February 2006 #1 Comparing Estimators of Quartiles Under Various Models
by Steven T. Garren
## Title

### by Steven T. Garren.

**Abstract:**
Some authors define the first quartile of a
data set to be the median of the ordered data strictly
to the left of the overall median. Other authors define
the first quartile to be the median of the ordered data
to the left of and including the overall median.
We propose a third estimator (of the population first quartile)
which has minimum mean squared error under normality,
among weighted averages of two order statistics, and
robustness against nonnormal distributions is examined.
Likewise, estimators of the third quartile may be defined.
The three estimators of first and third quartiles are
compared in terms of mean squared error under normality,
a t-distribution with 3 degrees of freedom, a uniform
distribution, and an exponential distribution.
The preferred estimator depends on the distribution and the
sample size modulus 4, for sample sizes no larger than 30. **Key Words: **
Exponential distribution, Normal distribution,
t-distribution, Uniform distribution

**Author:**

Steven T. Garren, garrenst@jmu.edu

**Editor:**
Richard Graf,rgraf@sunstroke.sdsu.edu

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