LOS E and F require us to:
e. calculate and interpret measures of central tendency, including the population mean, the sample mean, arithmetic mean, weighted average or mean, geometric mean, harmonic mean, median, and mode.
f. calculate and interpret quartiles, quintiles, deciles, and percentiles.
The measures of central tendencies are quantitative measures of location that specifies where the data is centered. Different measures of central tendencies are discussed below:
1. Arithmetic Mean
a. Arithmetic mean is the average of the data of numerical values. It is calculated using the following formula:
That is, the mean is the sum of all observations in data, divided by the number of observations.
b. The arithmetic mean could either be calculated for the entire population or the sample of the population.
c. The population mean can only be calculated if the population can be defined. It can be calculated by summing up all the data in the population and dividing it by the number of observations. Thus, a population mean can be calculated as follows:
The ‘µ’ here represents the parameter because it is the mean of all the data in the population and there is the least scope for the error.
d. The other mean that can be calculated is the sample mean. The sample is a smaller data, selected from within the population that is a reflection of the population behavior. The sample mean can be calculated as follows:
2. Median
a. Median is the value in the middle of the population or the sample.
b. Median can be calculated by first arranging the observations in ascending order, and then finding the middle value of the series of observations.
c. If the series has an odd number of observations, then the median is the observation that occupies the [(n + 1) / 2]th position.
d. If, however, the series has an even number of observations, the median is the average of observations occupying the [(n / 2]th and [(n + 2) / 2]th position
3. Mode
a. The mode is the most frequently occurring observation in the population or the sample.
b. The data could either have a single mode, i.e. it might be unimodal, or it might have multiple modes (such as bi-modal data).
c. The mode is the only measure of central tendency that can be used with nominal or categorical data.
4. Weighted Mean
The weighted mean is similar to the arithmetic mean, except that instead of each data point contributing equally to the average, the more influential data is given the higher weights.
Thus, if say a portfolio consists of two types of investments, i.e. equity and bonds. The stocks form 60% of the portfolio and bonds form 40%. If equities provide a return of 10% and bonds a return of 8%, then the arithmetic mean of the two would be:
(10 + 8) / 2 = 9%
This average does not reflect the true weight of the individual components in the portfolio; therefore, it would be more appropriate to have a mean that gives weights to the individual categories in the portfolio. Thus, we calculate the weighted average instead of the arithmetic mean as follows:
Thus, the formula for calculating the weighted average is:
Example:
Consider a portfolio with 60% investment in equities and 40% in bonds. The different returns on equities and bonds for a period of 5 years were:
|
Year |
Return on Equity |
Return on Bonds |
|
1 |
-2% |
9% |
|
2 |
8% |
10% |
|
3 |
27% |
-1.5% |
|
4 |
-9% |
8% |
|
5 |
-5% |
7.5% |
If we want to calculate the average returns earned by the investor, we should calculate the weighted average of the returns of each year and then take the arithmetic mean of the 5-year return.
|
Year |
Return on Equity |
Return on Bonds |
Weighted Average (WiXi) |
|
1 |
-2% |
9% |
2.4% |
|
2 |
8% |
10% |
8.8% |
|
3 |
27% |
-1.5% |
15.6% |
|
4 |
-9% |
8% |
-2.2% |
|
5 |
-5% |
7.5% |
0% |
And the average of five year’s returns would be:
5. Geometric Mean
a. It is the central number in a geometric progression. It is mostly used to average the ‘rates of change’, i.e. the growth rate of the variable.
b. The geometric mean is calculated by taking the nth root of the product of n numbers.
The geometric mean can be calculated as long as all the numbers in the data are greater than or equal to zero.
c. In the above example, if we were to calculate the geometric mean of the equity investments we can calculate as follows:
|
Year |
Return on Equity |
|
1 |
-0.02 |
|
2 |
0.08 |
|
3 |
0.27 |
|
4 |
-0.09 |
|
5 |
-0.05 |
Since the returns are negative in some of the years; we should add 1 and then calculate the geometric mean. Thus,
Or, the average return on equity stocks was 3.05%.
d. A general formula for calculation of geometric mean is:
or,
6. Comparison of Arithmetic and Geometric Mean
Suppose we begin our portfolio with a value of $ 100, at the end of year 1, it is worth $ 200 (i.e. 100% returns), and at the end of year 2, it is again worth $ 100 (i.e. a loss of 50% in the year 2).
So, if we calculate the arithmetic mean, the average returns would be:
And, if we calculate the geometric mean the average returns would be:
7. Harmonic Mean
a. Harmonic mean is another measure of central tendencies, but with very limited applications. They are mainly used for averaging the ratios, that too, only when they are repeatedly applied to a fixed quantity to yield a variable number of units.
b. The harmonic mean is usually calculated by taking the reciprocal of the reciprocals of the numbers in the set. The formula for calculating the harmonic mean is:
c. For example, if an investor has $ 1000 to invest every month and the price of a unit of the fund at the end of the first and second month is $ 10 and $ 15 respectively, so that he can buy 100 and 66.67 shares at the end of each month. To calculate the average price of the unit, the arithmetic mean is not appropriate here. Rather, we should calculate the harmonic mean of the price as follows:
Thus, the average price of the fund is $ 12.
d. While using the harmonic mean it is extremely important to understand the situations where the harmonic mean is used.
8. Quantiles
a. Quantiles refer to the set of variables that divide a frequency distribution into equal groups, having the same fraction of the total population.
b. There are different types of quantiles; some of them are:
i. Median: It divides the frequency distribution into two equal parts. It is the mid-point of the frequency distribution.
ii. Quartile: The quartiles divide the frequency distribution into four equal parts.
iii. Quintiles: They divide the frequency distribution into five equal parts.
iv. Deciles: They divide the frequency distribution into ten equal parts.
v. Percentiles: They divide the frequency distributions into the 100 equal parts
c. Thus, for example when we say that a student in a class has obtained 70 percentile; it means that 70% of the students in the class have marks less than this student.
d. Locating Percentiles:
The percentile points can be located using the following equation:
|
where y is the percentage point at which we are dividing the distribution and Ly is the location (L) of the percentile (Py) in the array sorted in ascending order. |
e. The point Ly may not be the whole number, and in such a case, we can locate the points using linear interpolation.
For example, if we know that observation 12 has the value of 20 and observation 13 has the value of 30. And if we have to find the value of observation 12.5, we can do so by interpolating the data as follows:
