CHEM 524 -- Course Outline (Part 11)—2005 minor modification

VII. Error and Statistical Sampling: Chapter 6 and Appendix A (Read both, esp. Append. A)

F. Statistical Sampling only applies to random error, Statistics yield evaluation of error

1. Systematic error more difficult-

a. Calibration—pure analyte, concentrations must bracket unknown, must be appropriate to analyte

b. Matrix, - Blank must be all but unknown, concentration not interacting, no interfering species

c. Sampling errors- e.g. uncalibrated pipette or aliquots with sequence effect

2. Sampling uses previous definitions, average and deviation:

a. Averaging data from multiple measurements:

E = S E_i/n as nà very large, then E à m

b. Standard deviation (rms excursion from mean):

s = [S (E_i-E)²/(n-1)]^1/2 nà very large, then sà s

3. Random distribution of error is Gaussian -- z test, large set

--- where: m = true mean; s = true S.D.

--- a-probability in interval:

expressed as P(z < z_a) = 1 - a, P(|z| < z_a) = 1 - 2a

--- values from table of z and a (Table A1)

4. Smaller samples: Student t-test (s is unknown) -- measure first (n < 30)

-- for a small number of data, the error (uncertainly) increases

--- s and s differ -- need table for a depend on n

--- t = (E - m)/(s/ n^1/2), where E - average of n samples table gives t (a, n)

same form P(t < tⁿ_a) = 1 - a, P(t > tⁿ_a) = a,

also P(|t |< tⁿ_a) = 1 - 2a, and P(0 < t < tⁿ_a) = 0.5 - a,

5. Hypothesis testing -- is difference between E and m significant?

--- test confidence interval m = E ± zs/ n^1/2 (or m = E ± ts/ n^1/2)

two-tailed, 1-2a level confidence

-- confidence (or probability) that an interval (error range) encloses the true mean

-- as confidence increases, interval must increase, as n increases, interval decrease

-- example problem

G. Concentration Sensitivity

1. Calibration curve gives E = f(c), (book uses S) calibration sensitivity: m = dE/dc = df(c)/dc

-- Concentration Confidence interval: m_c = c ± ts_c/ n^1/2 s_c = s/m

-- Actual confidence (error) also affected by calibration error

use t = (c-m_c)/(s_c/n^1/2)

-- Analytical sensitivity: g = m/s = 1/s_c corrects for gain, etc.

note smaller error more sensitivity

2. Detection Limit—smallest signal/conc. at some level of confidence

-- DL = k^.s_bk / m s_bk -- S.D. of blank, k -- confidence factor,

-- limited sampling use t-test: t = k / 2^1/2 2 from sample + blank measurement

(goal make measurements at >10*DL)