Calculate the Coefficient of Variation (Video)

0:15 Introduction

The coefficient of variation is the standard deviation of a set of measurements divided by their mean, multiplied by 100 and reported as a percentage, so it expresses spread as a fraction of the size of the measurement. The video has nine parts.

0:32 Why spread needs context

First, we will look at why a spread in replicate data needs context. Replicate measurements of the same sample rarely agree exactly, and whether a disagreement matters depends on the size of the measurement. For body weight, two grams is a large variation for a mouse that weighs 25 grams, at 8 percent of its mass, and a negligible variation for an elephant that weighs 4,000 kilograms, at 0.00005 percent. To compare variability across such different scales, Karl Pearson introduced the coefficient of variation in 1896.

1:22 Mean, deviation, variance and standard deviation

Second, we will define the mean, the deviation, the variance and the standard deviation. The coefficient of variation is calculated from the mean and the standard deviation, so you need both quantities defined first. The mean is the sum of the values divided by their number, and a deviation is the difference between one value and the mean. Squaring each deviation keeps positive and negative deviations from cancelling when you add the squared values. The variance is the sum of the squared deviations divided by n minus 1 for a sample, and the standard deviation is its square root, which carries the units of the data. Take the three values 4, 6 and 8. The mean is 6. The deviations are minus 2, 0 and plus 2, and their squares are 4, 0 and 4. The sum of the squares is 8. Divide by n minus 1, which is 2, and the variance is 4. The standard deviation is the square root of 4, which is 2. A sample divides by n minus 1 because the sample mean sits closer to the sample values than the true mean does, and the correction removes that bias.

2:31 Why the standard deviation alone is not enough

Third, we will see why the standard deviation alone is not enough. Because the standard deviation carries the units of the data, it cannot compare spreads of measurements in different units or at different levels. Five mice with body weights of 23, 25, 27, 24 and 26 grams have a mean of 25 grams and a standard deviation of 1.58 grams. Plasma glucose from the same five animals, at 132, 158, 145, 170 and 120 milligrams per deciliter, has a mean of 145 milligrams per deciliter and a standard deviation of 19.9 milligrams per deciliter. One standard deviation is in grams and the other is in milligrams per deciliter, so you cannot set the two numbers side by side. Divide each standard deviation by its own mean and the comparison becomes possible. The body weights give 6.3 percent and the glucose gives 13.7 percent. The glucose measurements vary about twice as much relative to their mean.

3:38 The coefficient of variation formula

The fourth part builds the coefficient of variation formula. Dividing the standard deviation by the mean removes the units, because both quantities carry the same units. The coefficient of variation, abbreviated CV, equals the standard deviation divided by the mean, multiplied by 100 to give a percentage. A CV of 6.3 percent says that the standard deviation is 6.3 percent of the mean. You can therefore compare a CV computed from grams with a CV computed from milligrams per deciliter. Report the mean and the number of replicates alongside the CV, since a percentage alone does not show the size of the measurement or the precision of the estimate.

4:21 Worked example with an ELISA triplicate

Fifth, we will work through an example with an ELISA triplicate. To apply the formula, take one sample loaded into three replicate wells of an ELISA plate, with illustrative optical densities of 0.82, 0.85 and 0.79. The mean is the sum, 2.46, divided by 3, which gives 0.82. The deviations are 0, plus 0.03 and minus 0.03. Their squares are 0, 0.0009 and 0.0009, and the sum of the squares is 0.0018. Divide by n minus 1, which is 2, and the variance is 0.0009. The standard deviation is the square root of 0.0009, which is 0.03. Divide 0.03 by the mean of 0.82 and multiply by 100. The CV is 3.66 percent, which you report as 3.7 percent.

5:26 Sample or population standard deviation

Sixth, we will compare the sample and the population standard deviation. Spreadsheet software offers two formulas for the standard deviation, and with three replicates the choice changes the answer. The sample formula divides the sum of the squared deviations by n minus 1, which is 2 here. The population formula divides by n, which is 3. With the population formula the standard deviation is 0.0245 and the CV is 2.99 percent, against 3.66 percent from the sample formula. The population formula gives a CV about a fifth lower. Laboratory practice uses the sample standard deviation, so state the formula you used when you report a CV.

