Use an Online Calculator for Complicated Error-Propagation Expressions

Statpages calculates how precision propagates through almost any expression involving one or two variables. It even handles the case of two variables with correlated fluctuations. You simply enter the following items:

• The expression, using a fairly standard algebraic syntax (JavaScript)

• The values of the variable or variables

• The corresponding SEs

Consider the example of estimating the SE of the area of a circle whose diameter is 2.3 cm, with a SE of 0.2 cm. The formula for the area of a circle, in terms of its diameter (d) is A = (π/4)r²

Credit: Screenshot courtesy of John C. Pezzullo, PhD

The expression must refer to the variable (the diameter, in this case) as x, and the squaring of x must be indicated as x * x, because JavaScript doesn't allow x². The web page knows what the value of Pi is. It calculates an area of 4.15 square centimeters and an SE of 0.72 square centimeter.

The same web page can also analyze error propagation through expressions involving two measured values. Suppose you want to calculate body mass index (BMI, in kilogramsper square meter) from a measured value of height (in centimeters) and weight (in kilograms), using the formula: BMI = 10,000weight/height².

Suppose the measured height is 175 ± 1 centimeter, and the weight is 77 ± 1 kilograms (where the ± numbers are the SEs). The BMI is easily calculated as 10,000 x 77/175², or 25.143 kg/m². But what's the SE of that BMI?

The problem would be entered into the web page as:

Credit: Screenshot courtesy of John C. Pezzullo, PhD

The page requires that height and weight be called x and y, respectively. Enter the square of the height as (x * x), because JavaScript doesn't allow x². Enter 0 for the error-correlation term because height and weight are two independent measurements (using different instruments). The resulting BMI produced by the web page was 25.1 ± 0.4 kilogramsper square meter.