Lecture 4
Basic Statistics
Dr. A.K.M. Shafiqul Islam
School of Bioprocess Engineering
University Malaysia Perlis
21.09.2011
PROPAGATION OF ERRORS
Addition and subtraction:
— errors add as the square root of the squares of the absolute values of the
uncertainties
3.067 ± 0.040
4.02 ± 0.01
- 2.9846 ± 0.3308
rabc
er 
2
ea

2
eb

2
ec
4.1024 ± 0.333
PROPAGATION OF ERRORS
Multiplication and division:
─ The relative uncertainties are additive, and the most probable error is
represented by the square root of the sum of the relative variances.
─ i.e., the relative variance of the answer is the sum of the individual
relative variances.
PROPAGATION OF ERRORS
Multiplication and division:
— relative errors add as the square root of the squares of the relative
uncertainties
ab
r
c
2
2
er
 ea   e b   ec 
      
r
a b c
2
PROPAGATION OF ERRORS
Find out the relative uncertainty of the calculation
PROPAGATION OF ERRORS
=356.0
= ±0.002566
= ±2.6 X 10 3
= ±356.0 X ±2.6 X 10
3
So the answer is 356.0 ± 0.9.
= ±0.93
CONTROL CHARTS
• A quality control chart is a time plot of a measured quantity
that is assumed to be constant (with a Gaussian distribution)
for the purpose of ascertaining that the measurement
remains within a statistically acceptable range.
• The control chart consists of a central line representing the
known or assumed value of the control and either one or two
pairs of limit lines, the inner and outer control limits.
CONFIDENCE LIMIT
• Calculation of the standard deviation for a set of data provides
an indication of the precision inherent in a particular
procedure or analysis.
• But unless there is a large number of data, it does not by itself
give any information about how close the experimentally
determined mean x might be to the true mean value m.
• Statistical theory allows us to estimate the range within which
the true value might fall, within a given probability, defined by
the experimental mean and the standard deviation.
CONFIDENCE LIMIT
• This range is called the confidence interval, and the limits of
this range are called the confidence limit.
• The likelihood that the true value falls within the range is
called the probability, or confidence level, expressed as a
percent.
• The confidence limit is given by
• where t is a statistical factor that depends on the number of
degrees of freedom and the confidence level desired.
CONFIDENCE LIMIT
CONFIDENCE LIMIT
• A soda ash sample is analyzed in the analytical chemistry
laboratory by titration with standard hydrochloric acid. The
analysis is performed in triplicate with the following results:
93.50, 93.58, and 93.43% Na2CO3. Within what range are you
95% confident that the true value lies?
• So you are 95% confident that, in the absence of a
determinate error, the true value falls within 93.31 to 93.69%.
CONFIDENCE LIMIT
SIGNIFICANT TESTING
•
Why we do testing to our experimental data?
1.
to compare data among friends with the intention of gaining
some confidence that the data observed could be accepted or
rejected.
2.
to decide whether there is a difference between the results
obtained using two different methods. All these can be
confirmed by doing some significant tests.
T TEST
• Is used to determine if 2 sets of
measurements are statistically different.
• The comparison between 2 set of
measurements which made by 2 method, one
will be test method and the other one will be
accepted method.
• By using T test, we can determine whether
these two method are significant difference.
F TEST
• This is a test designed to indicate whether
there is a significant difference between two
methods based on their standard deviations.
• F is defined in terms of the variances of the
two methods, where the variance is the
square of the standard deviation:
F TEST
• The F test evaluates differences between the
spread of results,
while the t test looks at differences between
means.
F TEST
• You are developing a new colorimetric procedure for determining the
glucose content of blood serum. You have chosen the standard Folin-Wu
procedure with which to compare your results. From the following two
sets of replicate analyses on the same sample, determine whether the
variance of your method differs significantly from that of the standard
method.
F TEST
• Solution
= 1.73
• The F value is > 1.
F TEST
F TEST
• The tabulated F value for v1 = 6 and V2 = 5 is 4.95.
• Since the calculated value is less than this, we
conclude that there is no significant difference in the
precision of the two methods,
i.e., the standard deviations are from random error
alone and don't depend on the sample.