Biostatistics for Animal Science

10 Biostatistics for Animal Science
If the population mean is unknown, the sample mean ( y ) is substituted for µ and skewness
is:
3
( )( ) ∑ n ⎛ yi
− y ⎞
sk =
⎜ ⎟ i
n −1
n − 2 ⎝ s ⎠
For a symmetric distribution skewness is equal to zero. It is positive when the right tail is
longer, and negative when left tail is longer (Figure 1.5).
a) b)
Figure 1.5 Illustrations of skewness: a) negative, b) positive
Kurtosis (kt) is a measure of flatness or steepness of a distribution, or a measure of the
heaviness of the tails of a distribution. If the population mean (µ) is known, kurtosis is:
4
1 ⎛ yi
− µ ⎞
kt = ∑ ⎜ ⎟ − 3
i n ⎝ s ⎠
If the population mean is unknown, the sample mean ( y ) is used instead and kurtosis is:
n(
n + 1)
( n −1)(
n − 2)(
n − 3)
4
2
3(
n −1)
( n − 2)(
− 3)
⎛ yi
− y ⎞
kt = ∑ ⎜ ⎟ −
i
⎝ s ⎠ n
For variables such as weight, height or milk yield, frequency distributions are expected to
be symmetric about the mean and bell-shaped. These are normal distributions. If
observations follow a normal distribution then kurtosis is equal to zero. A distribution with
positive kurtosis has a large frequency of observations close to the mean and thin tails. A
distribution with a negative kurtosis has thicker tails and a lower frequency of observations
close to the mean than does the normal distribution (Figure 1.6).
a) b)
Figure 1.6 Illustrations of kurtosis: a) positive, b) negative