Name: Degrees of Freedom and Effect Sizes: Crash Course Statistics #28
Uploaded: 2020-03-30T03:32:16.000Z
Duration: 13 min 30 s
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Hi, I'm Adriene Hill, and welcome back to Crash Course Statistics.

But sometimes we limit our choices in order to do something productive or meaningful.

Like being on a team project that needs a writer, director, host, camera person, and

If we have 5 different people who can be on that team, after assigning 4 of them positions...the

last person doesn't have any freedom to choose theirs.

If she's willing to give up the freedom to have a choice of positions and take on

the great feat of upper body strength that is holding a boom mic, then they have a team

Occasionally we have to give up some freedom--degrees of freedom--in order to do something useful

Degrees of freedom are the number of independent pieces of information we have and Degrees

of freedom are an important part of many of the models that we use.

In fact, we've also been leaving out another important component of the t-test: effect size.

Knowing what degrees-of-freedom and effect-size are and why they matter will help give our

In the last few episodes we've covered the general formula for test statistics.

And we've gotten pretty good at calculating t-statistics for all sorts of situations:

means, proportions, one sample, two sample, paired, unpaired but every time we've needed

a p-value, we've let the computer do the work.

But it's important to know that we're not using the same t-distribution every single time.

As we've previously discussed, the t-distribution is like the z-distribution, but it has fatter

tails, meaning that extreme t-values tend to be slightly more likely.

And that's because we don't know the population standard deviation when we calculate a t-statistic,

so we estimate it using the sample standard deviation.

This little bit of uncertainty means that we don't have a perfect normal--or z--distribution.

But with bigger sample sizes, we're better able to estimate population parameters like

the mean and standard deviation, so our t-distribution changes its shape to reflect that.

As n--our sample size--gets bigger, we're less and less uncertain about our estimate,

and the t-distribution will get closer and closer to z.

More information usually means we have a more accurate estimate.

Degrees of freedom can help us measure that accuracy.

We choose our t-distribution based on the number of degrees of freedom that we have.

Degrees of freedom are the number of pieces of independent information in our data.

After dinner with 2 friends, you all pull out your credit cards to split the bill.

Your friend Carmen, who's a bit of math savant, and a bit of a showoff, notices that if you took your credit

card numbers as a single 16 digit number, the mean of your three credit card numbers

She said this really loudly and you're a little nervous that an identity thief might

have been lurking nearby and overheard Carmen make her very public declaration.

Even though a potential thief has the mean of your credit card numbers, they won't

be able to figure out what any of your individual numbers are.

In other words, there's a lot of “freedom” around what those numbers could be.

And actually, you'd even be okay if the thief found out Carmen's credit card number.

At that point, they could figure out the sum or mean of your and your other friend Eli's

cards, but they still couldn't tell what your exact number was.

There's still freedom for your credit card number to take on different values.

BUT as soon as someone knows the mean of all three cards, Carmen's number, and Eli 's

number, they'll know exactly what your credit card number is.

It's no longer “free” to take on different values.

Then knowing the mean allows anyone to figure out that your number must be this:

So you should probably make sure that Eli keeps his number underwraps.

In that example, the three credit card numbers already existed before we started doing any math.

And they are three independent pieces of information.

Eli's credit card number has no effect on your credit card number, which has no effect

But, as soon as Carmen calculated the mean, she used up one of those independent pieces

Once the thief knows the mean, they only need TWO pieces of independent information.

In this case, once they know any two of the credit card numbers--and the mean--they know

So when they learn Carmen's number and Eli's number -- SUDDENLY those numbers can

The thief can figure out your exact credit card number.

Since it's no longer independent of the others.

To bring it back to our t-tests... when we calculate a mean, we're using up one degree

of freedom--or one piece of independent information.

The amount of information that we originally have depends on our sample size--n--which

is why you'll often see it in the formulas to calculate degrees of freedom.

The more data you have, the more independent information that you have.

But every time you make a calculation like a mean, you're using up one piece of independent

So, for example, we have data from 100 randomly sampled square miles of avocado orchard, and

we've painstakingly counted the number of bees spotted in each sampled square mile over

We need to be sure avocados are getting pollinated!

The owner of one avocado orchard says that she usually sees 15,000 bees per square mile.

So, you set out to analyze your data to see whether you think the bee population has changed.

You have 100 pieces of independent data--one measure from each square mile--so, when you

calculate the mean number of bees from all 100 square miles, you're using up 1 degree

Now that we know the mean number of bees is 16,838, you only need 99 of the bee counts

to figure out what the count for the 100th square mile would bee.

With a quick one sample t-test, we get our p-value from a t-distribution with 99 degrees

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Degrees of Freedom and Effect Sizes: Crash Course Statistics #28

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