Chapter 4
Statistics
Is my red blood cell count high today?
μLcells105.6 μL
cells 10
5.2
5.4
4.8
5.3
5.1
count sToday' days normal""
66
4.1 The Gaussian Distributions -11) Nerve cells muscle cells
(1991 Nobel Prize in Medicine & Physiology) Sakmann & Neher
absence neurotransmitter present neurotransmitter
4.1 The Gaussian Distributions -2
922 ion channels response
Typical lab measurements: Gaussian distribution
2
2
2σμx
e2πσ
1y
4.1 The Gaussian Distributions -3Gaussian distribution is characterized by
1) Mean:
2) Standard deviation:
μx
n21i
i
xxxn
1
n
xx
σS
1n
xxS i
2
i
4.1 The Gaussian Distributions -5
The smaller the s ,
the more
precise the results
reproducible
set. infinite an for σμ,
set. finite a for S,x
4.1 The Gaussian Distributions -4
Other terms• Median• Range
σ & probability• Table 4.1
4.2 Student’s t -1
n
tsxμ :intervale Confindenc
Student’s t is the statistical tool used to express confidence intervals & to compare results from different experiments.
confidence interval: allows us to estimate the range within which the true value () might fall, (given probability = confidence level) defined by mean and standard deviation.
4.2 Student’s t -3(ex)
In replicate analyses, the carbohydrate content of a glycoprotein (a protein with sugars attached to it) is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohydrate per 100 g of protein.
Find the 50 % and 90% confidence intervals for the carbohydrate content.
4.2 Student’s t -4
n
tsx90% μ
n
tsx50% μ
4.2 Student’s t -5
Smaller confidence intervals
Better measurement
For 90% sure that a quantity lies in the range
62.3 0.5 vs. 62.3 1.3
4.2 Student’s t -6
S
procedure expt.improve (2)n
1n
tsmeasuremenmore make (1)
tmeasuremen
your ofy reliabilitthe improving *n
tsxμ
4.2 Student’s t -7
t test : used to compare one set of measurements with another to decide whether or not they are different.
Three ways in which a t test can be used will be described.
4.2 Student’s t -8
Case 1 : comparing a measured result with a “known” value
Sample: 3.19 wt% (known value)
a new analytical method :
3.29, 3.22, 3.30, 3.23 wt%
= 3.260 S = 0.041X
4.2 Student’s t -9Does answer agree with the known answer ?
95% confidence tcalculate > ttable
result is different from the known value.
41
1
calculate
3.40.04
3.263.19
ns
xvalue knownt
4.2 Student’s t -10Case 2
comparing replicate measurements.
1904 Nobel Prize by Lord Rayleigh. for discovering Inert gas argon :
4.2 Student’s t -11
24
2
NONH
NO
ON
(s)2(s) CuOO2
1Cu
4.2 Student’s t -12t test for comparison of means :
where
2nn
1ns1nss
nn
nn
s
xxt
21
2
2
21
2
1pooled
21
21
pooled
21
4.2 Student’s t -13Case 3• Comparing individual differences
Cholesterol content (g/L)
Sample Method A Method B Different (di)
1 1.46 1.42 0.04
2 2.22 2.38 -0.16
3 2.84 2.67 0.17
4 1.97 1.80 0.17
5 1.13 1.09 0.04
6 2.35 2.25 0.10
= 0.060d
4.2 Student’s t -14
∴ two techniques are not significant different at the 95% confidence level
tablecal
2
0calculate
2
22
2
id
d
calculate
tt 1.20 60.12
0.06t
0.1216
0.060.160.060.04
1n
dds
ns
dt
4.3 Q test for bad data -1help decide whether to retain or discard a datum
range
gapQ :discarding for testQ
4.3 Q test for bad data -2
Qcalculate > Qt
discard
any datum from a
faulty procedure.
4.4 Finding the “Best” straight line -1
calibration methods prepare calibration curve.
4.4 Finding the “Best” straight line -2
2
i
2
i
iiii
2
i
iiii
2
i
2
i
iii
xxnD
by given is D, r,denominatothe where D
xyxyxb :intercept squares-Least
D
yxyxnm :slope squares-Least
only)(postive b)mx(yd
-)or ( b)(mx-yy-yd
bmxy
square least of Mrthod
4.5 Constructing a Calibration Curve -1
1) Blank standard soln
Table 4-6Spectrophotometer readings for protein analysis by the Lowry method
Sample
(μg)Absorbance of three independent samples
RangeCorrected absorbance ( after subtracting average blank )
0 0.099 0.099 0.100 0.001 -0.0003 -0.0003 0.0007
5 0.185 0.187 0.188 0.003 0.0857 0.0877 0.0887
10 0.282 0.272 0.272 0.010 0.1827 0.1727 0.1727
15 0.392 0.345 0.347 0.047 --- 0.2457 0.2477
20 0.425 0.425 0.430 0.005 0.3257 0.3527 0.3307
25 0.483 0.488 0.496 0.013 0.3837 0.3887 0.3967
Standard
soln
blank
4.5 Constructing a Calibration Curve -2
b
m
2) Finding the protein in an unknown
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