Download a) spread or range b) coefficient of variation c) Gaussian distribution

6-1 Define : a) spread or range
b) coefficient of variation c) Gaussian distribution
(a) The spread or range for a set of replicate data is the numerical difference between the
highest and lowest value.
(b) The coefficient of variation is the percent relative standard deviation
(c) A Gaussian distribution or normal distribution is described by the bell-shaped curve
obtained by plotting frequency versus deviation from the mean for measurements that
conform to such a distribution.
6-2 Differentiate between :
a) Population mean and sample mean
b) Accuracy and precision
c) Random error and systematic error
a) Both types of mean are obtained by summing the available replicate data and dividing
by the number of data. For the population mean the number of data is large. For the
sample mean the number of data are a small fraction of the population of data.
b) Accuracy represents the agreement between an experimentally measured value and the
true or accepted value. Precision describes the agreement among measurements that
have been performed in exactly the same way.
c) Random errors result from uncontrolled variables in an experiment while systematic
errors are those that can be ascribed to a particular cause and can usually be
determined.
6-5 From the Gaussian error curve, what is the probability that a result from a
population lies between 0 and +1σ of the mean? What is the probability of a result
occurring that is between +1 σ and +2 σ of the mean?
The population between 0 and +1σ
of the mean
(±
± σ)
½ x 0.683 = 0.342
The population between +1 σ and +2 σ
of the mean
(±
± 2 σ)
½ x (0.954-0.683) = 0.135
6-6 From the Gaussian error curve, find the probability that a result is outside the
limits of ±2σ of the mean? What is the probability that a result has a more negative
deviation from the mean than -2 σ
The population outside ±2σ
of the mean
1- 0.954 = 0.046
The population that the result has a more
negative deviation from the mean
than -2σ
½ (1- 0.954) = 0.023
6-7 Consider the following sets of replicate measurements :
A
B
C
D
E
F
3.5
70.24
0.812
2.7
70.65
0.514
3.1
70.22
0.792
3.0
70.63
0.503
3.1
70.10
0.794
2.6
70.64
0.486
0.900
2.8
70.21
0.497
3.3
2.5
3.2
0.472
For each set, calculate the a) mean, b) median; c) spread, d) standard deviation, e)
coefficient of variation.
6-17 Analysis of several plant-food preparations for potassium ion yielded the
following data:
sample
Percent K+
1
5.15, 5.03, 5.04, 5.18, 5.20
2
7.18, 7.17, 6.97
3
4.00, 3.93, 4.15, 3.86
4
4.68, 4.85, 4.79, 4.62
5
6.04, 6.02, 5.82, 6.06, 5.88
The preparations were randomly drawn from the same population.
a) Find the mean and standard deviation for each sample.
Sample 1 :
∑(
Sample 2, 3 …
−
= 5.12
)2 = 0.0254
=
0.0254
= 0.079687
4
Obtain the pooled value Spooled
∑(
−
)2 = 0.0254
∑(
−
)2 = 0.0281
∑(
−
)2 = 0.0325
∑(
−
)2 = 0.0459
=
.
.
.
.
!
∑(
.
−
)2 = 0.0461
= 0.11