Applied Univariate, Bivariate, and Multivariate Statistics. Daniel J. Denis
may also wish to compute what is known as the coefficient of variation, which is a ratio of the standard deviation to the mean. We can estimate this coefficient for
parent and child respectively in Galton's data:
> cv.parent <- sd(parent)/mean(parent) > cv.parent [1] 0.02616573 > cv.child <- sd(child)/mean(child) > cv.child [1] 0.03698044
Computing the coefficient of variation is a way of comparing the variability of competing distributions relative to each distribution's mean. We can see that the dispersion of child relative to its mean (0.037) is slightly larger than that of the dispersion of parent relative to its mean (0.026).
2.9 DEGREES OF FREEDOM
In our discussion of variance, we saw that if we wanted to use the sample variance as an estimator of the population variance, we needed to subtract 1 from the denominator. That is, S2 was “corrected” into s2:
We say we lost a degree of freedom in the denominator of the statistic. But what are degrees of freedom? They are the number of independent units of information in a sample that are relevant to the estimation of some parameter (Everitt, 2002). In the case of the sample variance, s2, one degree of freedom is lost since we are interested in using s2 as an estimator of σ2. We are losing the degree of freedom because the numerator,
A conceptual demonstration may prove useful in understanding the concept of degrees of freedom. Imagine you were asked to build a triangle such that there was to be no overlap of lines on either side of the triangle. In other words, the lengths of the sides had to join neatly at the vertices. We shall call this the “Beautiful Triangle” as depicted in Figure 2.9. You are now asked to draw the first side of the triangle. Why did you draw this first side the length that you did? You concede that the length of the first side is arbitrary, you were free to draw it whatever length you wished. In drawing the second length, you acknowledge you were also free to draw it whatever length you wished. Neither of the first two lengths in any way violated the construction of a beautiful triangle with perfectly adjoining vertices.
However, in drawing the third length, what length did you choose? Notice that to complete the triangle, you were not free to determine this length arbitrarily. Rather, the length was fixed given the constraint that the triangle was to be a beautiful one. In summary then, in building the beautiful triangle, you lost 1 degree of freedom, in that two of the lengths were of your free choosing, but the third was fixed. Analogously, in using s2 as an estimator of σ2, a single degree of freedom is lost. If
Figure 2.9 The “Beautiful Triangle” as a way to understanding degrees of freedom.
Degrees of freedom occur throughout statistics in a variety of statistical tests. If you understand this basic example, then while working out degrees of freedom for more advanced designs and tests may still pose a challenge, you will nonetheless have a conceptual base from which to build your comprehension.
2.10 SKEWNESS AND KURTOSIS
The third moment of a distribution is its skewness. Skewness of a random variable generally refers to the extent to which a distribution lacks symmetry. Skewness is defined as:
Skewness for a normal distribution is equal to 0, just as skewness for a rectangular distribution is also equal to 0 (one does not necessarily require a bell‐shaped curve for skewness to equal 0)
Skewness for a positively skewed distribution is greater than 0; these distributions have tails that stretch out into values on the abscissa of greatest value
Skewness for a negatively skewed distribution is less than 0; these distributions have tails that stretch out to values on the abscissa of least value
An example of a positively skewed distribution is that of the typical F density, given in Figure 2.10.
The fourth moment of a distribution is its kurtosis, generally referring to the peakness of a distribution (Upton and Cook, 2002), but also having much to do with a distribution's tails (DeCarlo, 1997):
Figure 2.10 F distribution on 2 and 5 degrees of freedom. It is positively skewed since the tail stretches out to numbers of greater value.
With regard to kurtosis, distributions are defined:
mesokurtic if the distribution exhibits kurtosis typical of a bell‐shaped normal curve
platykurtic if the distribution exhibits lighter tails and is flatter toward the center than a normal distribution
leptokurtic if the distribution exhibits heavier tails and is generally more narrow in the center than a normal distribution, revealing that it is somewhat “peaked”
We can easily compute moments of empirical distributions in R or SPSS. Several packages in R are available for this purpose. We could compute skewness for parent on Galton's data by:
> library(psych) > skew(parent) [1] -0.03503614
The psych package (Revelle, 2015) also provides a range of descriptive statistics:
> library(psych) > describe(Galton) vars n mean sd median trimmed mad min max range skew kurtosis parent 1 928 68.31 1.79 68.5 68.32 1.48 64.0 73.0 9 -0.04 0.05 child 2 928 68.09 2.52