Basic Statistics
Basic descriptive statistics operations.
Given a list with
elements
, the mean
Mean[list] is defined to be
.
The variance
Variance[list] is defined to be
, for real data. (For complex data
.)
The standard deviation
StandardDeviation[list] is defined to be
.
If the elements in
list are thought of as being selected at random according to some probability distribution, then the mean gives an estimate of where the center of the distribution is located, while the standard deviation gives an estimate of how wide the dispersion in the distribution is.
The median
Median[list] effectively gives the value at the halfway point in the sorted version of
list. It is often considered a more robust measure of the center of a distribution than the mean, since it depends less on outlying values.
The
quantile
Quantile
effectively gives the value that is
of the way through the sorted version of
list.
For a list of length
,
Mathematica defines
Quantile
to be
s[[Ceiling[n q]]], where
is
Sort[list, Less].
There are, however, about 10 other definitions of quantile in use, all potentially giving slightly different results.
Mathematica covers the common cases by introducing four
quantile parameters in the form
Quantile
. The parameters
and
in effect define where in the list should be considered a fraction
of the way through. If this corresponds to an integer position, then the element at that position is taken to be the
quantile. If it is not an integer position, then a linear combination of the elements on either side is used, as specified by
and
.
The position in a sorted list
for the
quantile is taken to be
. If
is an integer, then the quantile is
. Otherwise, it is
, with the indices taken to be
or
if they are out of range.
| {{0,0},{1,0}} | inverse empirical CDF (default) |
| {{0,0},{0,1}} | linear interpolation (California method) |
| {{1/2,0},{0,0}} | element numbered closest to  |
| {{1/2,0},{0,1}} | linear interpolation (hydrologist method) |
| {{0,1},{0,1}} | mean-based estimate (Weibull method) |
| {{1,-1},{0,1}} | mode-based estimate |
| {{1/3,1/3},{0,1}} | median-based estimate |
| {{3/8,1/4},{0,1}} | normal distribution estimate |
Common choices for quantile parameters.
Whenever
, the value of the
quantile is always equal to some actual element in
list, so that the result changes discontinuously as
varies. For
, the
quantile interpolates linearly between successive elements in
list.
Median is defined to use such an interpolation.
Note that
Quantile
yields quartiles when
and percentiles when
.
| Mean[{x1,x2,...}] | the mean of the  |
| Mean[{{x1,y1,...},{x2,y2,...},...}] | a list of the means of the  |
Handling multidimensional data.
Sometimes each item in your data may involve a list of values. The basic statistics functions in
Mathematica automatically apply to all corresponding elements in these lists.
This separately finds the mean of each "column" of data.
| Out[1]= |  |
Note that you can extract the elements in the
"column" of a multidimensional list using
list[[All, i]].
quantile
