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