 # On descriptive confidence intervals for the mean of asymmetric populations

Rousson, V; Choi, E (2003). On descriptive confidence intervals for the mean of asymmetric populations. Journal of Nonparametric Statistics, 15(4-5):553-563.

## Abstract

Given a sample of n observations with sample mean Xmacr and standard deviation S drawn independently from a population with unknown mean μ, it is well known that the skewness of the statistic n-1/2(Xmacr - μ)/S is of the opposite sign to the skewness of the population. As a consequence, an equal-tailed confidence interval for μ may be used for descriptive purposes, since the relative position of Xmacr in the interval provides visual information about the skewness of the population. In this paper, we are interested in confidence intervals for the mean which share this descriptive property. We formally define two simple classes of intervals where the degree of asymmetry around Xmacr monotonically depends on the sample skewness through a parameter λ with values between 0 and 1. These classes contain the symmetric ordinary-z (or the ordinary-t) confidence interval as a special case. We show how to determine this parameter λ in order to obtain an equal-tailed confidence interval for μ which is second order accurate. While our first solution has already been investigated in the literature and has serious drawbacks, our second solution appears to be new and sound. Moreover, our method provides the sample skewness with a new and concrete interpretation.

## Abstract

Given a sample of n observations with sample mean Xmacr and standard deviation S drawn independently from a population with unknown mean μ, it is well known that the skewness of the statistic n-1/2(Xmacr - μ)/S is of the opposite sign to the skewness of the population. As a consequence, an equal-tailed confidence interval for μ may be used for descriptive purposes, since the relative position of Xmacr in the interval provides visual information about the skewness of the population. In this paper, we are interested in confidence intervals for the mean which share this descriptive property. We formally define two simple classes of intervals where the degree of asymmetry around Xmacr monotonically depends on the sample skewness through a parameter λ with values between 0 and 1. These classes contain the symmetric ordinary-z (or the ordinary-t) confidence interval as a special case. We show how to determine this parameter λ in order to obtain an equal-tailed confidence interval for μ which is second order accurate. While our first solution has already been investigated in the literature and has serious drawbacks, our second solution appears to be new and sound. Moreover, our method provides the sample skewness with a new and concrete interpretation.

## Statistics

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