Variations are biological, measurement, or temporal. Time series analysis relates biological
to temporal variation. Analysis of variance (ANOVA) relates biological variation (inter- or between subject) to measurement
variation (intra- or within subject) variation. Biological variation is more common than measurement variation.
Temporal variation is measured in calendar time or in chronological time.
Measures of variation can be classified as absolute (range, inter-quartile range, mean
deviation, variance, standard deviation, quantiles) or relative (coefficient of variation and standardized z-score).
Some measures are based on the mean (mean deviation, the variance, the standard deviation,
z score, the t score, the stanine, and the coefficient of variation) whereas others are based on quantiles (quartiles, deciles,
2.0 MEASURES OF VARIATION BASED ON THE MEAN
2.1 Mean deviation is the arithmetic
mean of absolute differences of each observation from the mean. It is simple to compute but is rarely used because it is not
intuitive and allows no further mathematical manipulation.
2.2 The variance is the sum of
the squared deviations of each observation from the mean divided by the sample size, n, (for large samples) or n-1 (for small
samples). It can be manipulated mathematically but is not intuitive due to use of square units.
2.3 The standard deviation, the
commonest measure of variation, is the square root of the variance. It is intuitive and is in linear and not in square units.
The standard deviation is the most popular measure of variation.
The standard deviation, s,
is from a population but the standard error of the mean, s, is from a sample with s being more precise and smaller than s. The relation between the standard deviation, s,
and the standard error, s, is given by the expression s = s /(n-1) where n
= sample size.
The percentage of observations covered by mean +/- 1 SD is 66.6%, mean +/-
2 SD is 95%, and mean +/- 4 SD virtually 100%.
The standard deviation has the following advantages: it is resistant to sampling
variation, it can be manipulated mathematically, and together with the mean it fully describes a normal curve. Its disadvantage
is that it is affected by extreme values.
2.4 The standardized z-score defines
the distance of a value of an observation from the mean in SD units.
2.5 The coefficient of variation (CV)
is the ratio of the standard deviation to the arithmetic mean usually expressed as a percentage. CV is used to compare variations
among samples with different units of measurement and from different populations.
3.0 MEASURES OF VARIATION BASED ON QUANTILES
3.1 Quantiles (quartiles, deciles, and percentiles)
are measures of variation based on division of a set of observations (arranged in order by size) into equal intervals and
stating the value of observation at the end of the given interval. Quantiles have an intuitive appeal.
3.2 Quartiles are based on dividing observations
into 4 equal intervals. Deciles are based 10, quartiles on 4, and percentiles on 100 intervals. The inter-quartile range,
Q3 - Q1, and the semi inter-quartile range, ˝ (Q3 - Q1) have the advantages of
being simple, intuitive, related to the median, and less sensitive to extreme values. Quartiles have the disadvantages of
being unstable for small samples and not allowing further mathematical manipulation.
3.3 Deciles are rarely used.
3.4 Percentiles, also called centile scores,
are a form of cumulative frequency and can be read off a cumulative frequency curve. They are direct and very intelligible.
The 2.5th percentile corresponds to mean - 2SD. The 16th percentile
corresponds to mean - 1SD. The 50th percentile corresponds to mean + 0 SD. The 84th percentile corresponds
to mean + 1SD. The 97.5th percentile corresponds to mean + 2SD. The percentile rank indicates the percentage of
the observations exceeded by the observation of interest. The percentile range gives the difference between the values of
any two centiles.
4.0 THE RANGE
The full range is based on extreme values. It is defined by giving the minimum
and maximum values or by giving the difference between the maximum and the minimum values. The modified range is determined
after eliminating the top 10% and bottom 10% of observations. The range has several advantages: it is a simple measure, intuitive,
easy to compute, and useful for preliminary or rough work. Its disadvantages are: it is affected by extreme values, it is
sensitive to sampling fluctuations, and it has no further mathematical manipulation.
5.0 OPERATIONS / MANIPULATIONSAdding or subtracting a constant to each observation
has no effect on the variance. Multiplying or dividing each observation by a constant implies multiplying or dividing the
variance by that constant respectively. A pooled variance can be computed as a weighted average of the respective variances
of the samples involved.