There is no point reaching for a calculator that cannot show its working. This one starts from x where smp(x) = max (ml), maximum of the model (max), lower limit value of x (pxmin), upper limit value of x (xmax), power (p₁), power (p₂), x₁, x₁, x₁, x₂ and for an independent variable x = and reports p (x₁ < x < x₂), p (x > x₁) and p (x < x₁), so you can see how each answer follows from the numbers you entered.
What Is the SMp(x) Calculator?
The smp(x) calculator turns a set of measurements into p (x₁ < x < x₂), p (x > x₁) and p (x < x₁), with the uncertainty made explicit rather than left implied.
Statistics is easy to quote badly. A figure without its spread tells the reader far less than they assume it does, and this calculator puts both together.
How the SMp(x) Is Calculated
The calculation uses the standard relationship for this kind of estimate, applied to the figures you supply. The intermediate steps are exposed so you can see how much of the answer comes from each one.
Sample size matters more than most people expect. Doubling the number of observations improves the precision by a factor of about one and a half, not two.
What the Result Does and Does Not Tell You
A result describes the data you gave it. It does not tell you whether the sample was representative, whether the method was appropriate, or whether there is a confound in the way the figures were collected.
Report the result together with its uncertainty. A number on its own is much easier to misread than a number with a range attached.
A Worked Example
Working through an example, with x where smp(x) = max (ml) set to 10, maximum of the model (max) set to 10, lower limit value of x (pxmin) set to 10, the calculator returns smp(x) of -, and p (x > x₁) of -, and p (x < x₁) of -. Substitute your own figures and the same steps produce the same kind of answer.
Frequently Asked Questions
What do I need to work out p (x₁ < x < x₂)?
Enter x where smp(x) = max (ml), maximum of the model (max), lower limit value of x (pxmin), upper limit value of x (xmax), power (p₁), power (p₂), x₁, x₁, x₁, x₂ and for an independent variable x =. The calculator then works through the relationship and reports p (x₁ < x < x₂), p (x > x₁) and p (x < x₁).
What does the smp(x) actually calculate?
It takes x where smp(x) = max (ml), maximum of the model (max), lower limit value of x (pxmin), upper limit value of x (xmax), power (p₁), power (p₂), x₁, x₁, x₁, x₂ and for an independent variable x = and converts them into p (x₁ < x < x₂), p (x > x₁) and p (x < x₁) using the standard relationship for this problem. Each step is applied in turn, so every output is traceable back to the numbers you typed in.
How should I enter x where smp(x) = max (ml)?
Use the value you actually have, in the unit shown next to the field. Enter every value in the same unit system. The calculator assumes the figures you supply are consistent, so it cannot correct a mistake that happens before the calculation starts.
Why does a small change in one value move the answer so much?
That happens when an output depends on a ratio, because dividing by a small number magnifies any change in it. Check the units first, then check whether the value you entered is in the range the relationship was designed for.
Do I have to fill in every field?
Only the fields a particular result depends on. If you leave one blank, the results that need it simply show a note instead of a number, and everything else still works.