STATISTICAL ANALYSISPresented by:

Iswar Hazarika1st yr M. Pharm Pharmacology

The Oxford college of Pharmacy

Contents:1. Statistics - Introduction 2. Scope of statistic3. Normal Distribution4. Central Tendency

1. Arithmetic mean2. Median 3. Mode

5. Dispersion1. Standard deviation (SD)

6. Standard error of mean (SEM)7. Probability8. Test for significance

1. Student ‘t’ test 2. Chi square test

Statistics:“Statistics is a science which deals with the collection, classification and tabulation of numerical facts as the basis for explanation, description and comparison of phenomena.”

Here, the data are numbers which contain information.

Scope: Industries Medical Science Agricultural biology Social Science Planning and economics Space research

Normal Distribution When many independent random

factors act in an additive manner to create variability, the data set follows a bell shaped distribution called as normal distribution.

Mathematicians De Moivre and Laplace used this distribution in the 1700's.

In the early 1800's, German mathematician and physicist Karl Gauss used it to analyze astronomical data, and known as the Gaussian distribution.

Normal Distribution

Normal Distribution

When maximum frequency of distribution occurring at the centre of the curve and the remaining evenly distributed around it, it follows normal distribution.

Normal distribution is described by its mean (µ) and standard deviation (σ).

Central tendency Arithmetic mean Geometric mean Median Mode

Arithmetic Mean: It is defined as the sum of the all

variates of a variable divided by the total number of item in a sample.

It is expressed by the symbol

Where, = Arithmetic mean n = frequency Xi = all the varietes of Variable

Arithmetic Mean:

Example:

Geometric Mean: It is defined as the nth root of the

product of the n items in an ungrouped data.

When percentage increase or decrease is expressed over a period of time, the mean percentage is find out by using geometric mean.

If X1, X2, X3,…. Xn are the n variates of the variable X then,

Geometric Mean =

Geometric Mean: Example:

Following administration of a drug in a laboratory mammal, the blood glucose level increased by 5% in the first hour, by 8% in the second hour and 77% in the third hour. What is the mean percentage increase during the observation period?

Here, we assume that the glucose level at the beginning of every hour as 100mg%

Then the level of blood sugar

Geometric Mean: At the end of 1 hour= 100+5 =105mg

% At the end of 2 hour=100+8 = 108mg

% At the end of 3 hour=100+77 =

177mg%

So, geometric mean=

= 126.14 So the mean percentage increase

= 126.14 – 100 = 26.14

Median: It is the central value of all observations

arranged from the lowest to the highest.

Example: (1) For Odd number of variates

Weight of frog in gram. n = 7 75, 66, 55, 68, 71, 78, 72.

Data in ascending order of value: 55, 66, 68, 71, 72, 75, 78.

Here, Median is 71.

Median:

Example: (1) For Even number of variates

Height of Students in cm, n = 8

165, 175, 161, 155, 169, 171, 152, 166.

Data in ascending order of value: 152, 155, 161, 165, 166, 169, 171, 175.

Here, Median is = 165.5

Mode: It is defined as the value which

occurs most frequently in the sample.

ExampleWeight of tablet in mg:52, 48, 50, 51, 50, 51, 50, 49.

In the above data, 50 occurs 3 timesSo mode of above data = 50 mg

Dispersion:

Range Mean deviation Standard deviation Variance (σ2) Standard Error Mean (SEM)

Standard Deviation: It is defined as the square root of

the arithmetic mean of the squared deviations of the various items from arithmetic mean.

It is expressed as SD

It is calculated by the following formula

Standard Deviation: Weight in gram of 6 Frogs.

30, 90, 20, 10, 80, 70. For the above data: = 50.

X weight in gram

10 10 – 50 = - 40 +1600

20 20 – 50 = - 30 +900

30 30 – 50 = - 20 +400

70 70 – 50 = 20 +400

80 80 – 50 = 30 +900

90 90 – 50 = 40 +1600

= 300 = 5800

Standard Deviation:

SD =

=

=

= 34.05

Text Book : Basic Concepts and Methodology for the Health Sciences 21

Variance:It measure dispersion relative to the scatter of

the values about there mean. a) Sample Variance ( ) : ,where is sample meanx

2S

1

)(1

2

2

n

xxS

n

ii

Text Book : Basic Concepts and Methodology for the Health Sciences 22

b)Population Variance ( ) :

where , is Population meanExample: slide no:20

Varience=( )2

= 1160

2

N

xN

ii

1

2

2

)(

Standard Error Mean: In a small sample size the arithmetic

mean would be an approximation of the true mean of the whole population, and therefore subject to error.

In such cases the error of the observed mean is calculated.

The SE allows to find out the range in which the true mean would lie.

It gives an estimate of the extent to which the mean will vary if the experiment is repeated.

Standard Error of the Mean:

SE=

SE of the previous example.

SE=

= 13.05

Probability: The term probability means

“chance” or “likelihood” of the occurrence of the event.

It is defined as the symbol ‘P’.

Where, m= Number of favorable events

N= Total number of events

Test of Significance

In scientific research, a sample investigation produces results which are helpful in making decisions about a population

We are interested in comparing the characteristics of two or more groups.

