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PROC UNIVARIATE - Part 02: HISTOGRAM, QQPLOT, and Normality Testing

Overview

• Part 01 covered the core statistical summary produced by PROC UNIVARIATE: moments, percentiles, extreme observations, the OUTPUT= statement, and merging summary statistics back to detail data.
• This lesson focuses on the graphical and distributional testing capabilities: producing histograms, Q-Q plots, and formal normality test statistics.
• These tools are used to assess whether a variable follows a normal (Gaussian) distribution — an assumption required by many parametric statistical tests.
• Note: PROC UNIVARIATE produces traditional SAS/GRAPH or ODS Graphics output depending on your SAS environment. The code shown here uses ODS Graphics which is the modern default in SAS 9.2 and later.

/* Create a sample dataset with two variables: one approximately normal, one right-skewed */ data work.measures; call streaminit(42); do subject = 1 to 50; normal_val = round(rand('normal', 100, 15), 0.1); skewed_val = round(rand('exponential') * 30 + 10, 0.1); output; end; run; Copy Code View Log

• CALL STREAMINIT(42) sets a random number seed for reproducibility — running this code will produce the same dataset every time.
• RAND('normal', mean, std) generates values from a normal distribution with the specified mean and standard deviation.
• RAND('exponential') generates values from an exponential distribution which is inherently right-skewed — a useful contrast for demonstrating normality tests.
• Inspect WORK.MEASURES to see the two variables before running any analysis.

View Data

HISTOGRAM Statement

• The HISTOGRAM statement in PROC UNIVARIATE produces a graphical display of the distribution of a variable.
• By default, it uses automatic bin widths. You can overlay a normal distribution curve using the NORMAL option to visually assess how closely the data follow a normal shape.
• The MIDPOINTS= option specifies exact bin midpoints if you want to control the histogram bins manually.
• A histogram that is roughly bell-shaped and symmetric is consistent with normality; long tails, sharp peaks, or asymmetry suggest non-normality.

ods graphics on; proc univariate data=work.measures; var normal_val skewed_val; histogram / normal; inset mean std / position=ne; run; ods graphics off; Copy Code View Log

• ODS GRAPHICS ON enables the modern graphical output system. ODS GRAPHICS OFF closes it after the procedure to avoid leaving it active for subsequent steps.
• The NORMAL option on the HISTOGRAM statement overlays a fitted normal distribution curve on the histogram bars. This makes it easy to see whether the data shape matches the normal curve.
• INSET MEAN STD / POSITION=NE adds a small statistics box in the northeast corner of the plot showing the mean and standard deviation.
• Examine the histogram for normal_val — it should appear roughly bell-shaped. The histogram for skewed_val will show a long right tail, inconsistent with the overlaid normal curve.

Q-Q Plot (Quantile-Quantile Plot)

• A Q-Q plot compares the quantiles of your data against the theoretical quantiles of a normal distribution.
• If the data are normally distributed, the points on the Q-Q plot should fall approximately along a straight diagonal reference line.
• Deviations from the line indicate non-normality: an S-shaped curve suggests heavy tails or light tails; a curved pattern suggests skewness.
• The QQPLOT statement in PROC UNIVARIATE generates a Q-Q plot. The NORMAL option overlays the reference line for the fitted normal distribution.

ods graphics on; proc univariate data=work.measures; var normal_val skewed_val; qqplot / normal(mu=est sigma=est); run; ods graphics off; Copy Code View Log

• MU=EST and SIGMA=EST instruct SAS to estimate the mean and standard deviation from the data (rather than requiring you to specify them manually) and use those estimates for the reference line.
• For normal_val, the points should cluster closely around the diagonal reference line — minor deviations at the tails are expected with small samples.
• For skewed_val, the points will curve away from the reference line, especially in the upper tail, confirming the right-skewed distribution.
• Q-Q plots are generally more sensitive to distributional departures than histograms, particularly in the tails, making them a valuable complement to the histogram.

Formal Normality Tests with the NORMAL Option

• Visual assessment of histograms and Q-Q plots is subjective. PROC UNIVARIATE also provides formal statistical tests for normality.
• Add the NORMAL option to the PROC UNIVARIATE statement to produce four normality test statistics in the output.
• The four tests are: Shapiro-Wilk (recommended for n less than 2000), Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling.
• The null hypothesis for all four tests is that the data are normally distributed. A small p-value (typically less than 0.05) leads to rejecting the null hypothesis and concluding non-normality.
• With large sample sizes, these tests become very sensitive and may flag minor, inconsequential departures from normality — always combine test results with visual assessment.

proc univariate data=work.measures normal; var normal_val skewed_val; run; Copy Code View Log

• Look for the "Tests for Normality" table in the output for each variable.
• For normal_val, the p-values for all four tests should be large (well above 0.05), indicating no evidence against normality.
• For skewed_val, the p-values should be small (below 0.05), indicating that the data are not consistent with a normal distribution.
• The Shapiro-Wilk W statistic ranges from 0 to 1, where values close to 1 indicate normality. A W below 0.9 is often a practical flag of concern.

Combining Histogram and QQPLOT in One Step

• You can include both HISTOGRAM and QQPLOT in a single PROC UNIVARIATE call to get all distributional diagnostics at once.
• This is efficient for a standard normality check routine across multiple variables.

ods graphics on; proc univariate data=work.measures normal; var normal_val skewed_val; histogram / normal; qqplot / normal(mu=est sigma=est); inset n mean std skewness kurtosis / position=ne; run; ods graphics off; Copy Code View Log

• The INSET statement here adds n, mean, standard deviation, skewness, and kurtosis to the histogram plot — a quick distributional summary embedded in the graphic.
• Skewness near 0 and kurtosis near 3 (excess kurtosis near 0) are consistent with normality. Skewness above 1 or below -1 is generally considered a meaningful departure.
• This combined call produces histogram plots, Q-Q plots, and formal test statistics in a single PROC step — the standard approach for a normality assessment in a clinical or statistical analysis program.

Key Points

• HISTOGRAM / NORMAL overlays a fitted normal curve on the distribution bars — a quick visual check of distributional shape.
• QQPLOT / NORMAL(MU=EST SIGMA=EST) plots observed vs theoretical quantiles — points along the diagonal line confirm normality; curves or S-shapes indicate departures.
• Adding NORMAL to the PROC UNIVARIATE statement produces four formal normality test statistics including the Shapiro-Wilk W test which is recommended for sample sizes below 2000.
• A p-value below 0.05 in the normality tests rejects the null hypothesis of normality — but with large samples, even trivial departures become significant, so always use visual assessment alongside formal tests.
• Always enclose ODS GRAPHICS ON and ODS GRAPHICS OFF around PROC UNIVARIATE graphical output to control when the graphical system is active.