 observations. The distribution of a variable refers to the values a variable can take, and the number of observations associated with each of these values.
We shall cover two forms of descriptive statistics:
Those that describe the central trend of a data set (e.g., the average or mean)
Those that describe the dispersion or spread of the data (e.g., the standard deviation)
In addition to understanding whether the travel to work is getting slower (by comparing monthly averages), you might also want to discover a technique to determine how much variance there is between journey times from one day to another (by utilizing standard deviation).
Businesses have comparable demands to summarise data. Descriptive statistics quickly summarise information from vast quantities of data for the aim of comprehending the data and establishing comparisons across specific data sets for risk assessment. Measures of Frequency and Average
The goal of assessing the frequency of outcomes or 'central tendency' is to describe a set of individual data points with a single measurement. The value used to characterize the group will be the one value considered to be best representative of all the individual data points.
Measures of central tendency are important for drawing comparisons between groups of persons or between sets of figures. Such metrics reduce a huge number of measurements to a single figure. For instance, the mean or average temperature in Country X in July from 1961 to 2022 is calculated to be 16.1°C. Over the same period, the average temperature in Country X in September is 13.6°C. Because it is a long time series, you can legitimately assume that it is usually warmer in July than September in Country X.
The following are common metrics of central tendency:
Arithmetic mean
Geometric mean
Median Mode
The right measure for a given data collection depends on the features of the data and the aim of your calculation. These measures are examined in the following sections.
Arithmetic Mean
The arithmetic mean is the most often used measure of central tendency and is recognizable to most people. It is generally abbreviated to just ‘mean’ or ‘average’.
To calculate the mean, you put all the numbers in the data set together and divide by the number of observations (items in the data set). The arithmetic mean presupposes that each observation is equally likely to occur.
How to Calculate the Mean
The arithmetic mean return, or average yearly return during the 10-year period, is equal to 6.3% and is computed as follows:
Mean = (1.3 + 2.4 + 0.8 + 3.7 + 8.0 + 3.7 + 7.2 + 26.4 + 4.2 + 5.2)/10 = 6.3%
Geometric Mean
An alternative average to the arithmetic mean is the geometric average or geometric mean. Applied to investment returns, the geometric mean return is the average return assuming that returns are compounding.
Consider Tom Parker who recently sold his company and has USD10 million to invest. He invests in a fund that returns 8% the first year, 3% the second year, and 7% the third year. How much return did Tom Parker accumulate during these three years? The following slides illustrate how to calculate geometric mean.
Multiply 1 Plus Each Annual Return
The first step in computing the geometric mean return is to multiply 1 plus each annual return together, which gives you the amount Mike would have accumulated at the end of the three years per dollar of investment:
[(1 + 8%) × (1 + 3%) × (1 + 7%) ≈ 1.1903].
This figure of 1.1903 indicates three years of investment, however the geometric mean return should record an average rate of return for each of the three years.
Raise the Accumulation to the Power of 1 Over Periods Held
The second step entails raising the accumulation to the power of ‘one over the number of periods held’ (three in this specific case); this computation may alternatively be expressed as taking ‘the number of periods held’ root of the value ((1.1903)1/3 = 1.0598).
This number of 1.0598 comprises both the original investment and the average yearly return on the investment each year (1 plus the geometric mean return).
Subtract 1 from the Value in Step 2
The third step is to deduct 1 from this figure to get at the return that would have to be earned on average each year to attain to the total accumulation over the three years (1.0598 – 1 = 0.0598 or 5.98%).
The geometric mean return is 5.98%, which is less than the arithmetic mean return (6.0% = (8% + 3% + 7%)/3). Geometric mean is often the favored measure for the investing business.
Take Away Notes
An key element to observe is that the geometric mean is lower than the arithmetic mean, even though the yearly returns over the 10-year holding period are identical. This conclusion is because the returns are compounded when computing the geometric mean return. Compounding will result in a bigger value over time, so a lower rate of return is necessary to obtain the same amount. In fact, if the same set of numbers is used to calculate both means, the geometric mean return is never more than the arithmetic mean return and is generally lower.
