# More Than You Ever Wanted to Know About Calibrations, Part 5 – Accuracy, Precision and Detection Limits

1 May 2023In previous blog posts I’ve discussed the fundamentals of calibrations, from calibration types, curve fits, zero points, and calibration acceptance. In this post we’ll talk about how calibrations affect your results, focusing especially on low-level accuracy, precision, and detection limits.

Most people probably think of calibrations as being related to accuracy instead of precision. It’s easy to understand how calibrations reflect accuracy, the farther the calibration curve is from the true values the less accurate it is. Precision isn’t quite as obvious, but if you think about how a calibration curve works it becomes more clear. The variability in a measurement is going to be a combined result of the variabilities in your sample/standard prep, sample introduction, and detector response. All of this will lead to some variability in your final instrument response, which your calibration curve converts to a concentration. A linear calibration curve has the form of y=mx+b, where x is concentration, y is response, b is the y intercept, and m is the slope. If we solve for concentration, that gives us x=(y-b)/m. If we have variability in our response (y), then that will create variability in our concentration (x) that is inversely proportional to the slope of the calibration curve (m). Figure 1 below shows this, where a response of 4 ±10% is shown in the horizontal black lines (i.e., 4.4 and 3.6). You can see that as the slope decreases, the concentration precision, shown by the horizontal lines, increases. The concentration value also increases as the slope decreases, showing that the slope affects both the precision and accuracy of your calibration. Since slope is multiplicative, the relative effect on precision and accuracy is the constant over the curve, but the absolute effect increases as the concentration increases. For example, a change in slope of 10% would change a concentration of 1 ppm by 0.1 ppm, but a concentration of 100 ppm would change by 10 ppm.

**Figure 1** – Change in precision increasing as slope decreases.

In contrast, changing the intercept has an effect on the accuracy, but no effect on precision. You can see this in Figure 2, where a response range of 4 ±10% gives the same variation in concentration with three different y-intercepts.

**Figure 2** – No change in precision as the y-intercept increases.

Also, since the intercept is additive instead of multiplicative, the absolute effect on accuracy is constant while the relative effect decreases as concentration increases. For example, a change in intercept of 0.1 ppm would be a change of 10% at 1 ppm but would be a change of 0.1% at 100 ppm. This means that y-intercept changes have a greater effect on the % accuracy of your calibration at the low end.

Why is all this important? It’s common to calculate detection limits by doing low-level precision studies. For example, the method detection limit (MDL) guidance from the EPA involves 7 replicate blanks and 7 low-level spikes, and the standard deviation is used to calculate the MDL. This means that understanding what effects your low-level precision is important when trying to optimize your MDLs. To see how changes in calibration might affect this, let’s look at some real-world data.

If you are a consistent reader of the blog, you may have read a post by our intern Yannic Schneck on SPME fundamentals that included some nitrosamine data. As part of his work, he also did some instrument detection limit (IDL) studies with liquid injections using a similar method to the EPA MDL guidelines. Table 1 shows his initial results using an equal weighted linear calibration. While the precision measurements show IDL values from ~2-12 ng/mL, the accuracy of his spikes seemed to indicate that the instrument was not capable of measuring that low.

Name |
IDL (ng/mL) |
Avg. result (ng/mL) |
Spike Value (ng/mL) |
Avg. recovery |

NDMA |
2.4 |
5.8 |
16 |
36% |

NMEA |
12.4 |
4.1 |
16 |
26% |

NDEA |
1.7 |
8.0 |
16 |
50% |

NDIPA |
4.2 |
8.4 |
16 |
52% |

NPYR |
2.2 |
8.9 |
16 |
55% |

NDPA |
5.2 |
13.4 |
16 |
84% |

NMOR |
7.7 |
-2.7 |
16 |
-17% |

NPIP |
1.6 |
7.4 |
16 |
46% |

NDBA |
10.0 |
4.0 |
16 |
25% |

**Table 1** – IDL results using equal weighted linear calibration.

Knowing that equal weighted calibrations can be inaccurate at the low end, I helped him re-work the data using a 1/x^{2 }weighting. Table 2 shows the results, with the IDL values changing very little, still being in the ~2-12 ng/mL range, but the accuracy is much improved.

Name |
IDL (ng/mL) |
Avg. result (ng/mL) |
Spike Value (ng/mL) |
avg. recovery |

NDMA |
2.1 |
14.2 |
16 |
89% |

NMEA |
11.8 |
10.2 |
16 |
64% |

NDEA |
1.5 |
13.5 |
16 |
85% |

NDIPA |
4.1 |
12.7 |
16 |
80% |

NPYR |
2.0 |
13.2 |
16 |
82% |

NDPA |
5.3 |
14.6 |
16 |
91% |

NMOR |
7.0 |
9.2 |
16 |
58% |

NPIP |
1.5 |
12.8 |
16 |
80% |

NDBA |
9.0 |
12.1 |
16 |
75% |

**Table 2** – IDL results using 1/x^{2} weighted linear calibration.

Knowing that the IDL values are based on precision and only the slope of the curve effects precision, we can guess that the curve fits did not change the slope very much. However, the change in accuracy implies a significant change in the y-intercept. If we look at the calibration curve for NDMA we can see that this appears to be true. The unweighted curve has an equation of y=45360x + 457.2, while the weighted curve is y=52810x -37.73. The slope increased by 16%, while the intercept changed by over an order of magnitude. The 16% increase in slope corresponds to a 16% decrease in the IDL, but the average result changed from 6.4 ng/mL to 12.5 ng/mL, a change of almost a factor of 2.

Anecdotally, this pattern of slight change in slope and large change in intercept is what I normally see from changing curve fits. This means that in general, optimizing curve fits greatly improves your low-level accuracy, as detailed in the curve fits blog, but has little effect on your precision or detection limits. Detection limits derived from precision measurements will give you what you’re potentially capable of if you have an accurate calibration. If you find that you’re not capable of measuring accurately near your detection or quantitation limits, this may indicate that your curve fit isn’t ideal.

These blog posts have dealt a lot with the theoretical and mathematical considerations of calibrations, but starting with the next one we’ll be digging a bit more into the practical side of things, starting with choosing a calibration range and points, so stay tuned for that.

View all of the posts in the "More Than You Ever Wanted to Know About Calibrations" series.