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# Statistics in “APA Format”

Mathematics is an ancient discipline, developed through many cultures and with many different ways of representing numbers and operations. Fortunately, there is a accepted convention that now governs how mathematics is represented around the world and it is through this convention that many mathematicians are able to exchange ideas independent of whether they speak the same (non-mathematical) language. It is not surprising, therefore, that statistics (a branch of mathematics) also has a convention on how statistics are represented.

## Statistical Conventions

In the statistics discipline we are typically dealing with one of two groups, the population or the sample. The population we are working with should be denoted by a capital Roman letter (e.g.  $X$ ) and an individual from that population should be denoted with a lower case Roman letter (e.g.  $x$ ). When we take a sample from the population, we are getting a set of  $x$ ‘s out of our population  $X$  (i.e each  $x \in X$ )

Population parameters are denoted using Greek symbols ( $\alpha$ ,  $\beta$ ,  $\ldots$ ) whereas sample values are denoted using Roman symbols ( $a$ ,  $b$ ,  $\ldots$ ). Sample statistics are then denoted by adding an accent, ligature or special mark to the Roman symbol, such as adding a bar ( $\bar{\phantom{a}}$ ) above the symbol to indicate the mean (e.g.  $\bar{x}$  is the mean of the  $x$ ‘s).

These conventions, of course, preceded the advent of modern typesetting and certainly preceded the invention of the word processor. Many mathematicians, statisticians and scientists find that most current word processors (such a Microsoft’s Word and Apple’s Pages) make it difficult or cumbersome to typeset mathematics and hence many choose to use  $\LaTeX$  (as I am doing in this website) to typeset mathematical symbols and formulae. Unfortunately  $\LaTeX$ , whilst Open Source, immensely powerful and widely used, takes a little more effort to produce your document and so is not as readily used by those not needing to often typeset mathematical formulae. As such, Microsoft’s Word and Apple’s Pages tend to be accepted formats for many publications despite the limitations in typesetting mathematics.

## Publishing Statistics

Nearly all professional publications give their contributors guidelines on how to format their manuscripts to make them acceptable for publishing. In many cases, the publisher requires the author to follow certain standards for things such as referencing (e.g. Harvard). If the publication is allied with or related to Psychology, it will most likely ask the author to follow the American Psychological Association (APA) publication manual (currently in 6th edition).

The APA publication manual sets out guidelines for authors on not only how to reference, but also on the appearance of tables, text and even the appropriate formatting of headings. The manual is nearly 300 pages long (the 5th edition was over 400 pages long), but gives an invaluable guide to authors on how to write scholarly articles. Included in the many things this publication manual covers is a guide on the presentation of statistics. One can imagine that the original draftees of the APA Publication Manual found it just as difficult or cumbersome to typeset mathematics and so opted to simplify many of the common symbols. This attempted simplification has in fact led to some confusion, particularly for undergraduate students undertaking statistics courses when studying psychology as I well know from my own experience of teaching such students.

## Translating Statistics into APA Format

To help you translate between statistical symbology and the APA equivalent, the following table outlines some of the most common areas of confusion.

Statistical Symbol APA Format Meaning
$\bar{x}$  M Sample Mean (arithmetic mean)
$\mu$  M Population Mean
$\tilde{x}$  Mdn Sample Median
$\eta$  Mdn Population Median
$s$  SD Sample Standard Deviation
$\sigma$  SD Population Standard Deviation
$s^{2}$    Sample Variance
$\sigma^{2}$    Population Variance
$H_{0}$  H0 Null Hypothesis
$H_{A}$  H1 Alternate Hypothesis
$H_{1}$    First Alternate Hypothesis (used in Bayesian Analyses)
$n$  N Sample Size
$n_{1}$  n Sample Size for group 1 (or Sub-sample 1)
$n$    Population Size (typically unknown or assumed infinite)

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