When you are working with interest rates, percentage changes, or returns on investment portfolios, which can be volatile, it is better and more accurate to use geometric mean. And the extended term (10 years in this example) makes the compounding impact more essential, hence the use of a geometric mean. Conversely, for data sets in which the numbers are neither skewed or reliant on each other, the arithmetic mean is better appropriate because it is easy to use and understand.
Median
If you put data in ascending order of size from the smallest to the largest, the median is the midpoint value. If there is an even number of items in a data set, then you average the two middle observations to derive the median. Hence, in many circumstances (i.e., when the sample size is odd or when the two middle-ranked items of an even-numbered data set are the same) the median will be a number that actually occurs in the data set. The example below displays the calculation of the median for the sample 10 years of returns.
When the returns are ordered from low to high, the median value is the arithmetic mean of the fifth and sixth ordered observations.
The mode is the most often occurring value in a data set. The following explains how the mode is derived for the same set of data in the Median section above. We can see that one value occurs twice, 3.7%, as seen in the example below. This number is the mode of the data.
The mode can be used as a measure of central tendency for data that have been categorized into categories or groups. For example, if all the employees of a company were asked what method of transportation they used to come to work each day, it would be able to arrange the answers into groups, such as automobile, bus, train, bicycle, and walking. The category with the highest number would be the mode.
A difficulty with the mode is that it is often not unique, in which case there is no mode. Another difficulty with the mode is that the most frequently occurring observation may be far removed from the rest of the data and does not meaningfully reflect them.
As you have seen, the arithmetic mean is often used to locate the middle position of the distribution of a group of data points. But it is not always a reliable indicator. When there are outliers, the mode or median can better illustrate the core tendency of a set of data points than the mean.
In a positively-skewed data set, the median and mode are smaller than the arithmetic mean, as seen in the following graph.
As the name suggests, descriptive statistics are used to characterize data. Often, you are confronted by data that you need to structure in order to understand.
Let's take everyday commute time as an example. You have the sensation that the commute home from work is growing slower, and you are thinking of altering your route. How could you judge whether the travel actually is growing slower? Suppose you calculated and compared your average daily travel time each month over a year.
The first thing you need to address is: What is meant by average?
There are a lot of alternative approaches to calculate averages, as you shall see, each of which has advantages and downsides.
Types of Descriptive Statistics
In general, descriptive statistics are figures that summarise fundamental properties of a data set. A data set pertains to a certain variable — for example, the time it takes to travel home from work. The data set includes several observations – that is, observable values for the variable. For example, if you keep track of your everyday commute time for a year, you will end up with around 250 observations. The distribution of a variable refers to the values a variable can take, and the number of observations associated with each of these values.
We shall cover two forms of descriptive statistics:
Those that describe the central trend of a data set (e.g., the average or mean)
Those that describe the dispersion or spread of the data (e.g., the standard deviation)
In addition to understanding whether the travel to work is getting slower (by comparing monthly averages), you might also want to discover a technique to determine how much variance there is between journey times from one day to another (by utilizing standard deviation).
Businesses have comparable demands to summarise data. Descriptive statistics quickly summarise information from vast quantities of data for the aim of comprehending the data and establishing comparisons across specific data sets for risk assessment. Measures of Frequency and Average
The goal of assessing the frequency of outcomes or 'central tendency' is to describe a set of individual data points with a single measurement. The value used to characterize the group will be the one value considered to be best representative of all the individual data points.
Measures of central tendency are important for drawing comparisons between groups of persons or between sets of figures. Such metrics reduce a huge number of measurements to a single figure. For instance, the mean or average temperature in Country X in July from 1961 to 2022 is calculated to be 16.1°C. Over the same period, the average temperature in Country X in September is 13.6°C. Because it is a long time series, you can legitimately assume that it is usually warmer in July than September in Country X.
The following are common metrics of central tendency:
Arithmetic mean
Geometric mean
Median Mode
The right measure for a given data collection depends on the features of the data and the aim of your calculation. These measures are examined in the following sections.
Arithmetic Mean
The arithmetic mean is the most often used measure of central tendency and is recognizable to most people. It is generally abbreviated to just ‘mean’ or ‘average’.
To calculate the mean, you put all the numbers in the data set together and divide by the number of observations (items in the data set). The arithmetic mean presupposes that each observation is equally likely to occur.
