The arithmetic mean is the sum of the observations divided by the number of observations. It is the most frequently used measure of the middle or center of data.
The Population Mean
The population mean is the arithmetic mean computed for a population. It is expressed as:
N is the number of observations in the entire population and
Xi is the ith observation
Consider the stock returns of a company over the last 10 years: 2%, 5%, 4%, 7%, 8%, 8%, 12%, 10%, 8%, and 5%.
For this data set, the population mean can be computed as:
The Sample Mean
The sample mean is calculated like the population mean, except that we use the sample values. It is expressed as:
where: n is the number of observations in the sample.
If the sample data is: 8, 12, 10, 8 and 5, the sample mean can be calculated as:
The median is the midpoint of a data set that has been sorted into ascending or descending order.
For odd number of observations: 2,5,7,11,14 → Median = 7
For even number of observations: 3, 9, 10, 20 → Median = (9 + 10)/2 = 9.5
As compared to a mean, a median is less affected by extreme values (outliers).
The mode is the most frequently occurring value in a distribution.
For the following data set: 2, 4, 5, 5, 7, 8, 8, 8, 10, 12 → Mode = 8
A distribution can have more than one mode, or even no mode. When a distribution has one mode it is said to be unimodal. If a distribution has two or three modes, it is called bimodal or trimodal respectively.
The Weighted Mean
Here different observations are given different proportional influence on the mean. The formula for the weighted mean is:
where: the sum of the weights equals 1; that is
Consider an investor with a portfolio of three stocks. $40 is invested in A, $60 in B, and $100 in C. If returns were 5% on A, 7% on B, and 9% on C, compute the portfolio return using the weighted mean.
An arithmetic mean is a special case of a weighted mean where all observations are equally weighted by the factor 1/n.
The Geometric Mean
The most common application of the geometric mean is to calculate the average return of an investment. The formula is:
The return over the last four periods for a given stock is: 10%, 8%, -5% and 2%. Calculate the geometric mean.
Given the returns shown above, $1 invested at the start of period 1 grew to: $1.00 x 1.10 x 1.08 x 0.95 x 1.02 =$1.151. If the investment had grown at 3.58% every period, $1.00 invested at the start of period 1 would have increased to: $1.00 x 1.0358 x 1.0358 x 1.0358 x 1.0358 =$1.151. As expected, both scenarios give the same answer. 3.58% is simply the average growth rate per period.
In the reading on Discounted Cash Flow Applications, we used the geometric mean to calculate the time-weighted rate of return.
The Harmonic Mean
The harmonic mean is a special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude. The formula for a harmonic mean is:
where: Xi > 0 for i = 1, 2 … n, and n is the number of observations
The harmonic mean is used to find average purchase price for equal periodic investments.
An investor purchases $1,000 of a security each month for three months. The share prices are $10, $15 and $20 at the three purchase dates. Calculate the average purchase price per share for the security purchased.
The average purchase price is simply the harmonic mean of $10, $15 and $20.
The harmonic mean is:
A more intuitive way of solving this is total money spent purchasing the shares divided by the total number of shares purchased.
Total money spent purchasing the shares = $1,000 x 3 = $3,000
Total shares purchased = sum of shares bought each month
Average purchase price per share =
Comparison of AM, GM and HM
A quantile is a value at or below which a stated fraction of the data lies. Some examples of quantiles include:
The formula for the position of a percentile in a data set with n observations sorted in ascending order is:
y is the percentage point at which we are dividing the distribution.
n is the number of observations.
Ly is the location (L) of the percentile (Py) in an array sorted in ascending order.
Some important points to remember are:
Consider the data set:
47 35 37 32 40 39 36 34 35 31 44
Solution to 1:
First arrange the data in ascending order:
31, 32, 34, 35, 35, 36, 37, 39, 40, 44, 47
Location of the 75th percentile is the:
L75 = (11 + 1) (75/100) = 9th value. i.e. P75 = 40
With a small data set, such as this one, the location and the value is approximate. As the data set becomes larger, the location and percentile value estimates become more precise.
Solution to 2:
Location of the 1st quartile is the:
L25 = (11 + 1) (25/100) = 3rd value. i.e. P25 = 34
Solution to 3:
Location of the 5th decile is the:
L50 = (11 + 1) (50/100) = 6th value. i.e. P50 = 36
Solution to 4:
L60 = (11 + 1) (60/100) = 7.2
Use linear interpolation, which estimates an unknown value on the basis of two known values that surround it.
In this case, the 7th value is 37 and the 8th value is 39. The 6th decile is: P60 = 37+ 0.4 (0.2 times the linear distance between 37 and 39). P60 = 37.4