The UNIVARIATE Procedure
- Overview
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Getting Started
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Syntax
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DetailsMissing ValuesRoundingDescriptive StatisticsCalculating the ModeCalculating PercentilesTests for LocationConfidence Limits for Parameters of the Normal DistributionRobust EstimatorsCreating Line Printer PlotsCreating High-Resolution GraphicsUsing the CLASS Statement to Create Comparative PlotsPositioning InsetsFormulas for Fitted Continuous DistributionsGoodness-of-Fit TestsKernel Density EstimatesConstruction of Quantile-Quantile and Probability PlotsInterpretation of Quantile-Quantile and Probability PlotsDistributions for Probability and Q-Q PlotsEstimating Shape Parameters Using Q-Q PlotsEstimating Location and Scale Parameters Using Q-Q PlotsEstimating Percentiles Using Q-Q PlotsInput Data SetsOUT= Output Data Set in the OUTPUT StatementOUTHISTOGRAM= Output Data SetOUTKERNEL= Output Data SetOUTTABLE= Output Data SetTables for Summary StatisticsODS Table NamesODS Tables for Fitted DistributionsODS GraphicsComputational Resources
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ExamplesComputing Descriptive Statistics for Multiple VariablesCalculating ModesIdentifying Extreme Observations and Extreme ValuesCreating a Frequency TableCreating Basic Summary PlotsAnalyzing a Data Set With a FREQ VariableSaving Summary Statistics in an OUT= Output Data SetSaving Percentiles in an Output Data SetComputing Confidence Limits for the Mean, Standard Deviation, and VarianceComputing Confidence Limits for Quantiles and PercentilesComputing Robust EstimatesTesting for LocationPerforming a Sign Test Using Paired DataCreating a HistogramCreating a One-Way Comparative HistogramCreating a Two-Way Comparative HistogramAdding Insets with Descriptive StatisticsBinning a HistogramAdding a Normal Curve to a HistogramAdding Fitted Normal Curves to a Comparative HistogramFitting a Beta CurveFitting Lognormal, Weibull, and Gamma CurvesComputing Kernel Density EstimatesFitting a Three-Parameter Lognormal CurveAnnotating a Folded Normal CurveCreating Lognormal Probability PlotsCreating a Histogram to Display Lognormal FitCreating a Normal Quantile PlotAdding a Distribution Reference LineInterpreting a Normal Quantile PlotEstimating Three Parameters from Lognormal Quantile PlotsEstimating Percentiles from Lognormal Quantile PlotsEstimating Parameters from Lognormal Quantile PlotsComparing Weibull Quantile PlotsCreating a Cumulative Distribution PlotCreating a P-P Plot
- References
Calculating Percentiles
The UNIVARIATE procedure automatically computes the 1st, 5th, 10th, 25th, 50th, 75th, 90th, 95th, and 99th percentiles (quantiles), as well as the minimum and maximum of each analysis variable. To compute percentiles other than these default percentiles, use the PCTLPTS= and PCTLPRE= options in the OUTPUT statement.
You can specify one of five definitions for computing the percentiles with the PCTLDEF= option. Let n be the number of nonmissing values for a variable, and let represent the ordered values of the variable. Let the tth percentile be y, set , and let
where j is the integer part of np, and g is the fractional part of np. Then the PCTLDEF= option defines the tth percentile, y, as described in the following table.
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PCTLDEF |
Description |
Formula |
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1 |
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2 |
observation numbered closest to np |
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3 |
empirical distribution function |
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4 |
weighted average aimed |
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5 |
empirical distribution function with averaging |
Weighted Percentiles
When you use a WEIGHT statement, the percentiles are computed differently. The 100pth weighted percentile y is computed from the empirical distribution function with averaging:
where is the weight associated with and is the sum of the weights.
Note that the PCTLDEF= option is not applicable when a WEIGHT statement is used. However, in this case, if all the weights are identical, the weighted percentiles are the same as the percentiles that would be computed without a WEIGHT statement and with PCTLDEF=5.
Confidence Limits for Percentiles
You can use the CIPCTLNORMAL option to request confidence limits for percentiles, assuming the data are normally distributed. These limits are described in Section 4.4.1 of Hahn and Meeker (1991). When , the two-sided confidence limits for the th percentile are
where n is the sample size. When , the two-sided confidence limits for the th percentile are
One-sided confidence bounds are computed by replacing by in the appropriate preceding equation. The factor is related to the noncentral t distribution and is described in Owen and Hua (1977) and Odeh and Owen (1980). See Example 4.10.
You can use the CIPCTLDF option to request distribution-free confidence limits for percentiles. In particular, it is not necessary to assume that the data are normally distributed. These limits are described in Section 5.2 of Hahn and Meeker (1991). The two-sided confidence limits for the th percentile are
where is the jth order statistic when the data values are arranged in increasing order:
The lower rank l and upper rank u are integers that are symmetric (or nearly symmetric) around , where is the integer part of and n is the sample size. Furthermore, l and u are chosen so that and are as close to as possible while satisfying the coverage probability requirement,
where is the cumulative binomial probability,
In some cases, the coverage requirement cannot be met, particularly when n is small and p is near 0 or 1. To relax the requirement of symmetry, you can specify CIPCTLDF(TYPE = ASYMMETRIC). This option requests symmetric limits when the coverage requirement can be met, and asymmetric limits otherwise.
If you specify CIPCTLDF(TYPE = LOWER), a one-sided lower confidence bound is computed as , where l is the largest integer that satisfies the inequality
with . Likewise, if you specify CIPCTLDF(TYPE = UPPER), a one-sided lower confidence bound is computed as , where u is the largest integer that satisfies the inequality
Note that confidence limits for percentiles are not computed when a WEIGHT statement is specified. See Example 4.10.