Sample Moments
Mean and Standard Deviation
The following table summarizes notation and definitions related to the sample standard
deviation.
Y1, Y2, …, Yn
a random sample of size n
n
sample mean of Y
Y 
We think of the dispersion, or spread, of the
sample in terms of the deviations from the
mean.
Y
i 1
i
n
Y  Y , i  1, 2, ..., n
i
S YY   Yi  Y 
n
2
i 1
sum of squared deviations from the mean,
commonly called the “sum of squares”, and
denoted by SS or by or by SYY
 Y 

2
  Yi
2
i
n
  Yi 2  nY 2
df = n  1
degrees of freedom
 Y  Y 
n
sample variance of Y
s 
2
S yy
df

i 1
i
n 1
 Y  Y 
n
sample standard deviation of Y
s s 
sample standard error of the (sample) mean
Y , which is the standard deviation of Y
Golde I. Holtzman, Department of Statistics, Virginia Tech
2
S yy
df

SE  s Y 
i 1
i
n 1
s
n
Skewness and Kurtosis
The sample skewness is a function of the third
central moment, i.e., a function of the deviation
cubed. See note 1.
The sample kurtosis is a function of the fourth
central moment, i.e., a function of the deviation
raised to the 4th power. See note 1.
 z 
n
3
i
i 1
n
n 1 n  2 
 z 
n
i 1
4
i
 where
n  n 1) 
n 1 n  2  n 3 
zi 
Yi  Y
s
 3     
n 1 2
n  2 n 3
Note 1. The formulas for skewness an kurtosis are those that are used in JMP and SAS. There are
several variations (by various authors) on the adjustments made for sample size (and the
subtraction term in the kurtosis formula), but all are functions of the third and fourth moments,
respectively.
Note 2. The mean is the first moment (non-central), or center of gravity. The variance is the
second central moment. The skewness and kurtosis are essentially the third and fourth central
moments, respectively. One can compute up to the nth central (or non-central) moment for a
sample of size n. The central moments can be thought of as a transformation of the n data into
an alternative representation (like, for example, the coefficients that are produced by Fourier
transformation), that present the same information in a different form. The mean (1st moment)
is a summary of the data. The variance (2nd central moment), a measure of dispersion,
summarizes the sample giving information beyond that which is given by the mean. The
skewness (3rd central moment) gives information about shape, specifically symmetry, that is
beyond that which is given by the mean and variance. And so on, as we add successive
moments. Each moment gives further information, but each higher moment is in a sense giving
us less important information than the lower moments that precede it.
Golde I. Holtzman, Department of Statistics, Virginia Tech