If each observation in a climate
series is xi when i changes from 1 to n (the total
number of observations), then:
expected values of the difference between a random variable and its
expected value raised to a power (the moment number).
statistic that measures the center of a sample of data (the first
central moment) by adding up the observations and dividing by the
number of data points. It may be thought of as the center of mass
or balancing point for the data, i.e., that point at which a ruler
would balance if all the data values were placed along it at their
appropriate numerical values.
second central moment, and is a scale parameter of the variable distribution
positive square root of variance
statistic which measures the center of a set of data by finding that
value which divides the data in half. A technical definition is that
the median is the value which is greater than or equal to half of
the values in the data set and less than or equal to half the values.
statistic defined as the most frequently occurring data value. It
is sometimes used as an alternative to the mean or median as a measure
of central tendency
the statistics which divide the observations in a numeric sample into
3 intervals, each containing 33.333% of the data. The lower, middle,
and upper terciles are computed by ordering the data from smallest
to largest and then finding the values below which fall 1/3 and 2/3
of the data.
third central moment of a random variable. This statistic measures
symmetry of the variable distribution. If the variable has more numerous
lower values skewness is negative, for more numerous high values skewness
is positive. Skewness close to zero implies a symmetrical distribution
fourth central moment characterizes how peaked (positive kurtosis
or leptokurtic) or flat (negative kurtosis or platykurtic) the distribution
is. Kurtosis close to zero is for a balanced distribution with regard
|Figure 1. Histograms and
basic statistics for daily temperature (top) and precipitation (bottom)
at Olympia Airport, Washington.
Example of computation of basic
statistics and histograms is shown in Figure 1. Temperature (Figure 1
top) mean, median and mode are very similar to each other relative to
their magnitude, which is one of the characteristics of symmetric bell
shaped histograms. Skewness and kurtosis statistics reflect this as well.
Precipitation histogram (Figure 1 bottom) displays distinct positive skewness
and kurtosis, the mean is much greater than the median relative to their
magnitude, and the two are much greater than the mode.
von Storch, H., and Zwiers,
F.W., 1999. Statistical Analysis in Climate Research. Cambridge University
Wilks, D.S., 1995. Statistical Methods in the Atmospheric Sciences. Academic