Confidence interval for a median and other quantiles

This is a section from my text book An Introduction to Medical Statistics, Fourth Edition. I hope that the topic will be useful in its own right, as well as giving a flavour of the book.

8.10 Confidence interval for a median and other quantiles

In Section 4.5 we estimated medians and other quantiles directly from the frequency distribution. We can estimate confidence intervals for these using the Binomial distribution. This is a large sample method. The 95% confidence interval for the q quantile can be found by an application of the Binomial distribution (Section 6.4, Section 6.6) (see Conover 1980). The number of observations less than the q quantile will be an observation from a Binomial distribution with parameters n and q, and hence has mean nq and standard deviation √(nq(1-q)). We calculate j and k such that:
j = nq − 1.96 √(nq(1-q))
k = nq + 1.96 √(nq(1-q))
We round j and k up to the next integer. Then the 95% confidence interval is between the jth and the kth observations in the ordered data.

For example, the data in Table 4.4 are measurements of Forced Expiratory Volume in one second (FEV1) for 57 male medical students:

Table 4.4 FEV1 (litres) for 57 male medical students
2.85 3.19 3.50 3.69 3.90 4.14 4.32 4.50 4.80 5.20
2.85 3.20 3.54 3.70 3.96 4.16 4.44 4.56 4.80 5.30
2.98 3.30 3.54 3.70 4.05 4.20 4.47 4.68 4.90 5.43
3.04 3.39 3.57 3.75 4.08 4.20 4.47 4.70 5.00
3.10 3.42 3.60 3.78 4.10 4.30 4.47 4.71 5.10
3.10 3.48 3.60 3.83 4.14 4.30 4.50 4.78 5.10

**Table 4.4** FEV1 (litres) for 57 male medical students
2.85	3.19	3.50	3.69	3.90	4.14	4.32	4.50	4.80	5.20
2.85	3.20	3.54	3.70	3.96	4.16	4.44	4.56	4.80	5.30
2.98	3.30	3.54	3.70	4.05	4.20	4.47	4.68	4.90	5.43
3.04	3.39	3.57	3.75	4.08	4.20	4.47	4.70	5.00
3.10	3.42	3.60	3.78	4.10	4.30	4.47	4.71	5.10
3.10	3.48	3.60	3.83	4.14	4.30	4.50	4.78	5.10

For these FEV1 data the median is 4.1, the 29th value in the Table. For the 95% confidence interval for the median, n = 57 and q = 0.5, and
j = 57 × 0.5 − 1.96 × √(57 × 0.5 × (1−0.5)) = 21.10
k = 57 × 0.5 + 1.96 × √(57 × 0.5 × (1−0.5)) = 35.90
The 95% confidence interval is thus from the 22nd to the 36th observation, 3.75 to 4.30 litres from the Table. Compare this to the 95% confidence interval for the mean, 3.9 to 4.2 litres, which is completely included in the interval for the median. This method of estimating percentiles is relatively imprecise. Another example is given in Section 15.5.

Reference

Conover, W.J. (1980) Practical Nonparametric Statistics John Wiley and Sons, New York.

Adapted from page 110 of An Introduction to Medical Statistics by Martin Bland, 2015, reproduced by permission of Oxford University Press.

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