How to Calculate the Mean
The arithmetic mean return, or average yearly return during the 10-year period, is equal to 6.3% and is computed as follows:
Mean = (1.3 + 2.4 + 0.8 + 3.7 + 8.0 + 3.7 + 7.2 + 26.4 + 4.2 + 5.2)/10 = 6.3%
Geometric Mean
An alternative average to the arithmetic mean is the geometric average or geometric mean. Applied to investment returns, the geometric mean return is the average return assuming that returns are compounding.
Consider Tom Parker who recently sold his company and has USD10 million to invest. He invests in a fund that returns 8% the first year, 3% the second year, and 7% the third year. How much return did Tom Parker accumulate during these three years? The following slides illustrate how to calculate geometric mean.
Multiply 1 Plus Each Annual Return
The first step in computing the geometric mean return is to multiply 1 plus each annual return together, which gives you the amount Mike would have accumulated at the end of the three years per dollar of investment:
[(1 + 8%) × (1 + 3%) × (1 + 7%) ≈ 1.1903].
This figure of 1.1903 indicates three years of investment, however the geometric mean return should record an average rate of return for each of the three years.
Raise the Accumulation to the Power of 1 Over Periods Held
The second step entails raising the accumulation to the power of ‘one over the number of periods held’ (three in this specific case); this computation may alternatively be expressed as taking ‘the number of periods held’ root of the value ((1.1903)1/3 = 1.0598).
This number of 1.0598 comprises both the original investment and the average yearly return on the investment each year (1 plus the geometric mean return).
Subtract 1 from the Value in Step 2
The third step is to deduct 1 from this figure to get at the return that would have to be earned on average each year to attain to the total accumulation over the three years (1.0598 – 1 = 0.0598 or 5.98%).
The geometric mean return is 5.98%, which is less than the arithmetic mean return (6.0% = (8% + 3% + 7%)/3). Geometric mean is often the favored measure for the investing business.
Take Away Notes
An key element to observe is that the geometric mean is lower than the arithmetic mean, even though the yearly returns over the 10-year holding period are identical. This conclusion is because the returns are compounded when computing the geometric mean return. Compounding will result in a bigger value over time, so a lower rate of return is necessary to obtain the same amount. In fact, if the same set of numbers is used to calculate both means, the geometric mean return is never more than the arithmetic mean return and is generally lower.
When you are working with interest rates, percentage changes, or returns on investment portfolios, which can be volatile, it is better and more accurate to use geometric mean. And the extended term (10 years in this example) makes the compounding impact more essential, hence the use of a geometric mean. Conversely, for data sets in which the numbers are neither skewed or reliant on each other, the arithmetic mean is better appropriate because it is easy to use and understand.
Median
If you put data in ascending order of size from the smallest to the largest, the median is the midpoint value. If there is an even number of items in a data set, then you average the two middle observations to derive the median. Hence, in many circumstances (i.e., when the sample size is odd or when the two middle-ranked items of an even-numbered data set are the same) the median will be a number that actually occurs in the data set. The example below displays the calculation of the median for the sample 10 years of returns.
When the returns are ordered from low to high, the median value is the arithmetic mean of the fifth and sixth ordered observations.
The mode is the most often occurring value in a data set. The following explains how the mode is derived for the same set of data in the Median section above. We can see that one value occurs twice, 3.7%, as seen in the example below. This number is the mode of the data.
The mode can be used as a measure of central tendency for data that have been categorized into categories or groups. For example, if all the employees of a company were asked what method of transportation they used to come to work each day, it would be able to arrange the answers into groups, such as automobile, bus, train, bicycle, and walking. The category with the highest number would be the mode.
A difficulty with the mode is that it is often not unique, in which case there is no mode. Another difficulty with the mode is that the most frequently occurring observation may be far removed from the rest of the data and does not meaningfully reflect them.
As you have seen, the arithmetic mean is often used to locate the middle position of the distribution of a group of data points. But it is not always a reliable indicator. When there are outliers, the mode or median can better illustrate the core tendency of a set of data points than the mean.
In a positively-skewed data set, the median and mode are smaller than the arithmetic mean, as seen in the following graph.
In a negatively-skewed data set, the median and mode are bigger than the arithmetic mean, as demonstrated in the following graph.
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