qPCR Standard Curve & Efficiency Calculator
Enter the mean Cq values from a serial dilution to fit a qPCR standard curve. The calculator reports the slope, amplification efficiency, R squared, residual error, and an estimated quantity for an unknown sample.
Amplification efficiency100%
- Standard-curve slope
- -3.32
- Y-intercept
- 34.93
- Coefficient of determination
- 1
- Per-cycle amplification factor
- 2
- Expected Cq spacing per dilution
- 3.32
- Residual standard error
- 0
- Estimated unknown quantity
- 976.71
- Quantification status
- Interpolated within the standard curve range (10 to 1,000,000).
Standard curve fit
| Standard | Quantity | log10 quantity | Mean Cq | Fitted Cq | Residual |
|---|---|---|---|---|---|
| 1 | 1,000,000 | 6 | 15 | 15 | 0 |
| 2 | 100,000 | 5 | 18.32 | 18.32 | 0 |
| 3 | 10,000 | 4 | 21.64 | 21.64 | 0 |
| 4 | 1,000 | 3 | 24.97 | 24.97 | 0 |
| 5 | 100 | 2 | 28.29 | 28.29 | 0 |
| 6 | 10 | 1 | 31.61 | 31.61 | 0 |
Assumes each standard Cq is the mean of comparable technical replicates.
Quantities use the same units as the highest standard quantity.
How to use this calculator
- Choose how many consecutive standards from the dilution series to include.
- Enter the highest standard quantity and the constant serial dilution factor.
- Enter the mean Cq for each included standard in order from most concentrated to least concentrated.
- Enter the unknown sample mean Cq to estimate its quantity from the fitted curve.
- Review the efficiency, R squared, residual error, and interpolation status together.
How the standard curve is fitted
A qPCR standard curve relates measured Cq to the base-10 logarithm of standard quantity. For a constant serial dilution, the calculator builds each standard quantity from the highest standard quantity and the dilution factor, then fits a least-squares line through the entered mean Cq values.
For standards i = 0 through n - 1, the quantity is:
Q_i = Q_0 / D^i
The x value for the regression is x_i = log10(Q_i), and the y value is the standard's mean Cq. With xbar and ybar as the means, the fitted slope and intercept are:
m = sum((x_i - xbar)(y_i - ybar)) / sum((x_i - xbar)^2)
b = ybar - m * xbar
Each fitted Cq is yhat_i = m * x_i + b. The calculator reports R^2 = 1 - SSE / SST, where SSE is the sum of squared residuals and SST is the total Cq sum of squares. Residual standard error is sqrt(SSE / (n - 2)), so it has units of Cq cycles.
How efficiency and unknown quantity are calculated
The amplification factor per cycle comes from the slope:
A = 10^(-1 / m)
Amplification efficiency is then:
E = 100 * (A - 1)
A slope near -3.322 gives an amplification factor of 2.0 and an efficiency of 100%. The expected Cq spacing for the selected dilution factor is -m * log10(D). For a tenfold dilution and an ideal slope, that spacing is about 3.322 cycles.
The unknown sample is quantified by inverting the fitted line:
Q_unknown = 10^((Cq_unknown - b) / m)
The calculator labels the estimate as interpolated when the unknown Cq falls inside the fitted Cq range of the standards. It labels the estimate as extrapolated when the unknown Cq is outside that range.
What moves the result most
The slope is driven by the Cq spacing across the dilution series. Wider spacing produces lower apparent efficiency, while narrower spacing can produce efficiency above 100%. Outlier standards can move both the slope and the residual error, especially when only three standards are included.
Many qPCR workflows review efficiency near 90% to 110% and R squared near or above 0.99, but those checks are not enough by themselves. A low residual standard error helps show that the individual standards agree with the fitted line rather than merely producing a favorable slope.
What this calculator leaves out
This tool uses the mean Cq values you enter. It does not inspect replicate spread, pipetting records, no-template controls, melt curves, primer specificity, inhibition, limit of detection, limit of quantification, or plate effects. An excellent mathematical fit can still describe an assay that is biologically or experimentally unsuitable.
Worked example
Suppose a six-point tenfold series starts at 1,000,000 copies, with mean Cq values of 15.000, 18.322, 21.644, 24.966, 28.288, and 31.610. Fitting Cq against log10 quantity gives a slope of about -3.322 and an intercept of about 34.932.
The amplification factor is 10^(-1 / -3.322), or about 2.000 per cycle, so the efficiency is about 99.99%. R squared is 1.000 for these evenly spaced values, and the residual standard error is 0.000 Cq. An unknown Cq of 25.000 inverts to about 976 copies in the same units as the standard quantity, and it is interpolated because its Cq lies inside the fitted standard range.
Common questions
How is qPCR efficiency calculated from a standard curve?
Fit Cq as the y value against log10 quantity as the x value, then use the fitted slope. The per-cycle amplification factor is 10 raised to -1 divided by the slope, and efficiency is 100 times that factor minus 1. A slope near -3.322 corresponds to 100% efficiency.
What slope represents 100% PCR efficiency?
A slope of about -3.322 represents exact doubling each cycle when Cq is plotted against log10 quantity. A more negative slope means the apparent efficiency is below 100%. A less negative slope means the apparent efficiency is higher and should be reviewed carefully.
What are acceptable qPCR efficiency and R squared values?
Many assay reviews look for efficiency near 90% to 110% and R squared near or above 0.99. Those are screening checks, not proof that an assay is valid. Replicate variation, specificity, controls, and the intended quantification range also matter.
Should Cq be plotted against concentration or log concentration?
Use the logarithm of concentration or copy number for the x axis. qPCR amplification is exponential, so Cq changes roughly linearly with log10 quantity across a valid dilution series. Plotting raw quantity usually bends the relationship and gives the wrong slope.
Can copies, mass concentration, and relative quantity all be used?
Yes, as long as every standard uses the same unit basis and the dilution factor is constant. The estimated unknown quantity inherits the unit used for the highest standard. This calculator does not convert between copies, mass concentration, molarity, and relative quantity.
Can an unknown sample be quantified outside the standard range?
The fitted line can be inverted outside the standards, but that result is extrapolated. Extrapolated quantities are less reliable because the assay has not been demonstrated to behave linearly there. A diluted or concentrated rerun that lands inside the curve is usually easier to defend.