Skewness and Kurtosis on the TI-83/84

See also:  Descriptive Statistics of a Data Set on the TI-83/84
Normality Check on the TI-83/84 or TI-89

Skewness

A data set may be skewed left (negatively), skewed right (positively), or symmetric (neither positive nor negative skew). You can get a general impression of skewness by drawing a histogram or polygon, but there are also some common numerical measures of skewness.

This recalls the ways you assess the spread or dispersion of a data set. You can draw a histogram and get a visual impression, or you can calculate the standard deviation or variance and have a handy numerical measure of spread.

Computing

The moment coefficient of skewness of a data set is

where and

m₃ is called the third moment of the data set, x̄ is the mean, √m₂ is the standard deviation and therefore m₂ is the variance, and n is the number of data points.

You can compute skewness on your TI-83/84 with list formulas. Suppose you have the data set in L1. Then sum((L1-mean(L1))^3)/dim(L1) gives you the third moment m₃, and you get the variance by pressing [2nd ENTER makes ENTRY] and changing the 3 to 2. Those expressions are fairly gnarly, and they get worse if you have a grouped frequency distribution instead of a simple list. It's a lot simpler to just download and run the program presented later in this note.

Interpreting

Zero skewness corresponds to perfectly symmetric data. If skewness is positive (negative), the data are skewed right (left), and larger numbers mean greater skew.

But this data set is just one sample drawn from a population. How far must the sample be skewed before you can say that the population is likely skewed?

To answer this question, compute a test statistic:

where

SES is the standard error of skewness, and the formula is adapted from page 85 of Duncan Cramer’s Basic Statistics for Social Research (Routledge, 1997).

If skewness/SES is more than about 2 or less than about −2, you can say that the population very likely has some skewness in the same direction as the sample, though you can’t put a number to the population skewness. (This is a two-tailed test of skewness ≠ 0 at roughly the 0.05 significance level.) If that fraction is between −2 and 2, you can’t say whether the population is symmetric (skewness = 0) or skewed.

Kurtosis

Most data sets have a central peak, meaning that the majority of data points cluster fairly close to the center. The height and sharpness of that peak relative to the rest of the data are measured by a number called kurtosis. Higher values indicate a higher, sharper peak; lower values indicate a lower, less distinct peak.

While a histogram can give you some idea of the peakedness of the data, the kurtosis is a more precise measire.

Computing

The moment coefficient of kurtosis of a data set is

where and

m₄ is called the fourth moment of the data set, x̄ is the mean, √m₂ is the standard deviation and therefore m₂ is the variance, and n is the number of data points.

Kurtosis, like skewness, can be computed on your TI-83/84 with list operations. If you have the data set in L1, then sum((L1-mean(L1))^4)/dim(L1) is the fourth moment m₄, and the variance is computed the same way except using ^2 instead of ^4. Again, the expressions get worse if you have a grouped frequency distribution, so you're better off to download and run the program presented later in this note.

Interpreting

Kurtosis = 3 indicates the same degree of peakedness as a normal distribution (the famous “bell curve”); such a data set is called mesokurtic. A distribution with a higher, sharper peak has kurtosis > 3 and is called leptokurtic; a distribution with a lower, less distinct peak has kurtosis < 3 and is called platykurtic.

Some authors subtract 3 from kurtosis and talk about excess kurtosis. This lets them describe excess kurtosis as =0 (mesokurtic), >0 (leptokurtic), or <0 (platykurtic). It also sets the excess kurtosis of a normal distribution to 0, which seems like a more satisfying reference point than 3.

But this data set is just one sample drawn from a population. How far must the sample kurtosis be from 3, or how far must excess kurtosis be from 0, before you can say that the population is likely to have a higher or lower peak than a normal distribution?

To answer this question, compute a test statistic:

where

SEK is the standard error of kurtosis, and the formula is adapted from page 89 of Duncan Cramer’s Basic Statistics for Social Research (Routledge, 1997).

If (kurtosis−3)/SEK is more than about 2 or less than about −2, you can say that the population very likely has some kurtosis in the same direction as the sample, though you can’t put a number to the population kurtosis. (This is a two-tailed test of kurtosis ≠ 3 at roughly the 0.05 significance level.) If that fraction is between −2 and 2, you can’t say whether the population is peaked about the same as a normal distribution (“bell curve”, mesokurtic, kurtosis=3), more sharply peaked, or less.

TI-83/84 Program SKURT

The preceding shows basic calculations for skewness and kurtosis, but doesn’t deal explicitly with grouped frequency distributions. And in any event there are a lot of keystrokes to make the calculations.

I’ve developed a TI-83/84 program that handles all the calculations, working with either a simple data list or a frequency distribution.

