# How to calculate the detection limit?

I am having trouble figuring out how to calculate the detection limit. Here is the question:

You and your friend working in a different lab have been given an unknown water sample that contains potassium. You are both asked to measure the potassium concentration in the solution (10 replicate measurements). The methods that you and your friend are using are different. The results that were obtained are listed below. Determine:

(a) the concentration, standard deviation and relative standard deviation for the unknown as measured using the two methods (check for outliers!),

(b) calculate the detection limit (3sigma) for each method,

(c) compare the standard deviations and evaluate whether the two averages are significantly different (or not) at the 95% confidence level.

RESULTS:

Standard    Method 1    Method 2
Concentration   Intensity   Intensity
(mg/L)  (nA)    (mV)
0.000   0.624        1.955
0.000   0.488        2.490
0.000   0.522        2.166
0.000   0.355        1.500
5.000   9.245       15.644
10.000  17.069      31.220
15.000  26.200      44.266
20.000  33.881      62.394
25.000  43.826      75.611

Replicate
1       26.544      46.977
2       25.449      49.120
3       21.053      50.998
4       24.353      46.615
5       23.899      49.326
6       24.010      46.666
7       25.554      45.291
8       23.549      42.995
9       26.008      43.678
10      24.404      49.012

So my problem is with part (b). I know the detection limit is calculated with the formula DL = 3sigma/slope and that I can find the slope by calculating Sxy/Sxx but I'm not sure what points I'm supposed to use? What points do I use for the standard deviation (sigma) and for Sxy/Sxx?

I've tried looking at examples in the textbook but nothing was really similar. I also tried searching online, but I couldn't find anything that helped me.

Have a look at this: http://www.chem.utoronto.ca/coursenotes/analsci/stats/LimDetect.html

It is possible to calculate the limit of detection from the standard error of the regression:

$$C_{\mathrm{LOD}} = \frac{3s_{y/x}}{b}$$ These values are calculated from the regression of all the data points, not the standard deviation of any subset of data.

However, in this case we have replicate blank measurements (four measurements at C=0), so we can also calculate the limit of detection as simply 3 times the standard deviation of the blank measurements:

$$y_{\mathrm{LOD}} = y_{\mathrm{blank}}+3s_{\mathrm{blank}}$$

(in instrument response units—convert to concentration from a known standard or from the regression equation)

• So I find the standard deviation of the intensity values with concentrations of 0, right? But I'm not quite sure what is yblank? – 302Laya Oct 16 '14 at 0:47
• Oh, it's the mean, correct? – 302Laya Oct 16 '14 at 0:48
• Yeah, the mean blank response. – Michael DM Dryden Oct 16 '14 at 0:48
• So what's the point of the other measurements (the non-zero concentrations)? Those are what really threw me off :S – 302Laya Oct 16 '14 at 0:49
• The other measurements establish the response of the instrument at different concentrations. It's not possible to answer question a) or get a regression equation without them. – Michael DM Dryden Oct 16 '14 at 0:50