# 2-sided tolerance interval

Is there a way to find out which percentiles are being bounded by the interval? Say I calculate a 2-sided tolerance interval for 95% of the population with 95% confidence interval. How do I find out whether this interval is using 0-95%, 1-96%, 2-97%, and so on percentiles?

The end game is that I want to calculate a (sort of) margin of error on a non-parametric tolerance interval, by calculating the difference between the upper tolerance interval and the percentile it's bounding.

EDIT: am I missing the point entirely? Is the 2-sided interval always centered around the median? So in the case of the 95%, is it always estimating the 2.5th and 97.5th percentiles?

Thank you!

data (only a subset of my original data; does not fit any parametric distribution well, whether transformed or not).

df <- structure(list(Val = c(7.8, 11.5, 9, 9.3, 9.2, 8.4, 8.6, 8.6,
9.3, 8.6, 8.6, 6.6, 7.9, 5.6, 7.9, 8.4, 8.7, 9.5, 9.3, 8.8, 7.7,
8.8, 8.8, 9.6, 7.9, 9.1, 10.3, 9, 8.1, 6.6, 10.4, 7.6, 9.1, 8.2,
8.4, 8.1, 9.1, 15.4, 7.8, 8.1, 11.2, 8.5, 6.6, 9.3, 9.7, 7.6,
5.7, 8.5, 8.9, 10, 8.3, 9.1, 8.2, 7, 8.2, 8.5, 6.8, 8.7, 6.4,
8.3, 7.8, 7, 8.2, 5.6, 8.6, 8.9, 8.5, 7.6, 10.5, 12.3, 13.3,
11.1, 11, 6.6, 7.1, 7, 8.2, 6.9, 10, 11, 9, 11, 10, 17, 13, 16,
9, 8, 13, 9, 11, 11, 8, 13, 10, 11, 11, 14, 10, 11)), .Names = "Val", row.names = c(480L,
484L, 632L, 634L, 653L, 654L, 656L, 658L, 659L, 660L, 663L, 671L,
674L, 676L, 681L, 684L, 685L, 690L, 691L, 692L, 693L, 695L, 697L,
699L, 700L, 704L, 705L, 706L, 708L, 710L, 713L, 715L, 722L, 723L,
724L, 725L, 726L, 728L, 731L, 732L, 746L, 747L, 748L, 751L, 752L,
754L, 756L, 757L, 758L, 759L, 760L, 762L, 763L, 766L, 770L, 774L,
776L, 780L, 783L, 785L, 786L, 788L, 789L, 790L, 793L, 794L, 796L,
798L, 819L, 824L, 878L, 910L, 926L, 1066L, 1079L, 1083L, 1084L,
1085L, 1479L, 1480L, 1544L, 1545L, 1546L, 1580L, 1601L, 1610L,
1611L, 1612L, 1615L, 1616L, 1617L, 1619L, 1651L, 1653L, 1658L,
1662L, 1663L, 1668L, 1672L, 1673L), class = "data.frame")

library(tolerance)
interval <- normtol.int(x = df$Val, alpha = 0.05, P = 0.95, side = 2, method = "HE", log.norm = FALSE)  subset of possible quantiles (0-5 in top row, 95-100 in bottom row, just for illustration) matrix(quantile(df$Val, probs = c(seq(0, 0.05, 0.01), seq(0.95, 1, 0.01))),
nrow = 2, byrow = T)