# Skewness and Kurtosis of a t-distribution

#### plaztik-love21

##### New Member
What is the skewness and kurtosis of a t-distribution, as compared to the underlying population distribution? Is it larger, smaller, or is there no definite rule/pattern?

Was asked this in an interview and didn't know the answer. Thank you!

#### spunky

##### Doesn't actually exist
as compared to the underlying population distribution?
This aspect is key. What is assumed about the "underlying population distribution"?

#### plaztik-love21

##### New Member
it is a normal distribution

#### spunky

##### Doesn't actually exist
Well... if I were asked this on the spot I'd first I'd first qualify I'm assuming a population mean of 0 (aka the distribution is centered). With this in mind, my reasoning would've been that the skewness would be the same as the normal, which is 0, because all symmetric distributions centered at 0 have 0 odd moments. The kurtosis would have to be greater than that of the normal because the tails of the t distribution are fatter BUT, as the degrees-of-freedom parameter increases, the kurtosis would become closer and closer to the normal, which is 3. (or an excess kurtosis of 0 if people prefer that).

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Yeah, I always remember this from back in the day, knowing the 95% interval for the standard normal uses 1.96 and it is something like 2.12 for the t-distributions. So in finite samples their shapes may vary in tails and peakedness, but as @spunky stated they converge, much like many distribution can approach standard normal as sample sizes approaches infinity.

#### spunky

##### Doesn't actually exist
like many distribution can approach standard normal as sample sizes approaches infinity.
But not the Cauchy... NEVER THE CAUCHY.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Well, you are the one mentioning tails after I had link earlier to the article saying, how do you even define what a tail is (e.g., uniform distribution).

#### Miner

##### TS Contributor
Actually, that would be correct for the first one. My company uses a lot of electronic components. Component suppliers will sell different levels of precision for components, but can't actually produce different levels of precision, so they sort them. If you order a high precision component, you get the first distribution. If you order non-precision components, you get the second one, which is the leftover tails.

#### spunky

##### Doesn't actually exist
In my world (psychometrics) truncated normals are the default assumption when talking about range restriction. Say you want to only admit students into a prestigious graduate program that score above X mark in their GRE. You usually only end up with the tails

#### hlsmith

##### Less is more. Stay pure. Stay poor.
The latter almost reminds me of limit of detection data.

#### spunky

##### Doesn't actually exist
Yeah, what's the 2nd one if only the 1st one is a truncated normal?

#### Dason

Yeah, what's the 2nd one if only the 1st one is a truncated normal?
The most hallowed of all distributions. The hole-y normal.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
I believe it was inspired by Excite Bike on NES Classic. I am not going to tell you what inspired the first one.

#### spunky

##### Doesn't actually exist
The most hallowed of all distributions. The hole-y normal.
I'm not gonna lie. This made me laugh more than it should have.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
But I thought these were
D-distribution and V-distribution, respectively

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#### hlsmith

##### Less is more. Stay pure. Stay poor.
Since I got a like on my prior crass post. What do you think I call their joint distribution?

A good time.

Just kidding, I will go with C-distribution, for conception. Since it could resemble a baby bump.

#### spunky

##### Doesn't actually exist
GOD SHTOP THIS, IT'S KILLING MEEEEE!

I luuuuuuv bad-pun jokes. I dunno why but they really make me laugh.