1
Chapter 4
Statistics
Dr. Elizabeth Binamira-Soriaga
Department of Chemistry
Texas A&M University
Gaussian Distribution
x
=
x
i
i
=
1
n
"
n
mean, x:
Standard
deviation, s:
s
=
x
i
"
x
(
)
2
i
=
1
n
#
n
"
1
For an infinite set of data, mean x approaches
μ
and s,
!
(no systematic error). Recall variance and rsd
Distribution ~ Gaussian for finite data with errors purely random
Gaussian Curves
μ
= ±
1
"
μ
= ±
2
"
μ
= ±
3
"
Range
Percentage of
Measurements
68.3
95.5
99.7
What fraction of observations in a Gaussian distribution is
expected to be below
μ
- 3
!
?
Confidence Intervals
Çonfidence interval expresses the certainty that the true
population mean,
μ
, lies within a certain distance from
the measured mean, x. The confidence interval of
μ
is:
μ
=
x
±
ts
n
s =measured standard deviation
n = number of observations
t = Student’s t
Student’s t is a statistical tool to express confidence
intervals and to compare results from different
experiments to see if they are essentially the same
Confidence interval:
Values of Student’s t
Degrees of freedom = n-1
Calculation of Confidence Intervals
•
Find the 50% and 90% confidence intervals for
carbohydrate content if the carbohydrate content of a
glycoprotein is determined to be 12.6, 11.9, 13.0, 12.7
and 12.5 per 100 g of protein in replicate analyses
•
First, calculate mean and standard deviation for set of
data: x = 12.5 and s = 0.4
•
From Table 4-2, values of Student’s t are : 0.741 (50%
confidence level and 2.132 (90% confidence level)
•
Use t’s to solve for
μ
for each confidence level:
μ
=
x
±
ts
n
=
12.54
±
(0.741)(0.40)
5
=
12.5
±
0.1
μ
=
x
±
ts
n
=
12.54
±
(2.132)(0.40)
5
=
12.5
±
0.3

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Calculation of Confidence Intervals
Confidence Intervals (CI) as Estimates
of Experimental Uncertainty
x = 6.374
6
mL
s = 0.001
8
mL
(90% CI at +/- 0.001
7
mL)
(90% CI at +/- 0.0007
mL)
The Meaning of Confidence Intervals
•
50% CI: 50% of error bars would include
μ
•
90% CI: 90% of error bars would include
μ
μ = 10000
and
!
= 1000
Filled squares are data points that do not include
the true population mean of 10000
Comparison of Means with Student’s t
•

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