Instrumental Analysis Statistics II

IV. Normal Error Law The fraction of a population of measurements, dN/N, whose values lie in the region x to (x+dx) is given by : Where: s = pop. std. dev. m = pop. Mean N = # of measurements x – m = Abs. Deviation from mean Relative frequency, dN / N +1s -1s m = 50 s = 5 x - m 10 20 30 40 -40 -30 -20 -10 +1s -1s m = 50 s = 10

Normalize the curve by expressing in terms of relative deviation from the mean. 10 20 -10 -20 s = 5 x - m Relative Frequency, dN/N 2 4 -2 -4 -3 -1 1 3 Then: z Relative Frequency, dN/N Substitute: z = (x – m)/s and dz = dx/s

Relative frequency, dN / N 68.3% of measurements will fall within ± s of the mean. +3s -3s +2s -2s +1s -1s Relative frequency, dN / N 95.5% of measurements will fall within ± 2s of the mean. -1 1 2 3 4 -2 -3 -4 99.7% of measurements will fall within ± 3s of the mean. z

V. Confidence Limits In the absence of any systematic errors, the limits within which the population mean (m) is expected to lie with a given degree of probability. 2s 4s -2s -4s -3s -1s 1s 3s dN/N 50% +0.67s -0.67s 2s 4s -2s -4s -3s -1s 1s 3s dN/N 80% +1.29s -1.29s 2s 4s -2s -4s -3s -1s 1s 3s dN/N 95% -1.96s +1.96s

A. Confidence Limits (CL) when s is a good estimate of s 1. When s is determined from ³20 replicate measurements. a. CL for a single measurement (xi) CL of m = xi ± zs Confidence Interval a. CL for multiple measurements (x) Confidence Interval CL of m = x ± zs N

A. Confidence Limits (CL) when s is a good estimate of s 2. Obtaining a good estimate of s by pooling data. Use a series of identical analyses on a similar set of samples. Degrees of freedom ³ 20 N1, N2, etc. = Number of measurements in each subset nt = Number of subsets

B. Confidence Limits (CL) when s is unknown When s is determined from <20 replicates Use the t parameter rather than the z parameter. t = (x-m) s z = (x-m) s t depends on confidence level and degrees of freedom (N-1) Confidence Interval CL of m = x ± ts N

VI. Propagation of Error Typical experimental methods of analysis involve several steps, each of which has an uncertainty associated that contributes to the net uncertainty of the analysis. A. Addition and subtraction x = p + q - r Where p, q, & r each have uncertainties sp, sq, & sr sx = sp2 + sq2 + sr2

VI. Propagation of Error Typical experimental methods of analysis involve several steps, each of which has an uncertainty associated that contributes to the net uncertainty of the analysis. A. Multiplication and division x = p · q / r Where p, q, & r each have uncertainties sp, sq, & sr sp sq sr p q r + 2 sx = x