Getting the Program

There are three methods to get the program into your calculator:

• If a classmate has the program on her calculator (any model TI-83/84), she can transfer it to yours with the short cable with headphone-style plugs that comes with all TI calculators. On your calculator, press [2nd x,T,θ,n makes LINK] [►] [ENTER], and then on hers press [2nd x,T,θ,n makes LINK] [3], select the program, then press [►] [ENTER].
• Or, download SKURT.ZIP (21 KB, updated Jun 22, 2008), unzip it, and transfer file SKURT.8XP to your calculator. (This requires TI-Connect or TI-GraphLink software and a cable.)
• Or, as a last resort, key in the program. See SKURT.PDF and SKURT_HINTS.HTM in the SKURT.ZIP file.

Using the Program

If you have a frequency distribution, put the data points or class marks in one statistics list and the frequencies in another. If you have a simple list of numbers, put them in a statistics list.

Then press [PRGM], scroll down to SKURT, and press [ENTER] [ENTER].

Example: College students’ heights

Here are grouped data for heights of 100 randomly selected male students:

The histogram shows the data are skewed left, not symmetric. But how highly skewed are they? And how does the central peak compare to the normal distribution for height and sharpness? To answer these questions, you have to compute the skewness and kurtosis.

Assuming you’ve downloaded the program, enter the x’s in one statistics list and the f’s in another. If you’re not sure how to create statistics lists, please see Descriptive Statistics of a Data Set on the TI-83/84.

Then press [PRGM] and scroll to SKURT, then press [ENTER] twice. When prompted, enter the list that contains the x’s, press [9] for Yes, and enter the list that contains the f’s.

(I used L5 and L6, but you could use any lists. You could even enter the numbers directly, delimited by curly braces — {61,64,67,70,73} — but if you discover you’ve made a mistake, there’s no way to correct it and you have to start over.)

The program gives its results on three screens of data.

The first screen shows some basic statistics: the sample size, the mean, the standard deviation, and the variance. M and V are not the proper symbols for mean and variance, as you know, but numeric variables on the TI-83/84 can have only single-letter names. The program stores key results in variables in case you want to do any further computations with them. (Standard deviation isn’t stored in a variable because you can always get it by √V.)

The second screen shows results for skewness. The third moment is just a way station on the way to computing the skewness, which is about −0.11. The distribution is therefore negatively skewed (skewed to the left), but can you say that the population is also negatively skewed? The standard error of skewness is 0.24, and the skewness divided by standard error is −0.45.

Recall the rule of thumb: if skewness is more than about two standard errors either side of zero, you can say that the population is skewed in that direction. Here, the last number tells you that skewness is only 0.45 standard errors below zero, so you can’t say whether the population is skewed or symmetric.

The third screen shows results for kurtosis. The fourth moment divided by the square of the variance gives the kurtosis, which is 2.74, so the sample is platykurtic. Comparing this to the “standard” kurtosis of 3, you see that the central peak is a little flatter and less distinct than the central peak of a normal distribution (“bell curve”). This may be easier to see if you subtract 3 and consider the excess kurtosis of −0.26. It certainly wasn’t obvious from the histogram!

What can you say about the kurtosis of the population from which the sample was taken? The rule of thumb is that an excess kurtosis of at least two standard errors is significant. The standard error of kurtosis is 0.48, and the excess kurtosis is only 0.54 standard errors below zero. (Or, the kurtosis is only 0.54 standard errors below 3.) Therefore you can't say whether the population is peaked like a normal distribution, more than normal, or less than normal.

Program Variables

The program stores its results in several variables, which are left afterward for your use:

• E = standard error of skewness
• F = standard error of kurtosis
• K = kurtosis (to find excess kurtosis, subtract 3)
• M = mean
• N = sample size (number of data points, or total of frequencies for a grouped frequency distribution
• S = skewness
• V = variance (for standard deviation, take the square root)

To access any of them, press the [ALPHA] key and then the key for the letter. Example: [ALPHA 6 makes V].

If you want to delete them, press [2nd + makes MEM] [2] [2], cursor to each one, and press [DEL].

The program also creates or overwrites two statistics lists, LD for the data points and LF for the frequencies. (If you don't have a frequency distribution, the program fills LF with 1’s to make the logic simpler.) You can access them by pressing [2nd STAT makes LIST] and then scrolling up.

If you want to add them to the statistics list editor, press [STAT] [ENTER] [▲], cursor to the L6 column heading, and cursor right once more to open a new list slot. You’ll see the A indicator in the upper right corner, so don’t press [ALPHA] but press [plain x-1 makes D] or [plain COS makes F].

If you want to delete these lists from your calculator’s memory, press [2nd + makes MEM] [2] [4], scroll down to find each one, and press [DEL].

Finally, the program stores your graphics settings in GDB0 and then automatically deletes that variable after restoring your settings. You don’t care about this unless you're using GDB0 for your own purposes, which would be quite unusual.