The UNIVARIATE Procedure
- Overview
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Getting Started
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Syntax
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Details
Missing Values Rounding Descriptive Statistics Calculating the Mode Calculating Percentiles Tests for Location Confidence Limits for Parameters of the Normal Distribution Robust Estimators Creating Line Printer Plots Creating High-Resolution Graphics Using the CLASS Statement to Create Comparative Plots Positioning Insets Formulas for Fitted Continuous Distributions Goodness-of-Fit Tests Kernel Density Estimates Construction of Quantile-Quantile and Probability Plots Interpretation of Quantile-Quantile and Probability Plots Distributions for Probability and Q-Q Plots Estimating Shape Parameters Using Q-Q Plots Estimating Location and Scale Parameters Using Q-Q Plots Estimating Percentiles Using Q-Q Plots Input Data Sets OUT= Output Data Set in the OUTPUT Statement OUTHISTOGRAM= Output Data Set OUTKERNEL= Output Data Set OUTTABLE= Output Data Set Tables for Summary Statistics ODS Table Names ODS Tables for Fitted Distributions ODS Graphics Computational Resources
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Examples
Computing Descriptive Statistics for Multiple Variables Calculating Modes Identifying Extreme Observations and Extreme Values Creating a Frequency Table Creating Plots for Line Printer Output Analyzing a Data Set With a FREQ Variable Saving Summary Statistics in an OUT= Output Data Set Saving Percentiles in an Output Data Set Computing Confidence Limits for the Mean, Standard Deviation, and Variance Computing Confidence Limits for Quantiles and Percentiles Computing Robust Estimates Testing for Location Performing a Sign Test Using Paired Data Creating a Histogram Creating a One-Way Comparative Histogram Creating a Two-Way Comparative Histogram Adding Insets with Descriptive Statistics Binning a Histogram Adding a Normal Curve to a Histogram Adding Fitted Normal Curves to a Comparative Histogram Fitting a Beta Curve Fitting Lognormal, Weibull, and Gamma Curves Computing Kernel Density Estimates Fitting a Three-Parameter Lognormal Curve Annotating a Folded Normal Curve Creating Lognormal Probability Plots Creating a Histogram to Display Lognormal Fit Creating a Normal Quantile Plot Adding a Distribution Reference Line Interpreting a Normal Quantile Plot Estimating Three Parameters from Lognormal Quantile Plots Estimating Percentiles from Lognormal Quantile Plots Estimating Parameters from Lognormal Quantile Plots Comparing Weibull Quantile Plots Creating a Cumulative Distribution Plot Creating a P-P Plot
- References
| Robust Estimators |
A statistical method is robust if it is insensitive to moderate or even large departures from the assumptions that justify the method. PROC UNIVARIATE provides several methods for robust estimation of location and scale. See Example 4.11.
Winsorized Means
The Winsorized mean is a robust estimator of the location that is relatively insensitive to outliers. The 
-times Winsorized mean is calculated as
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where 
is the number of observations and 
is the 
th order statistic when the observations are arranged in increasing order:
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The Winsorized mean is computed as the ordinary mean after the smallest observations are replaced by the 
st smallest observation and the largest observations are replaced by the st largest observation.
For data from a symmetric distribution, the Winsorized mean is an unbiased estimate of the population mean. However, the Winsorized mean does not have a normal distribution even if the data are from a normal population.
The Winsorized sum of squared deviations is defined as
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The Winsorized 
statistic is given by
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where 
denotes the location under the null hypothesis and the standard error of the Winsorized mean is
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When the data are from a symmetric distribution, the distribution of 
is approximated by a Student’s distribution with 
degrees of freedom (Tukey and McLaughlin; 1963; Dixon and Tukey; 1968).
The Winsorized 
confidence interval for the location parameter has upper and lower limits
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where 
is the 
th percentile of the Student’s distribution with degrees of freedom.
Trimmed Means
Like the Winsorized mean, the trimmed mean is a robust estimator of the location that is relatively insensitive to outliers. The -times trimmed mean is calculated as
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where is the number of observations and is the th order statistic when the observations are arranged in increasing order:
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The trimmed mean is computed after the smallest and largest observations are deleted from the sample. In other words, the observations are trimmed at each end.
For a symmetric distribution, the symmetrically trimmed mean is an unbiased estimate of the population mean. However, the trimmed mean does not have a normal distribution even if the data are from a normal population.
A robust estimate of the variance of the trimmed mean 
can be based on the Winsorized sum of squared deviations 
, which is defined in the section Winsorized Means; see Tukey and McLaughlin (1963). This can be used to compute a trimmed test which is based on the test statistic
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where the standard error of the trimmed mean is
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When the data are from a symmetric distribution, the distribution of is approximated by a Student’s distribution with degrees of freedom (Tukey and McLaughlin; 1963; Dixon and Tukey; 1968).
The "trimmed" 
confidence interval for the location parameter has upper and lower limits
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where is the th percentile of the Student’s distribution with degrees of freedom.
Robust Estimates of Scale
The sample standard deviation, which is the most commonly used estimator of scale, is sensitive to outliers. Robust scale estimators, on the other hand, remain bounded when a single data value is replaced by an arbitrarily large or small value. The UNIVARIATE procedure computes several robust measures of scale, including the interquartile range, Gini’s mean difference 
, the median absolute deviation about the median (MAD), 
, and 
. In addition, the procedure computes estimates of the normal standard deviation 
derived from each of these measures.
The interquartile range (IQR) is simply the difference between the upper and lower quartiles. For a normal population, can be estimated as IQR/1.34898.
Gini’s mean difference is computed as
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For a normal population, the expected value of is 
. Thus 
is a robust estimator of when the data are from a normal sample. For the normal distribution, this estimator has high efficiency relative to the usual sample standard deviation, and it is also less sensitive to the presence of outliers.
A very robust scale estimator is the MAD, the median absolute deviation from the median (Hampel; 1974), which is computed as
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where the inner median, 
, is the median of the observations, and the outer median (taken over ) is the median of the absolute values of the deviations about the inner median. For a normal population, 
is an estimator of .
The MAD has low efficiency for normal distributions, and it may not always be appropriate for symmetric distributions. Rousseeuw and Croux (1993) proposed two statistics as alternatives to the MAD. The first is
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where the outer median (taken over ) is the median of the medians of 
, 
. To reduce small-sample bias, 
is used to estimate , where 
is a correction factor; see Croux and Rousseeuw (1992).
The second statistic proposed by Rousseeuw and Croux (1993) is
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where
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In other words, is 2.2219 times the th order statistic of the 
distances between the data points. The bias-corrected statistic 
is used to estimate , where 
is a correction factor; see Croux and Rousseeuw (1992).