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Variance and standard deviation: Use and misuse
(Use for skewed data, corrections for bias, repeatability, within-subject standard deviation)
Statistics courses, especially for biologists, assume formulae = understanding and teach how to do statistics, but largely ignore what those procedures assume, and how their results mislead when those assumptions are unreasonable. The resulting misuse is, shall we say, predictable...
Use and Misuse
The variance provides a measure of spread or dispersion of a population. The population variance is computed as the average of the squared deviations of the observations from their mean, hence its alternative name 'mean square error'. If you take a sample, you will under-estimate the true value of the population variance. You correct for this bias by dividing by n - 1 (where n is the number of observations), rather than by n. The standard deviation of a population is simply the square root of the population variance. Similarly the standard deviation of a sample is the square root of the sample variance.
In our review of the literature we found that the standard deviation suffers from the same problem as other measures of dispersion - namely that it is usually just quoted or used to obtain a confidence interval - and then forgotten about. This is because of the tendency to focus excessively on the effects of a treatment on the mean value of the response variable, and pay no attention to the effect of a treatment on dispersion or other characteristics of the distribution. The standard deviation is also overused at the expense of other measures of dispersion. For example, its use with the arithmetic mean (as mean ▒ SD) is misleading for data with a skewed distribution. This is because errors are no longer distributed symmetrically around the mean. Similarly neither the mean nor the standard deviation are appropriate measures for ordinal variables. For skewed or ordinal data a box and whisker plot of the five quantile summary (minimum, lower quartile, median, upper quartile, and maximum) is much more informative. Often researchers admit their distribution is not remotely normal, and consequently use a non-parametric statistical test to compare medians - yet still (misleadingly) present their results as means with standard deviations.
Much more seriously, the standard deviation (of the observations) is still regularly confused with the standard deviation of the mean (also known as the standard error). The latter is an estimate of the variability of the mean. If a researcher quotes 'mean ▒ standard deviation', we do not know if the 'standard deviation' is of the observations or of the mean. In general medical researchers nearly always use the standard deviation (of the observations) as their 'default' measure of dispersion. Researchers in other disciplines may use either, which can create confusion!
Another important measure of spread is the within-subject standard deviation. This describes the random component of measurement error and hence provides a useful measure of both reproducibility (if the same test material sent to different laboratories) and repeatability (same test material analyzed by same person in same laboratory). For instance you might wish to assess repeatability of a technician for PCV readings by geting her to measure the PCV of each of ten blood samples on 5 consecutive occasions. Because the ten samples are not identical, the results you obtain will include the variation between cows - in addition to the measurement error. So simply pooling the results, and calculating the overall standard deviation, will overestimate the variation arising from measurement error. Instead the set of individual standard deviations is combined into a single, overall measure of measurement error - the within-subject standard deviation.
This measure tends to be used rather little reflecting a lack of interest on measurement error, or sometimes an apparent refusal to admit it even exists. Where within-subject standard deviation was estimated, we found that sometimes the standard deviation was not independent of the mean, which is a requirement for the validity of this measure. Under such circumstances, the data should have been transformed in an attempt to normalize distributions.
What the statisticians sayWoodward (1999) covers measures of location and dispersion in Chapter 2. He recommends the five quantile summary for general use, with the mean and standard deviation reserved for variables with a symmetrical distribution. Armitage & Berry (2002) and Bland (2000) introduces the variance and standard deviation in Chapters 2 and 4 respectively. Sokal & Rohlf (1995) cover the variance and standard deviation in Chapter 4. However, little attention is given to the best descriptive statistics to use for skewed distributions. Bart et al. (1998) introduce the standard deviation in Chapter 2.
Altman & Bland (2005) give a useful review of the difference between the standard deviation (of the observations) and the standard error (of the mean). They point out that the standard deviation is a valid measure of variability regardless of distribution - even though one may choose a different summary statistic for a skewed distribution. Lehmann et al. (1996) point out that whilst the mean and standard deviation (or standard error) are appropriate if a variable has a normal distribution, populations with skewed distributions cannot be adequately represented in this way. Bland & Altman (1996) (1) (2) (3) provide a clear account of how to use the within-subject standard deviation as a measure of repeatability. MassÚ (1997) and other letter writers comment on the assumptions made when estimating the within-subject standard deviation. Benedetti-Cecchi (2003) stresses the importance of the variance around the mean effect size of ecological processes. Anderson et al. (2001) provide a number of helpful suggestions to wildlife biologists for presenting the results of data analyses, in particular the need to distinguish between standard deviation and standard error! Good (1973) attempts to explain the meaning of the term degrees of freedom, following up the much earlier paper by Walker (1940). Wikipedia provides sections on the standard deviation, the variance and degrees of freedom. Stephen Gorard argues the advantages of the average absolute deviation over the standard deviation. Gerard Dallal takes a practical approach to explaining degrees of freedom.