6:11 Intra-assay and inter-assay CV

Seventh, we will separate intra-assay from inter-assay precision. Replicates on one plate show the precision within a run, and the same control measured across days shows the precision between runs. The intra-assay CV is the CV of replicates within one run or plate. The inter-assay CV is the CV of the same control across separate runs, plates or days. Consider a control sample measured in triplicate on three plates on three days. Run 1 reads 1.02, 1.05 and 0.99, with a CV of 2.94 percent. Run 2 reads 1.10, 1.12 and 1.07, with a CV of 2.29 percent. Run 3 reads 0.95, 0.98 and 0.96, with a CV of 1.59 percent. The mean intra-assay CV is 2.27 percent. The run means are 1.020, 1.097 and 0.963, and their CV is 6.52 percent, which is the inter-assay CV. The inter-assay CV is usually larger because it adds day-to-day sources such as reagent lots, calibration, temperature and operator. Common guidance for immunoassays sets the intra-assay CV below about 10 percent and the inter-assay CV below about 15 percent. Each assay sets its own limits during method validation, so treat those figures as guidance.

7:46 When the CV misleads

Eighth, we will look at cases where the CV misleads. The CV is a ratio, so it works only when the mean itself is meaningful as a size. A ratio scale has a true zero, such as mass or concentration, and ratios on such a scale carry meaning. An interval scale does not have a true zero, and a ratio of two readings on an interval scale depends on the unit you chose. Temperatures of 2, 4 and 6 degrees Celsius give a CV of 50 percent. The same three temperatures in kelvin, at 275.15, 277.15 and 279.15, give 0.72 percent. The physical spread is the same in both units, yet the CV changes, because zero on the Celsius scale is arbitrary. A mean near zero makes the CV unstable. Values of 0.01, 0.03 and 0.05 give a CV of 67 percent, although the standard deviation is only 0.02. The denominator is small, so a small change in the mean moves the percentage a long way. On a logarithmic scale the CV also misleads, because Ct values of 24.1, 24.3 and 24.2 in a qPCR run give a CV of 0.41 percent although one Ct cycle corresponds to about a twofold difference in template. Laboratories therefore judge Ct replicates by their standard deviation, which is 0.10 cycles here and corresponds to a difference in template of about 7 percent.

9:26 Troubleshooting a high CV

The last part covers troubleshooting a high CV. When replicates exceed the acceptance limit, the cause usually lies in how you prepared or read the wells. Start with the pipette, which you calibrate and use with a consistent technique. Pre-wet each tip, and change tips between samples. Mix samples and reagents thoroughly, and remove bubbles before you read the plate. Wells at the plate edge evaporate and warm differently from the rest of the plate, so keep samples away from the edge or seal the plate and incubate it evenly. Add reagents in the same order and at the same pace, and bring reagents to room temperature first. A deviating well needs investigation against the run record before you exclude it, and you report any exclusion.

10:11 Recap and quiz

Here is what you learned in this video. Replicates differ by some amount, and the mean and the standard deviation give their center and their spread. The coefficient of variation divides the standard deviation by the mean and reports the result as a percentage, which removes the units and allows comparison across scales. Replicates within a run give the intra-assay CV, and the same control across runs gives the inter-assay CV. The CV misleads on interval scales such as Celsius and on logarithmic values such as Ct, and it becomes unstable when the mean is near zero. Here is a question for you to test what you learned. One sample gives triplicate optical densities of 1.10, 1.14 and 1.62, and the acceptance limit is 10 percent. Calculate the CV and decide whether the replicates pass. Pause the video and work it out. The answer follows. The mean is 1.29, the standard deviation is 0.289, and the CV is 22.5 percent, so the replicates fail the 10 percent limit and the reading of 1.62 needs investigation.