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NAG[g01aac] NAG[nag_summary_stats_1var] - Mean, variance, skewness, kurtosis, etc., one variable, from raw data
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Calling Sequence
g01aac(x, nvalid, xmean, xsd, xskew, xkurt, xmin, xmax, wsum, 'n'=n, 'wt'=wt, 'fail'=fail)
nag_summary_stats_1var(. . .)
Parameters
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x - Vector(1..n, datatype=float[8]);
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On entry: the sample observations, ![x[i]](/support/helpjp/helpview.aspx?si=10148/file08355/math23.png) , for  .
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nvalid - assignable;
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Note: On exit the variable nvalid will have a value of type integer.
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On exit: the number  of valid observations – see Section [Description] above.
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xmean - assignable;
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Note: On exit the variable xmean will have a value of type float.
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On exit: the mean,  .
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xsd - assignable;
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Note: On exit the variable xsd will have a value of type float.
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On exit: the standard deviation,  .
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xskew - assignable;
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Note: On exit the variable xskew will have a value of type float.
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On exit: the coefficient of skewness, ![gamma[1]](/support/helpjp/helpview.aspx?si=10148/file08355/math72.png) .
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xkurt - assignable;
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Note: On exit the variable xkurt will have a value of type float.
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On exit: the coefficient of kurtosis, ![gamma[2]](/support/helpjp/helpview.aspx?si=10148/file08355/math83.png) .
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xmin - assignable;
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Note: On exit the variable xmin will have a value of type float.
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On exit: the smallest value in the sample.
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xmax - assignable;
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Note: On exit the variable xmax will have a value of type float.
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On exit: the largest value in the sample.
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wsum - assignable;
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Note: On exit the variable wsum will have a value of type float.
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On exit: the sum of the weights in the array wt, that is ![(∑)[i=1]^nw[i]](/support/helpjp/helpview.aspx?si=10148/file08355/math114.png) . This will be  if weighted estimates are not used.
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'n'=n - integer; (optional)
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Default value: the first dimension of the arrays x, wt.
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On entry: the number of observations,  .
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Constraint:  . .
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'wt'=wt - Vector(1..n, datatype=float[8]); (optional)
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On entry: if weighted estimates are required, then wt must contain the weights ![w[i]](/support/helpjp/helpview.aspx?si=10148/file08355/math148.png) , for  . Otherwise, wt need not be defined and the corresponding argument must be set to the null pointer, ((double *)0).
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_summary_stats_1var (g01aac) 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.
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Description
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The data consist of a single sample of  observations, denoted by ![x[i]](/support/helpjp/helpview.aspx?si=10148/file08355/math182.png) , with corresponding weights, ![w[i]](/support/helpjp/helpview.aspx?si=10148/file08355/math184.png) , for  .
If no specific weighting is required, then the array  need not be defined.
The quantities computed are as follows.
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a. The sum of the weights ![W=(∑)[i=1]^nw[i]](/support/helpjp/helpview.aspx?si=10148/file08355/math196.png)
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b. Mean ![x=((∑)[i=1]^nw[i]x[i])/W](/support/helpjp/helpview.aspx?si=10148/file08355/math199.png)
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c. Standard deviation ![s=sqrt(((∑)[i=1]^nw[i](x[i],-,x)^2)/d)](/support/helpjp/helpview.aspx?si=10148/file08355/math202.png)
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where ![d=W-((∑)[i=1]^nw[i]^2)/W](/support/helpjp/helpview.aspx?si=10148/file08355/math206.png)
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d. Coefficient of skewness ![gamma[1]=((∑)[i=1]^nw[i](x[i],-,x)^3)/(s^3)](/support/helpjp/helpview.aspx?si=10148/file08355/math209.png)
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e. Coefficient of kurtosis ![gamma[2]=((∑)[i=1]^nw[i](x[i],-,x)^4)/(s^4)-3](/support/helpjp/helpview.aspx?si=10148/file08355/math212.png)
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f. Maximum and minimum elements of the sample.
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Error Indicators and Warnings
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"NE_CASES_ONE"
The number of valid cases is one. In this case, standard deviation and coefficients of skewness and of kurtosis cannot be calculated.
"NE_CASES_ZERO"
The number of valid cases is zero.
"NE_INT_ARG_LE"
On entry,  must not be less than or equal to 0:  .
"NE_REAL_ARG_LT"
On entry, ![wt[< >+1]](/support/helpjp/helpview.aspx?si=10148/file08355/math253.png) must not be less than  : ![wt[< >+1]=< >](/support/helpjp/helpview.aspx?si=10148/file08355/math257.png) .
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Accuracy
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A single pass updating algorithm is used, which is believed to be stable.
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Further Comments
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The time taken by nag_summary_stats_1var (g01aac) is approximately proportional to  .
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Examples
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>
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n := 24:
x := Vector([193, 215, 112, 161, 92, 140, 38, 33, 279, 249, 473, 339, 60, 130, 20, 50, 257, 284, 447, 52, 67, 61, 150, 2200], datatype=float[8]):
wt := Vector([], datatype=float[8]):
NAG:-g01aac(x, nvalid, xmean, xsd, xskew, xkurt, xmin, xmax, wsum, 'n' = n, 'wt' = wt):
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