How to use
- Enter the standards into the table — concentration (x) and measurement (y). Two columns copied from a spreadsheet can be pasted straight in, and a header row is filtered out automatically.
- The slope, intercept, R² and the equation are calculated, and the graph is drawn alongside.
- Enter the measurement from an unknown below and its concentration is read back off the line. Several at once is fine.
- The intercept can be forced through zero (y = ax) where that is appropriate.
- Enter titles for the X and Y axes and they carry through the table, the graph and the back-calculation. Optional.
- The x axis can be switched to a logarithmic scale. The regression is unchanged; only the display differs.
Exponents can be entered with e — for example, 1.5×10⁻⁵ is entered as 1.5e-5.
Formula & notes
A standard curve fits a line to samples of known concentration and then reads unknown concentrations back off it. It is the basic procedure behind Bradford protein assays, ELISA and qPCR.
The line comes from least squares: the line for which the sum of the squared vertical distances from the points is smallest.
- Slope
- a = Σ(xᵢ − x̄)(yᵢ − ȳ) ÷ Σ(xᵢ − x̄)²
- Intercept
- b = ȳ − a × x̄
- Equation
- y = ax + b
R², the coefficient of determination, states on a scale of 0 to 1 how closely the data follow the line. The nearer to 1, the closer the points sit to it.
Reading concentrations back is what the tool is actually for. Given a measurement y, the concentration x comes from this.
- Back-calculation
- x = (y − b) ÷ a
Extrapolation warning
A back-calculated concentration outside the x range of the standards is flagged.
The line has only been demonstrated across the range the standards covered. Nothing guarantees it continues beyond that, and in practice most assays saturate at high concentration and bend away from the line. Treat a value outside the range as indicative, and either dilute the sample or extend the standards and measure again.
Practical notes
- Use at least three standards, and preferably five or more. With few points, a single outlier moves the whole line.
- Design the assay so the unknowns fall inside the standard range. Where they do not, the answer is to dilute and measure again.
- A high R² does not establish that the relationship is linear. Curved data can return an R² above 0.99. Look at the graph and check that the points scatter evenly about the line; a systematic bend means the linear model does not fit.
- Force the intercept through zero only where there is a reason to. Data with the blank already subtracted, which should pass through the origin in theory, is the usual case. Turning it on to improve R² is a misuse.
- The logarithmic scale changes the display only. The regression and every calculated result stay the same. A straight line looking curved on a log axis is expected. Points at x of zero or below cannot be plotted there and are excluded, with a note of how many.