# Kurtosis and Degrees of Freedom

hey guys I’m just wondering if these too are linked. In other words as the degrees of freedom increases does that mean that the distribution exhibits more of a leptokurtic bell shape? Thanks

Yes, they’re linked. When some says “degrees of freedom,” I think of sample size. The golden rule: as sample size increases, you approach normal distribution. Kurtosis helps you understand the skewness of your data. This means, the bigger the sample size, the more normal your distribution is, and the less skewed the spread is. I have seen some data on this, and in general, Kurtosis is a very ineffective metric, especially for smaller sample sizes. Something that you thought was leptokurtic @ sample size = 30 turns out to be platykurtic @ sample size = 300. Of course, as the sample size gets even bigger, you approach normal distribution (kurtosis = 0 i.e. no skewness). As with everything in statistics, you have to be extremely careful when deciphering data based on kurtosis. It does reflect on the variance of the data, but you must also be aware of your sample size (degrees of freedom) to a certain extent. As usual, others please correct me if I wrong.

'as the sample size gets even bigger, you approach normal distribution (kurtosis = 0 i.e. no skewness). ’ doesn’t normal distribution have a kurtosis of 3? I also seem to think that kurtosis measures the fatness and not the skewness of a distribution?

i was also wondering…normal distribution has a kurtosis of 3…that’s why we calculate the excess kurtosis for example

'as the sample size gets even bigger, you approach normal distribution (kurtosis = 0 i.e. no skewness). ’ Clarification - I shouldn’t have said *excess* kurtosis = 0. Kurtosis for ND is, in fact, three. Also, my previous explanation was based on my statistics classes - mostly, kurtosis and skewness go hand-in-hand. You never look at kurtosis without looking at the skewness. Chances are if the distribution is more peaked or less peaked, it is also skewed, and not normal. All my understanding aside, for the exam, I think it is safer to stay with what the book says regarding this topic: Kurtosis of ND = 3 *Excess* Kurtosis of ND = 0 Kurtosis is used to measure the peakedness of your distribution While we’re on the topic: Leptokurtic (Fat-tailed - more observations in tail compared to normal dis.) - EXCESS Kurtosis > 0, Kurtosis > 3 Mesokurtic (Normally Distributed) - EXCESS Kurtosis = 0, Kurtosis = 3 Platykurtic (fewer observations in tail compared to normal dis.) - EXCESS Kurtosis < 0, Kurtosis < 3 Apologies for any confusion I might have caused.

Oyster: I think the leptokurtic has thick tails…because it is peaked, the tails become thicker whereas the platikurtic will have fat tails…

They have 1 important difference. leptokurtic - MORE peaked and fatter tails than a normal distribution degrees of freedom refers to the student-t distribution t-distribution - LESS peaked and fatter tails than a normal distribution. Sample size only affect the shape of the t-distribution.

Oyster Wrote: ------------------------------------------------------- > Kurtosis helps you understand the skewness of your > data. This means, the bigger the sample size, the > more normal your distribution is, and the less > skewed the spread is. I’m really not sure this is correct.

Cinderella Wrote: ------------------------------------------------------- > Oyster Wrote: > -------------------------------------------------- > ----- > > > Kurtosis helps you understand the skewness of > your > > data. This means, the bigger the sample size, > the > > more normal your distribution is, and the less > > skewed the spread is. > > I’m really not sure this is correct. See my second post, Cinderella.

I just saw your second post. Even if you say kurtosis and skewness and kurtosis usually go hand in hand (I have no stats background so I cant comment on that) I feel quite confident saying that that degrees of freedon do influence kurtosis but not skewness. Just thought I’d clear that up for the benefit of the OP and other visitors. Oyster Wrote: ------------------------------------------------------- > Cinderella Wrote: > -------------------------------------------------- > ----- > > Oyster Wrote: > > > -------------------------------------------------- > > > ----- > > > > > Kurtosis helps you understand the skewness of > > your > > > data. This means, the bigger the sample size, > > the > > > more normal your distribution is, and the > less > > > skewed the spread is. > > > > I’m really not sure this is correct. > > See my second post, Cinderella.

