Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
G01AAF calculates the mean, standard deviation, coefficients of skewness and kurtosis, and the maximum and minimum values for a set of ungrouped data. Weighting may be used.
The data consist of a single sample of n observations, denoted by xi, with corresponding weights, wi, for i=1,2,…,n.
If no specific weighting is required, then each wi is set to 1.
The quantities computed are:
(a)
The sum of the weights
W=∑i=1nwi.
(b)
Mean
x-=∑i=1nwixiW.
(c)
Standard deviation
s2=∑i=1nwixi-x-2d, where d=W-∑i=1nwi2W.
(d)
Coefficient of skewness
s3=∑i=1nwixi-x-3d×s23.
(e)
Coefficient of kurtosis
s4=∑i=1nwixi-x-4d×s24-3.
(f)
Maximum and minimum elements of the sample.
(g)
The number of observations for which wi>0, i.e., the number of valid observations. Suppose m observations are valid, then the quantities in (c), (d) and (e) will be computed if m≥2, and will be based on m-1 degrees of freedom. The other quantities are evaluated provided m≥1.
On entry: the sample observations, xi, for i=1,2,…,n.
3: IWT – INTEGERInput/Output
On entry: indicates whether weights are to be supplied by you or not. In the latter case, the weights will be assumed equal and assigned the value 1.0 in the routine.
IWT=0
Indicates no user-supplied weights.
IWT=1
Indicates user-supplied weights are required, and they will be supplied in the array WT.
On exit: IWT is used to indicate the number of valid observations, m; see (g) in Section 3 above.
On entry: if IWT=1, then the elements of WT must contain the weights associated with the observations, wi, for i=1,2,…,n.
If IWT=0, then the elements of WT need not be set.
On exit: if IWT=1 the elements of WT are unchanged.
If IWT=0 each element of WT will be assigned the value 1.0.
5: XMEAN – REAL (KIND=nag_wp)Output
On exit: the mean, x-.
6: S2 – REAL (KIND=nag_wp)Output
On exit: the standard deviation, s2.
7: S3 – REAL (KIND=nag_wp)Output
On exit: the coefficient of skewness, s3.
8: S4 – REAL (KIND=nag_wp)Output
On exit: the coefficient of kurtosis, s4.
9: XMIN – REAL (KIND=nag_wp)Output
On exit: the smallest value in the sample.
10: XMAX – REAL (KIND=nag_wp)Output
On exit: the largest value in the sample.
11: WTSUM – REAL (KIND=nag_wp)Output
On exit: the sum of the weights in the array WT, that is ∑i=1nwi.
This will be N if IWT was 0 on entry.
12: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL=1
On entry,
N<1.
IFAIL=2
The number of valid cases, m, is 1. In this case, standard deviation and coefficients of skewness and of kurtosis cannot be calculated.
IFAIL=3
Either the number of valid cases is 0, or at least one weight is negative.
7 Accuracy
The method used is believed to be stable.
8 Further Comments
The time taken by G01AAF is approximately proportional to n.
9 Example
This example summarises a number of datasets. For each dataset the observations and, optionally, weights are read and printed. G01AAF is then called and the calculated quantities are printed.