# The UNIVARIATE Procedure

### Example 4.10 Computing Confidence Limits for Quantiles and Percentiles

This example, which is a continuation of Example 4.9, illustrates how to compute confidence limits for quantiles and percentiles. A second researcher is more interested in summarizing the heights with quantiles than the mean and standard deviation. He is also interested in computing 90% confidence intervals for the quantiles. The following statements produce estimated quantiles and confidence limits for the population quantiles:

```title 'Analysis of Female Heights';
ods select Quantiles;
proc univariate data=Heights ciquantnormal(alpha=.1);
var Height;
run;
```

The ODS SELECT statement restricts the output to the "Quantiles" table; see the section ODS Table Names. The CIQUANTNORMAL option produces confidence limits for the quantiles. As noted in Output 4.10.1, these limits assume that the data are normally distributed. You should check this assumption before using these confidence limits. See the section Shapiro-Wilk Statistic for information about the Shapiro-Wilk test for normality in PROC UNIVARIATE; see Example 4.19 for an example that uses the test for normality.

Output 4.10.1: Normal-Based Quantile Confidence Limits

 Analysis of Female Heights

The UNIVARIATE Procedure
Variable: Height (Height (in))

Quantiles (Definition 5)
Level Quantile 90% Confidence Limits
Assuming Normality
100% Max 70.0
99% 70.0 68.94553 70.58228
95% 68.6 67.59184 68.89311
90% 67.5 66.85981 68.00273
75% Q3 66.0 65.60757 66.54262
50% Median 64.4 64.14564 64.98770
25% Q1 63.1 62.59071 63.52576
10% 61.6 61.13060 62.27352
5% 60.6 60.24022 61.54149
1% 60.0 58.55106 60.18781
0% Min 60.0

It is also possible to use PROC UNIVARIATE to compute confidence limits for quantiles without assuming normality. The following statements use the CIQUANTDF option to request distribution-free confidence limits for the quantiles of the population of heights:

```title 'Analysis of Female Heights';
ods select Quantiles;
proc univariate data=Heights ciquantdf(alpha=.1);
var Height;
run;
```

The distribution-free confidence limits are shown in Output 4.10.2.

Output 4.10.2: Distribution-Free Quantile Confidence Limits

 Analysis of Female Heights

The UNIVARIATE Procedure
Variable: Height (Height (in))

Quantiles (Definition 5)
Level Quantile   Order Statistics
90% Confidence Limits
Distribution Free
LCL Rank UCL Rank Coverage
100% Max 70.0
99% 70.0 68.6 70.0 73 75 48.97
95% 68.6 67.5 70.0 68 75 94.50
90% 67.5 66.6 68.6 63 72 91.53
75% Q3 66.0 65.7 66.6 50 63 91.77
50% Median 64.4 64.1 65.1 31 46 91.54
25% Q1 63.1 62.7 63.7 13 26 91.77
10% 61.6 60.6 62.7 4 13 91.53
5% 60.6 60.0 61.6 1 8 94.50
1% 60.0 60.0 60.5 1 3 48.97
0% Min 60.0

The table in Output 4.10.2 includes the ranks from which the confidence limits are computed. For more information about how these confidence limits are calculated, see the section Confidence Limits for Percentiles. Note that confidence limits for quantiles are not produced when the WEIGHT statement is used.

A sample program for this example, uniex07.sas, is available in the SAS Sample Library for Base SAS software.