Degrees of Freedom
Not many people understand what Degrees for Freedom (DOF) is all about. But this
is a concept you must used in the CFA exam. However, you are not required to
understand what DOF is as well as a statistician. All you have to know is DOF is a measure of the number of independent
information.
If you take n observations, you have
n-DOF because there is n independent information. Once you have used the
information to estimate the mean, and you continue to estimate the standard
deviation, you only have n-1 DOF left. Why? It is because, to calculate the
standard deviation, you have to use the mean m, which was calculated from the n
observations already. So, you only have n-1 pieces of information left for the
standard deviation. (Or think it in this way, from n-1 observation and m, you can determine the nth
information.)
To understand what DOF has to use in
different equations, it can be very demanding. Therefore, as long as you
understand what was discussed above, it would be enough to just bear the
followings in mind:
- For samples, you have to determine DOF.
However for population, always use n. This is a trap you will find in the
exam.
- Memorize the DOF to use in different equations
|
Population |
Samples |
|
Mean – n |
Mean – n |
|
Standard Deviation – n |
Standard Deviation – n-1 |
|
Co-variance – n |
Co-variance – n-1 |
|
|
Standard deviation of sample mean
– n |
|
|
t-, z- test statistics – n |
|
|
Chi statistics – n-1 |
|
|
Test
of regression – n-2 |
|
|
Standard
Error of Estimate – n -2 |
Here is a way to help you memorize:
if you have only 1 sample, does it make sense to use n? For
mean, yes. For standard deviation and co-variance, no – so you
have to use n-1 so it will be divided by 0 (to show that it doesn’t make
sense). For the standard deviation of sample mean, t, z
statistics – yes. For Chi, no, because Chi is a comparison of
variance, it needs at least 2 samples to have variance.
E.g.
Q1. The stock
market has return of 4%, 3% and 4% in 1980, 1985 and 1995. What is the standard
deviation of the market return?
Since the
data given are the samples of the market return. n = 3-1 =2 has to be used in
the calculation.
Q2. A
3-year portfolio has a return of 3%, 6% and 4% in 2004, 2005 and 2006. What is
the standard deviation of the return?
The
portfolio data given is the population of
the portfolio (because it only has 3 years). Therefore, n=3 has to be used in
the calculation.