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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)  ### Definition

The coefficient of variation of the observations is used to describe the level of variability within a population independently of the absolute values of the observations. If absolute values are similar, populations can be compared using their standard deviations. But if they differ markedly (for example, the weights of mice and elephants), or are of different variables (for example, weight and height), then you need to use a standardized measure - such as the coefficient of variation.

The coefficient of variation (CV) for a sample is the standard deviation of the observations divided by the mean:

#### Algebraically speaking -

 Coefficient of variation of the observations (CV) = s  Where:
• s is the standard deviation of the sample,
• is the mean of the sample.
More usually, the coefficient of variation is presented as a percentage by multiplying it by 100 :  CV (%) = 100 s The estimate we get from this equation is a biased estimator of the population coefficient of variation. Hence we should use the following equation to give CVcor, which is the sample CV corrected for bias:

#### Algebraically speaking -

 CVcor = ( 1 + 1 ) × CV 4n
where:
• n is the number of observations in your sample

#### Note :

When you use a sample standard deviation to estimate the population coefficient of variation, you should not correct the sample standard deviation for bias, as this correction is included in the equation above. It only differs slightly from the formula for correcting the standard deviation for bias. ### Intra-assay and inter-assay coefficients of variation

These are frequently used by microbiologists to describe the precision of an assay (such as an ELISA) when applied to portions of the same specimen. In other words, each specimen is divided into a number of aliquots, and the set of aliquots are subject to an identical procedure - in order to measure the reliability of that procedure. Intra-assay and inter-assay precision are two distinct measures of this. The formulae used for these are given below:

• #### Intra-assay precision

Intra-assay precision reflects variability among replicate determinations within the same assay run. Each specimen is divided into a set of (usually n = 2) aliquots - and a number of sets of aliquots (N) are assayed on the same ELISA plate - in other words, a total of N × n aliquots. The intra-assay coefficient of variation or within assay coefficient of variation is best calculated by working out the coefficient of variation for each set of n aliquots, squaring them, taking the mean, and then taking the square root of the mean. This has been termed the root mean square approach. The intra-assay coefficient of variation is given by this formula:

#### Algebraically speaking -

 CVintra = √ [ΣCVi2 /N]
where :
• CVi2 is the squared coefficient of variation of the set of measurements on specimen i, or [si / i]2 - where si2 is the variance of the ith set, and i is its mean,
• N is the number of specimens.

An alternative method is to divide the within-assay standard deviation (given by the square root of the mean of the individual variances - see ) by the overall mean as follows:

#### Algebraically speaking -

 CVintra = √ [Σsi2 / N]  where :
• si2 is the variance of replicated measurements upon the ith specimen,
• n is the number of specimens,
• is the overall mean of all the measurements.

Despite being widely used, this second method has been criticized by a number of statisticians - it is therefore safer to use the first method. Some researchers take the mean or median of the individual CVs, or the mean of the individual standard deviations divided by the overall mean. However, both these methods will give biased estimates, and are not recommended. Yet another method (known as the log method) is to calculate the within subject standard deviation of the log transformed data; the coefficient of variation is then given by the antilog of this quantity minus one. This apparently gives a very similar estimate to the root mean square method.

• #### Inter-assay precision

This estimates how the mean precision varies with time. Here a number of aliquots (n) of the same specimen are assayed on (M) different days. For each day, the mean of that subset of n results is calculated - then the standard deviation of those M means. The inter-assay coefficient of variation, or between assay coefficient of variation is then calculated from the formula:

#### Algebraically speaking -

 CVinter = √(ns2)  i
where :
• n is the number of aliquots run each day,
• s2 is the variance of the mean of each day's results,
• is the overall mean of specimen i.

Be aware that, where n is assumed to be 2, many authors omit this term. Also, if results from a number of different specimens (n) need to be combined, rather than using the above formula with their overall mean, it is better to use the root mean of the square of their individual CVinter (i)'s - in other words: √Σ[CVinter (i)2/n] - where CVinter (i) is the inter-assay coefficient of variation of the ith specimen.   ### Uses:

1. #### As a measure of precision

The most common use of the coefficient of variation is to assess the precision of a technique.

Target values for intra- and interassay coefficients of variation are generally 5% and 10% respectively. For assays conducted over long period, coefficients of 7% and 15% are more typical. If the intra-assay coefficient of variation exceeds 10% or the interassay coefficient of variation exceeds 20%, then it is time to identify the source of the variation.

Similarly, when agricultural field trials are carried out on maize, the coefficient of variation of the yield is usually between 5 and 15%. If a researcher gets a markedly larger figure than this in a yield experiment, he should investigate the reasons. 2. #### As a measure of variability when the standard deviation is proportional to the mean

The most widely used measure of variability when the standard deviation is proportional to the mean is the standard deviation of the log transformed observations. This generally removes the dependence of the variance on the mean. But it has the disadvantage that when there are zeros in the data, one has to add a constant (usually 1) prior to taking logarithms. This results in a serious underestimate of the true value of the standard deviation.

The coefficient of variation largely overcomes these problems. It is independent of the mean, and is unaffected by zeros - although if the mean value is near zero, the coefficient of variation is unduly sensitive to small changes in the mean.

3. #### As a means to compare variability of measurements made in different units

For example, in animal productivity studies one may wish to assess which of a wide range of variables (such as milk yield, time to first calving, and age at maturity) is most variable. Since these are measured in different units, the measure of variability must be standardized. The coefficient of variation achieves this by dividing the standard deviation by the mean.   ### Assumptions and requirements

The coefficient of variation is only applicable for measurement (continuous or discrete) variables where measurements are made on a ratio scale. In other words the scale should have a non-arbitrary zero value. It is not appropriate for variables which can take negative values. It should also only be calculated on untransformed data. For it to function as a standardized measure, the standard deviation should be directly proportional to the mean. When being used as a measure of reliability - for example if you are estimating the repeatability of measurements on different individuals - it is assumed that these measurements all vary to a similar degree after allowing for the difference between their means.