Cinderella wrote: “I feel quite confident saying that that degrees of freedon do influence kurtosis but not skewness.” If you meant to type “not skewness,” I don’t agree with you. I’ll appreciate if someone else can confirm my understanding, but in order to answer the question, “Does sample size impact kurtosis or skewness?,” all you have to do is look at their respective formulas. Mind you, there are different formulas for skewness and kurtosis, but I want to stick to the ones in the CFA curriculum: Skewness = {n/[(n-1)(n-2)]} * [Sum(Xi - Xbar)^3/s^3] Kurtosis = {[n(n+1)]/[(n-1)(n-2)(n-3)]} * {[Sum(Xi - Xbar)^4/s^4] - [(3(n-1)^2)/((n-2)(n-3))]} The key here is that *both* formulas have “n” in them - “n” being the sample size. Also, for a normal distribution, skewness = 0, excess kurtosis = 0, and kurtosis = 3. Ultimately, sample size impacts both skewness and kurtosis. As sample size gets larger, your distribution approaches the normal. As a side note, even if you ignore all the “n’s” surrounding those formulas, you still have standard deviation in the formula, which, in turn, uses “n” in its denominator. Here is an external reference: http://www.tc3.edu/instruct/sbrown/stat/shape.htm

Oyster Wrote: ------------------------------------------------------- > > > > I’ll appreciate if someone else can confirm my > understanding, but in order to answer the > question, “Does sample size impact kurtosis or > skewness?,” all you have to do is look at their > respective formulas. Mind you, there are different > formulas for skewness and kurtosis, but I want to > stick to the ones in the CFA curriculum: > > Skewness = {n/[(n-1)(n-2)]} * > > Kurtosis = {/[(n-1)(n-2)(n-3)]} * { - > [(3(n-1)^2)/((n-2)(n-3))]} > > The key here is that *both* formulas have “n” in > them - “n” being the sample size. Also, for a > normal distribution, skewness = 0, excess kurtosis > = 0, and kurtosis = 3. Ultimately, sample size > impacts both skewness and kurtosis. As sample size > gets larger, your distribution approaches the > normal. > > As a side note, even if you ignore all the “n’s” > surrounding those formulas, you still have > standard deviation in the formula, which, in turn, > uses “n” in its denominator. > > Here is an external reference: > http://www.tc3.edu/instruct/sbrown/stat/shape.htm From what I understand, sample size does affect both the calculated excess kurtosis and the calculated skewness. Check out the abridged formulas for both, they have ‘n’ in their denominator. So greater the value of ‘n’, smaller the calculated excess kurtosis and skewness.

OK I feel there are two separate discussions here. If we are dealing with a t-distribution (the mention of degrees of freedom by the OP led me to think in terms of the t-distribution) then degrees of freedom have an impact on kurtosis, but NOT on skewness as the t-distribution is symmetric that is has zero skewness. If we are just relating n to skewness then there’s an obvious connection as pointed our by yourself and Anish. Do let me know if I am in the wrong on either of these fronts. Oyster Wrote: ------------------------------------------------------- > Cinderella wrote: > “I feel quite confident saying that that degrees > of freedon do influence kurtosis but not > skewness.” > > If you meant to type “not skewness,” I don’t agree > with you. > > > I’ll appreciate if someone else can confirm my > understanding, but in order to answer the > question, “Does sample size impact kurtosis or > skewness?,” all you have to do is look at their > respective formulas. Mind you, there are different > formulas for skewness and kurtosis, but I want to > stick to the ones in the CFA curriculum: > > Skewness = {n/[(n-1)(n-2)]} * > > Kurtosis = {/[(n-1)(n-2)(n-3)]} * { - > [(3(n-1)^2)/((n-2)(n-3))]} > > The key here is that *both* formulas have “n” in > them - “n” being the sample size. Also, for a > normal distribution, skewness = 0, excess kurtosis > = 0, and kurtosis = 3. Ultimately, sample size > impacts both skewness and kurtosis. As sample size > gets larger, your distribution approaches the > normal. > > As a side note, even if you ignore all the “n’s” > surrounding those formulas, you still have > standard deviation in the formula, which, in turn, > uses “n” in its denominator. > > Here is an external reference: > http://www.tc3.edu/instruct/sbrown/stat/shape.htm

