Coefficient of variation from duplicate measurements

Command:

Statistics Method comparison & evaluation Coefficient of variation from duplicate measurements

Description

The calculation of Coefficient of Variation (CV) from duplicate measurements made on a number of different subjects or materials is used to determine the reproducibility of the measurements as an alternative to making a large number of observations on a single subject or material to estimate the within-run imprecision directly (Jones & Payne, 1997).

Required input

In the dialog box you select the variables that contain the data for the two measurements. The order is not important.

You can select one of three methods to calculate the Coefficient of Variation:

• Root mean square method
• Logarithmic method
• Within-subject standard deviation method

The Root mean square and Logarithmic methods allow the calculation of a confidence interval for the CV and are the recommended methods.

The Within-subject standard deviation method can only be used when the standard deviation can be assumed reasonable constant across the concentration interval.

Results

With n being the number of data pairs and x₁ and x₂ duplicate measurements, the overall Mean is given by:

$$ \text{Mean} = \frac {\sum_{}^{}{(x_1+x_2)} } {2n} $$

Root mean square method

In this method (Hyslop & White, 2009), the CV is calculated as:

$$ \text{CV(%)} = 100 \times \sqrt{\frac{\sum_{}^{}{(d / m)^2}}{2n}} $$

where d is the difference between two paired measurements and m is the mean of paired measurements.

The Root mean square method cannot be used when the mean of one or more pairs of measurements is 0.

For the calculation of the 95% Confidence Interval see Bland, 2006.

Logarithmic method

In this method (Bland & Altman, 1996; Bland, 2006), first the sum of squared differences between the logarithms of observations is calculated:

$$ sl = \sum_{}^{}{(\ln(x_1)-ln(x_2))^2} $$

Next, the CV is calculated as:

$$ \text{CV(%)} = 100 \times ( \exp{\sqrt{\frac{sl}{2n}}}-1 ) $$

The Logarithmic method cannot be used when any value is 0 or negative.

Within-subject standard deviation method

The within-subject standard deviation is given by (Jones & Payne 1997; Synek 2008):

$$ \text{SD} = \sqrt{\frac{\sum_{}^{}{(x_1-x_2)^2}}{2n}}$$

The coefficient of variation is the standard deviation divided by the mean (× 100):

$$ \text{CV(%)} = 100 \times \frac{\text{SD}}{\text{Mean}} $$

In this method, no confidence interval is reported.

The Within-subject standard deviation method cannot be used when the overall mean of measurements is 0.

Literature

• Bland M (2006) How should I calculate a within-subject coefficient of variation? https://www-users.york.ac.uk/~mb55/meas/cv.htm
• Bland M, Altman DG (1996) Statistics Notes: Measurement error proportional to the mean. British Medical Journal 313:106.
• Hyslop NP, White WH (2009) Estimating precision using duplicate measurements. Journal of the Air & Waste Management Association 59:1032-1039.
• Jones R, Payne B (1997) Clinical investigation and statistics in laboratory medicine. London: ACB Venture Publications.
• Synek V (2008) Evaluation of the standard deviation from duplicate results. Accreditation and Quality Assurance 13:335-337.

See also

• Sample size calculation for coefficient of variation from duplicate measurements