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Usually when you are doing your masters, what are the minimum points that you can have when doing a calibration curve?

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There is not really a simple answer to this—it depends on what you're doing and what level of error your application can tolerate. If you've characterized the response of your instrument and you know that it's linear within a certain range but the response has a variable offset, even a single-point calibration may be sufficient. On the other hand, if you just built an instrument and have no idea of its response characteristics but need very accurate results, you're best to take measurements at many different points (and over a length of time to see how stable the response is) to accommodate the possibility of non-linear response in the range of interest.

Occasionally, if you need to follow a measurement standard (e.g. the ITS-90 standard for temperature), you may be constrained to doing it a certain way so your data can be directly compared with that of other labs but in general, do enough to get the results you need, but not so much that you're just wasting your time. This might mean taking a few well-separated calibration points to get a ball-park value for an unknown sample, then bracketing the value with more calibration points near it to get a better measurement.

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Your question contains very little about the analytical method you deploy; hence I agree with the answer by Michael "it depends on what you aim for".

On the other hand: assuming you know (almost) nothing about the relationship between the concentration of the analyte (like an $x$ in mathematics), on one side, and the response by the method deploy (like $f(x)$ in mathematics) I recommend to collect data for at least five different analyte concentrations that are not "too close to each other" to construct your calibration curve.

Why at least five? For me it appears to be the smallest number of data points to figure out if there is a linear relationship between $x$ and $f(x)$, like $f(x) = ax + b$ with a direct proportion factor $a$ (slope) and an absolute term of $b$ (intersection with the ordinate) including to calculate an estimate for the standard deviation of them and a regression coefficient.

But perhaps the relationship between $f(x)$ and $x$ is not strongly linear and another model fits better? This could be spotted for example by plotting the data in a diagram, where the attempt to draw a straight line across all of them with a ruler fails. Then, with five different values on your abscissa you may attempt transformations into a linear form and check if this significantly fits better For example:

  • $f(x) = y = a \cdot e^{bx}$ transformed into a form of: $\lg y = \lg{a} + bx$
  • $y = a^{bx}$ transformed into a form of: $\lg{y} = bx \cdot \lg{a}$
  • $y = a \cdot x^b$ transformed into a form of: $\lg{y} = \lg{a} + b \cdot \lg{x}$

This still is limited (on a shoestring) and of course with the limitation to cover only about the region covered by the data, being descriptive rather than predictive. Collecting more data (both repeated at one abscissa value, as well as at different abcissa values) may then become the next thing to do to support your analytical model.

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