Statistical Analysis of Data
Measured data is carefully collected.
A Sample distribution organizes the data as a bar chart or histogram.
Gaussian Distribution Fit for Sample Distribution of
Zinc Core U.S. Mint Penny Mass 1983-2007
n = 565, x = 2.503 g, s = 0.022, median = 2.501 g, mode = 2.515 g
max = 2.622 g, min = 2.414 g, 3s = 0.06 5 g
215453 = C2 > Critical Value = 33.9
160
140
120
Frequency
100
Sample Distribution
Gaussian Distribution
80
60
40
20
62
0
61
0
60
0
59
0
58
0
57
0
56
0
55
0
54
0
63
0
2.
2.
2.
2.
2.
2.
2.
2.
2.
53
0
52
0
51
0
49
0
48
0
47
0
46
0
45
0
44
0
43
0
50
0
Mass (grams)
2.
2.
2.
2.
2.
2.
2.
2.
2.
2.
2.
42
0
2.
2.
2.
41
0
0
The sample distribution may approximate a Gaussian distribution (also known as a Normal distribution). Notice
that some of the bars in the Sample distribution are inside and outside of the Gaussian distribution.
The center of a perfect Gaussian distribution represents the mean (x)-the sum of the measured values divided by the
number of measurements. This particular center is also the mode-the most frequent measured value. Obviously,
perfect Gaussian distributions are not obtained for a small sample of data, so, in reality, the mean and mode are
different.
The median value indicates that half the data is above this value and half is below it.
The standard deviation (s) is a measure of the width of the Gaussian distribution or distance from the center of the
Gaussian distribution. The standard deviation also indicates the precision of a measured quantity.
s =
i (xi -x )2
n-1
n = number of data points
n - 1 = degrees of freedom
One can always calculate the
nth data point from the mean
and n - 1 data points
For a finite set of measurements, x and s are known as the sample mean and sample standard deviation,
respectively.
For an infinite set of data,  and  represent the population mean and population standard deviation,
respectively.
Percentage of observations in a Gaussian distribution:
±  contains 68.3% of the observations
± 2 contains 95.5% of the observations
The standard of excellence is the 95.5% (95%) probability level.
± 3 contains 99.7% of the observations
A confidence interval is a range of values in which there is a particular probability of finding the population mean.
Better measurements (smaller s) give smaller confidence levels. Alternatively, more measurements (larger n) have
the same effect.
 x
ts
+
n
Student's t (the t in the equation above): Find the value of t from the table below for the desired probability (%) and
degrees of freedom (n-1). If a sample mean is 2.546 +/- 0.002 g for 3 measurements, then the population mean at
the 95 % probability level is 2.546 +/- 0.005 g ( 4.303  0.002 /
within the range 2.541 g –2.551 g.
3 ). This indicates that the population mean lies
Table adapted from
Harris, D. C. “Exploring Chemical
Analysis” 4th edition, W. H. Freeman and
Company, 2009
Rejection of Questionable Data: The Q-test
For the given set of data calculate the Gap and the Range.
83.5 g/mol, 136.1 g/mol, 138.5 g/mol
Range = 138.5-83.5 = 55.0
Gap = 136.1-83.5 = 52.6
(Gap = difference between the questionable data and the nearest value)
Calculate Q calc: Q calc = gap/range = 0.956 and compare to Q95 (Q at the 95% confidence level) or Q90 (Q
at the 90% confidence level): Table adapted from Harris, D. C. “Exploring Chemical Analysis” 4th edition,
W. H. Freeman and Company, 2009:
n
Q95
Q90
3 observations:
4 observations:
5 observations:
6 observations:
7 observations:
8 observations:
9 observations:
10 observations:
0.970
0.829
0.710
0.625
0.568
0.526
0.493
0.466
0.941
0.765
0.642
0.560
0.507
0.468
0.437
0.412
If Qcalc > QX the questionable data can be rejected with X% confidence. For this example 83.5 g/mol cannot be
rejected at the 95% level but can at the 90% level.