1. Instrumental Analysis
Statistics II

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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