2. " normal" days Today' s count
5.1
Is my red blood cell
5.3
count high today?
4.8 ×106 cells 5.6 ×106 cells
μL μL
5.4
5.2
3. 4.1 The Gaussian Distributions -1
• Nerve cells muscle cells
(1991 Nobel Prize in Medicine & Physiology)
Sakmann & Neher
absence neurotransmitter present neurotransmitter
4. 4.1 The Gaussian Distributions -2
922 ion channels
response
Typical lab
measurements:
Gaussian distribution
1 − ( x −μ )
y= e 2σ 2
σ 2π
5. 4.1 The Gaussian Distributions -3
Gaussian distribution is characterized by
3) Mean: x( μ)
∑x 1 i
x= i
= ( x1 + x 2 + + x n )
n n
7) Standard deviation: S( σ )
∑ (x ) 2
i −x
S= i
n −1
6. 4.1 The Gaussian Distributions -5
The smaller the s ,
⇒ the more
precise the results
reproducible
( )
x, S for a finite set.
( μ, σ ) for an infinite set.
7. 4.1 The Gaussian Distributions -4
Other terms
• Median
• Range
σ & probability
• Table 4.1
8. 4.2 Student’s t -1
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 (µ) might fall,
(given probability = confidence level) defined by
mean and standard deviation.
ts
Confindence interval : μ = x ±
n
9. 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.
10. 4.2 Student’s t -4
ts
μ ( 50% ) = x ±
n
ts
μ ( 90% ) = x ±
n
11.
12. 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
13. 4.2 Student’s t -6
ts
μ=x±
n
* improving the reliability of your
measurement
(1) make more measurements
( n ↑) ∝ 1
n
(2) improve expt. procedure
( ↓ S)
14. 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.
15. 4.2 Student’s t -8
Case 1 :
a. 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%
X = 3.260 S = 0.041
16. 4.2 Student’s t -9
Does answer agree with the known answer ?
known value − x
t calculate = n
s
3.19 − 3.26
= 4 = 3.41
0.041
95% confidence tcalculate > ttable
⇒ result is different from the known value.
17. 4.2 Student’s t -10
Case 2
• comparing replicate measurements.
1904 Nobel Prize by Lord Rayleigh.
for discovering Inert gas argon :
18. 4.2 Student’s t -11
1 N2 O
Cu (s) + O 2 → CuO (s)
← NO
2
NH NO
4 2
19. 4.2 Student’s t -12
t Test for comparison of means :
x1 − x 2 n1n 2
t=
s pooled n1 + n 2
s1 ( n1 − 1) + s 2 ( n 2 − 1)
2
where s pooled = 2
n1 + n 2 − 2
20. 4.2 Student’s t -13
Case 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
d = 0.060
21. 4.2 Student’s t -14
d
t calculate = n
sd
∑ (d ) 2
i −d
sd =
n −1
=
( 0.04 − 0.06 ) 2 + ( − 0.16 − 0.06 ) 2 = 0.12 2
6 −1
0.06 0
∴ t calculate = 6 = 1.20 t cal < t table
0.12 2
∴ two techniques are not significant different at the
95% confidence level
22. 4.3 Q test for bad data -1
help decide whether to retain or discard a datum
gap
Q test for discarding : Q =
range
23. 4.3 Q test for bad data -2
Qcalculate > Qt
discard
any datum from a
faulty procedure.
24. 4.4 Finding the “Best” straight line -1
calibration methods
prepare calibration curve.
25. 4.4 Finding the “Best” straight line -2
Mrthod of least square
y = mx + b
di = y i - y = y i - (mx + b) ( + or -)
di2 = (y i − mx − b) 2 (postive only)
n∑ ( x i y i ) − ∑ x i ∑ y i
Least - squares slope : m =
D
Least - squares intercept : b=
∑ ( x )∑ y − ∑ ( x y ) ∑ x
2
i i i i i
D
where the denominato r, D, is given by
( )
D = n∑ x − ( ∑ x i )
2
i
2
26. 4.5 Constructing a Calibration Curve -1
1) Blank standard soln
Spectrophotometer readings for protein analysis by the
Table 4-6
Lowry method
Sample Absorbance of three Corrected absorbance
Range ( after subtracting average
(μg) independent samples
blank )
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
Standard 10 0.282 0.272 0.272 0.010 0.1827 0.1727 0.1727
soln 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
27. 4.5 Constructing a Calibration Curve -2
m =
b =
1) Finding the protein in an unknown
