value (e.g., scatter-plot) may be amenable to single CV calculation using a maximum-likelihood estimation approach. Examples In the examples below, we will take the values given as randomly chosen from a larger population of values. The data set [100, 100, 100] has constant values. Its standard deviation is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0 The data set [90, 100, 110] has more variability. Its standard deviation is 10 and its average is 100, giving the coefficient of variation as 10 / 100 = 0.1 The data set [1, 5, 6, 8, 10, 40, 65, 88] has still more variability. Its standard deviation is 32.9 and its average is 27.9, giving a coefficient of variation of 32.9 / 27.9 = 1.18 In these examples, we will take the values given as the entire population of values. The data set [100, 100, 100] has a population standard deviation of 0 and a coefficient of variation of 0 / 100 = 0 The data set [90, 100, 110] has a population standard deviation of 8.16 and a coefficient of variation of 8.16 / 100 = 0.0816 The data set [1, 5, 6, 8, 10, 40, 65, 88] has a population standard deviation of 30.8 and a coefficient of variation of 30.8 / 27.9 = 1.10 Estimation When only a sample of data from a population is available, the population CV can be estimated using the ratio of the sample standard deviation to the sample mean : But this estimator, when applied to a small or moderately sized sample, tends to be too low: it is a biased estimator. For normally distributed data, an unbiased estimator for a sample of size n is: Log-normal data In many applications, it can be assumed that data are log-normally distributed (evidenced by the presence of skewness in the sampled data). In such cases, a more accurate estimate, derived from the properties of the log-normal distribution, is defined as: where is the sample standard deviation of the data after a natural log transformation. (In the event that measurements are recorded using any other logarithmic base, b, their standard deviation is converted to base e using , and the formula for remains the same.) This estimate is sometimes referred to as the "geometric CV" (GCV) in order to distinguish it from the simple estimate above. However, "geometric coefficient of