Example, with R
The standard deviation is the Root of the Mean Squared-deviation (or RMS deviation) from the mean - assuming your values contain the entire 'population' of interest. In other words it summarizes variation from their mean.
Given which the standard deviation of these 6 values equals 1:
1   -1   -1   1   -1   1
the mean of 1 -1 -1 1 -1 1 is 0, so these values are also their deviations from that mean.
Their squared deviations are equally simple since -12 equals 12 equals 1
So the mean deviation is 1, and the square-root of 1 equals 1.
You could find this population standard deviation with R
Definition and Use
- The standard deviation of y is a measure of variation in y, and is usually the square root of y's variance.
- The root mean squared deviation from the mean is generally known as the 'population standard deviation'.
- The population standard deviation (along with the population mean) are all that are needed to define 'normal' populations - and their 'true' values are known as population parameters.
- But the population standard deviation formula, when applied to random samples of infinite populations, gives a biased estimate of that population's standard deviation - on average.
- For random samples a modified formula, known as the 'sample standard deviation', is used in which the sum of the squared deviations from the mean is divided by (n - 1) rather than n. This gives a less-biased estimate of that parameter.
- Nevertheless, the population standard deviation formula is the maximum-likelihood estimator.
The sample standard deviation is often used (sometimes unwisely) to indicate variation of samples.
Assuming y contains n numbers, and that e (the deviation of each value from the mean) equals y - mean(y):
the 'population' standard deviation of y is squareroot(sum(e2)/n)
Note, sum(e2)/n is known as the mean squared error, or 'population variance'.
the 'sample' standard deviation of y is squareroot(sum(e2)/(n-1))
Tips and Notes
- Owing to its importance in simple statistical models, standard deviations are exceedingly popular.
- But summarizing data using its mean +/-sd is only meaningful if values are approximately normal, or at least more-or-less symmetrically distributed about the mean. When applied to strongly-skewed data, the standard deviation (sd) gives a misleading picture of sample variation!
- For real data other measures of range, such as interquartile range(the upper and lower quartiles), make fewer assumptions, are more robust and are often much less misleading.
With R you can easily convert the results of sample standard deviation formulae to RMS deviation.
- Altman, D.G. & Bland, J.M. (2005). Standard deviations and standard errors BMJ 351, 903 (15 October). Full text
- Argues that the standard deviation is a valid measure of variability regardless of distribution.
- Anderson, D.R. et al. (2001). Suggestions for presenting the results of data analyses. Journal of Wildlife Management 65 (3), 373-378. Full text
- Provides 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!
- Stephen Gorard (2004). Revisiting a 90-year-old debate: the advantages of the mean deviation. Paper presented at the British Educational research Association Annual Conference, University of Manchester, 16-18 September, 2004.Full text
- Do we really have to use the standard deviation to quantify variation?
- Lehmann, K.G. et al. (1996). Contributions of frequency distribution analysis to the understanding of coronary restenosis. A reappraisal of the Gaussian curve. Circulation 93 (6), 1123-1132. Full text
- Point out that whilst the mean and standard deviation are appropriate if a variable has a normal distribution, populations with skewed distributions cannot be adequately represented in this way.
- Stark, P.B.. Measures of location and spread. University of California Full text
- A comprehensive and thought-provoking account of measures of location and spread.
- Wikipedia Standard Deviation Full text