When I tell people that we use n - 1 in the denominator for estimating
population standard deviation, many people want to know why. This video will
explain the mathematics behind why we use n - 1, and what we mean by
"degrees of freedom." The degrees of freedom refer to how many options exist
for variation in our data. Specifically, it means "degrees of freedom to vary."
Let me illustrate using an example of making choices when choosing positions
for a softball team. Degrees of freedom will refer to the number of people who
get to make a choice. There are nine positions on the softball team: pitcher,
catcher, shortstop, first base, etc., and there are nine of us. We draw straws to
see who gets to go first.
The first person picks a position to play on the team.
The second person chooses from among the remaining eight positions. The third
person chooses from among the seven remaining positions, and so on. Each
person gets to choose a position, but who does not get a choice? When we get to the
eighth person, that person chooses between two remaining positions, but the
last person is stuck with whatever position remains. So that person doesn't
actually get to make a choice. When choosing among nine positions, only eight
people get to actually choose. The last person's "choice" is chosen for him.
Based upon the choices of others, you have n - 1 choices, or degrees of
freedom, to choose. Now let's apply this to a sample. We have chosen a sample of
nine subjects from an unknown population. The mean of our sample equals 5.
What do you estimate will be the mean of the population? Because the sample mean
is the best estimate of the
population mean, we estimate that the mean of the population
will be 5. So what is the formula for the mean? The mean is the sum of x
divided by n, which we estimate must equal 5. So we are now going to
select 9 new scores from this population, with an estimated mean of 5.
The sum of x divided by 9 = 5
So what is the sum of X? It has to be 45, because 45 is the only number that can
be divided by 9 (our sample size) to equal the mean of 5. The values in the
population can vary, but they must still average to the mean. We have chosen our
nine people and we know that their scores have to add up to 45, so that we
get that mean of 5.
Well, let's see what their scores were. If you like,
get out your calculator and enter value of 45. As we see the numbers from our
sample, subtract them from 45. The first person got a 5. That leaves us with 40.
The second person got a 6.
That leaves us with 34. The third person got a 4. That leaves us with 30. Next is
5. That leaves us with 25. Next is 4. Leaves us with 21. Next is a 7. That brings us down to 14.
Next is 3. That leaves us with 11. And the next-to-last person got a 5.
Ee only have one more person left.
What does his score have to be? The last one has to be a 6. You can vary any or
all of the population values (n), but the last value (n - 1)
will always be defined for you. You have n - 1 degrees of freedom to vary.
Population estimates using n tend to underestimate the true population values.
So n - 1 is a corrective factor.