The two samples drawn from the same population will show some difference

Difference can be controlled by “Test of significance”

Procedure for Test of Significance1. Laying down Hypothesis:a) Null hypothesis: Hypothesis which is to be actually

tested for acceptance. b) Alternative hypothesis: Hypothesis which is

complementary to the to the null hypothesis. Eg. avg of gene length is 170 kbpHo:µ=170

H1:µ=170

i.e, µ>170 or µ<170

2. Two types of error in testing of hypothesisa) Type I error: Rejection of null hypothesis which is trueb) Type II error: Acceptance of null hypothesis which is

false

3. Level of significance Minimize Type I & II error Level of significance is denoted by α α is conventionally chosen as 0.05 (moderate

precision) or 0.01 (high precision) In most biostatistical test α is fixed at 5%, means

probability of accepting a true hypothesis is 95%

4. One & two tailed tests of hypothesis In a test the area under probability curve is divided

into Acceptance region Critical/ rejection region

Types of test of Significance Two types of test used in

interpretation of results.

(1)Parametric test:- It involves normal distribution.

It includes: Student’s t-test Analysis of variance(ANOVA)

Regression Correlation

Z- test

Test of Significance

(2)Non-Parametric test:- It involves when the sample

data does not follow normal distribution.

It includes: Chi-squared test Wilcoxon Signed-

rank test Kruskal-Wallis test

Student ‘t’ test:

This test is applied to assess the statistical significance of difference between two independently drawn sample means obtained from two series of data with an assumption that the two mean are from normal distribution population, with no significant variation

t= (difference of means of two samples)/(std error of difference)

Standard error of difference(Sd) = √{(S12/n1)+

(S22/n2)}

t= |X1 – X2|/ √{(S12/n1)+(S2

2/n2)}

Degrees of freedom = (n1+n2-2)

Ex. Following data related to disintegration time(DT) of Chloroquine tablets using diluent, Lactose monohydrate(LM), dibasic calcium phosphate (DCP).Determine whether the two means are significantly different.

Lactose Monohydrate

DCP

n 3o 35

mean 32 38

variance 9.62 14.23

Null hypothesis: Ho: There is no significant difference between the mean DT in choroquine tablets between LM & DCP

Sd = √{(S12/n1)+(S2

2/n2)} = √(9.62/30)+(14.23/35)=√0.73 = 0.85

Difference between mean = 38-32 = 6

t= |X1 – X2|/ √{(S12/n1)+(S2

2/n2)}

= |32-38|/ √{(9.62/30)+(14.23/35)}= 6/√o.73 = 7.06

Degrees of freedom = (n1+n2-2)= (30+35-2)=63

Conclusion: Calculated value of t(7.06)> tabulated value of t for

63(at 1%=2.66)So the two mean are very much differentSo the null hypothesis is rejected at p=0.01The difference between the two sample means is a

real difference because the level of significance is very high

Chi-square test:-In biological research apart from quantitative

characters one has to deal with qualitative data like flower color or seed color

Results of breeding experiments and genetical analysis comes under chi-square test

The quantity x2 describes the magnitude of difference between the observed & the expected frequency

x2 = ∑(fo - fe)2/fefo – observed frequencyfe – effective frequency

Determination of value of x2

1. Calculate the expected frequency(fe)2. Find out the difference between the observed

frequency(fo) and expected frequency(fe)3. Square the value of (fo-fe) i.e (fo-fe)2

4. Divide each value of fe & obtain the total ∑(fo - fe)2/fe value

5. The calculated value of x2 is compared with the table value for the given degrees of freedom(d.f)

d.f= (r-1) (c-1)where, r- no. of rows in table

c- no. of columns in table

Examples of x2 testIn F2 generation, Mendel obtained 621 tall

plants & 187 dwarf plants out of the total of 808. test whether these two types of plants are in accordance with the Mendelian monohybrid ratio of 3:1 or they deviate from ratio

Solution:Tall plants

Dwarf plants

Total

Observed frequency(fo)

621 187 808

Expected frequency(fe)

606 202 808

Deviation(fo-fe) 15 -15

Formula appliedx2 = ∑(fo - fe)2/fe

=(15)2/606+(-15)2/202= 225/606+ 225/202= 0.3713+ 1.1139= 1.4852

Tabulated value is 3.84 at 5% level of probability for d.f= 2-1 =1Therefore the difference between the observed

& expected frequencies is not significantHence the null hypothesis is true

Application of x2 test1. To test the goodness of fit2. To test the independence of attributes3. To test the homogeneity of independent

estimates of the population varience4. To test the detection of linkage

References Khan IA, Khatum A. Fundamentals of

Biostatistics. 3rd revised edition. Ukazz publication, Hyderabad

Brahmankar DM, Jaiswal SB. Biopharmaceutics & Pharmacokinetics.

Kulkarni SK. Textbook of Experimental pharmacology. Khan IA, Khatum A. Biostatistics in Pharmacy. 3rd edition.

Ukazz publikation, HydrabadJeffery GH, Bassett J,Mendham J, Denney RC. Textbook of

quantitative chemical analysis. Fifth edition. Vogel’s publication.