Cinderella Wrote: “If we are dealing with a t-distribution (the mention of degrees of freedom by the OP led me to think in terms of the t-distribution) then degrees of freedom have an impact on kurtosis, but NOT on skewness as the t-distribution is symmetric that is has zero skewness.” Again, I can’t seem to digest your logic (please note that I am simply making intelligent conversation here, not fighting - it is hard to express emotions electronically). A t-distribution is symmetric when the sample size is large. If you look at the CFA curriculum, you can utilize the t-distribution when: 1) the sample size is large (n>=30), 2) when the sample size is large and population is non-normal, and 3) when the population is normal, but sample size is small. You can’t use a t-test when the population is small and the population is non-normal. In this scenario, you can’t infer that the distribution is normal (symmetric). When you utilize a t-test, you still have to deal with sample size. Skewness of zero (and excess kurtosis of zero) essentially means that the distribution is normal. You, more or less, use skewness and kurtosis to “eyeball” whether or not a given distribution is normal.

Oyster: I understand you’re only trying to make sense of this for all of us and I appreciate your contributions to this discussion. Take a looks at http://en.wikipedia.org/wiki/Student’s_t-distribution The t-distribution is symetrical irrespective of the dof. ample size has an influence on when it is appropriate to use the t-dist and on the level of peakness (kurtosis). If you look at the diagrams on the right on the wikipedia page you’ll see that it is always symmetric and therefore has zero skewness. Oyster Wrote: ------------------------------------------------------- > Cinderella Wrote: > > “If we are dealing with a t-distribution (the > mention of degrees of freedom by the OP led me to > think in terms of the t-distribution) then degrees > of freedom have an impact on kurtosis, but NOT on > skewness as the t-distribution is symmetric that > is has zero skewness.” > > > Again, I can’t seem to digest your logic (please > note that I am simply making intelligent > conversation here, not fighting - it is hard to > express emotions electronically). > > A t-distribution is symmetric when the sample size > is large. If you look at the CFA curriculum, you > can utilize the t-distribution when: > 1) the sample size is large (n>=30), > 2) when the sample size is large and population is > non-normal, and > 3) when the population is normal, but sample size > is small. > > You can’t use a t-test when the population is > small and the population is non-normal. In this > scenario, you can’t infer that the distribution is > normal (symmetric). > > > When you utilize a t-test, you still have to deal > with sample size. Skewness of zero (and excess > kurtosis of zero) essentially means that the > distribution is normal. You, more or less, use > skewness and kurtosis to “eyeball” whether or not > a given distribution is normal.

Cinderella Wrote: Take a looks at http://en.wikipedia.org/wiki/Student’s_t-distribution The t-distribution is symetrical irrespective of the dof. ample size has an influence on when it is appropriate to use the t-dist and on the level of peakness (kurtosis). If you look at the diagrams on the right on the wikipedia page you’ll see that it is always symmetric and therefore has zero skewness. Well, you may want to read through the body of that wiki. Here is a quote: “The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. AS THE NUMBER OF DEGREES OF FREEDOM GROWS, the t-distribution approaches the normal distribution with mean 0 and variance 1.” This is accurate because the CFA curriculum also states it this way (I don’t think Wikipedia is an authoritative source because I don’t know who last touched the wiki entry). Search for the above quote and look at the pictures that follow. Skewness of zero doesn’t imply that the distribution of your population is not impacted by the sample size (or degrees of freedom). Ultimately, I am not fighting your claim of the t-distribution being symmetric. What I am fighting is your claim of the sample size not impacting skewness. Again, regardless of the metric, when it comes to distributions (the ones within the realm of the CFA), your sample size will almost always come into play. As a side note, I hope you realize that degrees of freedom indirectly points to the sample size?

I am glad of this discussion. Mind you, once we have established what is what here, we are unlikely to forget it in the exam. @Cinderella: Let me quote you here “The t-distribution is symmetrical irrespective of the dof. ample size has an influence on when it is appropriate to use the t-dist and on the level of peakness (kurtosis). If you look at the diagrams on the right on the wikipedia page you’ll see that it is always symmetric and therefore has zero skewness.” My thoughts: DOF refers to the sample size and you cannot use t-dist if the sample size is small because the distribution will not be symmetric. The wiki article says: “Student’s t-distribution … arises when estimating the mean of a normally distributed population in situations where the sample size is small” They have specified that it should be normally distributed when n is small. When n is large, it is anyway approximately normally distributed. That is why the diagram shows symmetric t dist because it is used in only symmetric situations (small n normal or large n non-normal). The excess kurtosis of t dist is negative and approaches 0 as dof (or n) increases. So as n (or dof) increases, the distribution approaches normal.