This document describes various statistical validation methods used to analyze finite sample data, including measures of central tendency, dispersion, skewness, correlation, and regression. It also discusses different types of statistical tests like the t-test, F-test, and ANOVA that are used to test hypotheses and determine statistical significance. The document provides examples and formulas for calculating various statistical measures and performing tests on sample data sets.

1. STATISTICAL VALIDATION METHOD
VIZ. STATISTICAL TREATMENT OF
FINITE SAMPLE
By:
Sachin kumar
M.Pharm. (Pharmacology)
Deptt. of Pharma. Sciences
M.D.U. Rohtak, 124001

2. CONTENTS
• INTRODUCTION
• ANALYSIS OF DATA
1. MEASURES OF CENTRAL TENDENCY
2. MEASURES OF DISPERSION
3. SKEWNESS
4. CORRELATION
5. REGRESSION
• TEST OF SIGNIFICANCE
1. T-TEST
2. F-TEST
3. ANOVA
• REFERENCES

3. INTRODUCTION
• Biostatistics :- It is defined as the application of
statistical method to the data derived from
biological sciences.
• Statistics :- It is the collection of methods used in
planning an experiment and analyzing data in order
to draw accurate conclusions.
- It include collection, organization,
presentation, analysis and interpretation of
numerical data.
• Data :- Facts or figures from which conclusion can
be drawn.
- It may be qualitative or quantitative.

4. ANALYSIS OF DATA
• Analysis can be done through different
statistical techniques:-
1. Measures of central tendency
2. Measures of dispersion
3. Skewness
4. Correlation
5. Regression

5. 1. MEASURES OF CENTRAL
TENDENCY
• The observation of set of data exibit a tendency
to cluster around a specific value. This
characterstic of data is central tendency.
• The value around which individual observation
are clustered is called central value.
• Three main measures of central tendency.
1. Mean
2. Median
3. Mode

6. • MEAN- It is the mathematical average denoted
by x-bar.
(a) Arithmetic mean (simple mean)-
for ungrouped data-
for grouped data-

7. (b) Geometric mean-
(c ) Harmonic mean-

8. • Median- Central value when arranging in ascending
or descending order. Denoted by ‘M’.
for ungrouped data-
n is odd→ M = [(n+1)/2]th value
n is even →
M = [ (n/2)th value + (n/2 +1)th value ] /2
for grouped data-
M = L+ [ (n/2-F)/f ] x c
L= lower limit of median class
F= frequency of the class preceding
the median class
c= width of the median interval

9. MODE- Most commonly occuring value
-for ungrouped data-
which occurs maximum no. of times
-for grouped data-
mode= L+ [( f₁-f₀) / 2f₁- f₀-f₂]x c
L= lower limit of mode class
f₀= frequency of class preceding the m.c.
f₁ = frequency of class succeeding the m.c.
f₂= frequency of mode class
c= width of mode class
mode class= class which have maximum frequency

10. 2. MEASURES OF DISPERSION
• For comparing two set of data sets, we require a
measures of dispersion. Dispersion indicate the
extent to which a distribution is squeezed.
• There are five main measures of dispersion:
- Range
- Interquartile range
- Mean deviation
- standard deviation
- variance

11. • RANGE- It is the simplest measure of dispersion.
Range= L-S
L= largest observation
S= smallest observation
• INTERQUARTILE RANGE- Problem with range
such as instability from one sample to another or
when added new sample. So we calculate I.R.
I.R.= Q₃-Q₁
Q₁ = first quartile
Q₂= second quartile
Q₃= third quartile

12. • MEAN DEVIATION- The average absolute deviation
from the central value of a data set is called mean
deviation.
-For grouped data-
M.D. about mean = (∑|xᵢ-x̅|) /n
M.D. about median = (∑|xᵢ-M|) /n
M.D. about mode = (∑|xᵢ-Z|) /n
- For grouped data-
M.D. about mean = (∑fᵢ|xᵢ-x̅|) / ∑fᵢ
M.D. about median = (∑fᵢ|xᵢ-M|) / ∑fᵢ
M.D. about mode = (∑fᵢ|xᵢ-Z|) / ∑fᵢ

13. • STANDARD DEVIATION- It tell us how much
scores deviate from the mean. Denoted by sigma
or S.
Standard error of mean(SEM)= S/ n

14. • VARIANCE- It tell us how far a set of numbers
are spread out from their mean.
-Variance is the square root of standard
deviation.

15. 3. SKEWNESS
• It is the measure of degree of asymmetry of the
distribution.
(a) Symmetric- Mean, median, mode are the
same.
(b) Skewed left- Mean to the left of the median,
long tail on the left.
(c) Skewed right- Mean to the right of the
median, long tail on the left.
• Coefficient of Skewness = (mean-mode)/ S.D

17. 4. CORRELATION
• In correlation we study the degree of relationship
between two variables.
- Types of correlation:
(a) positive or negative correlation
(b) simple or multiple correlation
• Correlation coefficient- It is a measure of
correlation . Denoted by ‘r’.
when r=1 (+ve correlation)
when r= -1 (-ve correlation)
when r=0 (no correlation)

19. 5. REGRESSION
• It is the functional relationship between two
variable.
-We take variable whose values are known as
independent variable and the variable whose
values are to predicted as the dependent
variable.
Line of regression of Y on X- It is used for
estimation of the variable Y for a give value of
the variable X.
X= Independent variable
Y= Dependent variable

20. Line of regression of X on Y- It is used for
estimation of the variable X for a give value of
the variable Y.
Y= Independent variable
X= Dependent variable
Regression coefficient- It is measure or regression.
Denoted by ‘b’.
bxy(X on Y) = ( n∑xy - ∑x∑y )/ n∑y²-(∑y)²
byx(Y on X) = ( n∑xy - ∑x∑y )/ n∑x²-(∑x)²

21. TEST OF SIGNIFICANCE
• It is the formal procedure for comparing
observed data with a claim (also called a
hypothesis) whose truth we want to assess.

22. 1. T-TEST
• Two types of t-test
(a) Unpaired t-test
(b) Paired t-test
Unpaired t-test- If there is no link between the
data. Data is independent.
- Testing the significance of single mean-

23. - Testing the significance of difference between
two mean-
- Degree of freedom= n₁ +n₂ -2

24. • PAIRED T-TEST- When the two samples were
dependent. Two samples are said to be
dependent when the observation in one sample is
related to those in other.
- When the samples are dependent, they have equal
sample size.

25. 2. F-TEST
• Used to compare the precision of two set of data.

26. 2. ANOVA
• Developed by Sir Ronald A. Fisher in 1920.
• A statistical technique specially designed to the
test whether the means of more than two
quantitative population are equal.
• Types of ANOVA
(a) One way ANOVA
(b) Two way ANOVA
-ONE WAY ANOVA- There is only one factor or
independent variable.
-TWO WAY ANOVA- There are two independent
variable.

27. ONE WAY ANOVA
• Suppose we have three different groups.
• There are 5 steps:
1. Hypothesis- Two hypothesis.
Null hypothesis H₀ = All mean are equal.
Alternate hypothesis = At least one difference
among the mean
Group- A Group- B Group-c
1 2 2
2 4 3
5 2 4

28. 2. Calculate degree of freedom(d.f)-
Between the group= k-1
k= No. of level
3-1= 2
With in the group= N-k
N= total no. of observation
9-3= 6
Total d.f.= 8
F- critical value- 5.14

30. 3. Sum of squared deviation from mean-
Calculate mean - X̅ᴀ= 2.67
X̅ʙ= 2.67
X̅ᴄ= 3.00
Grand mean X= sum of all observation/ total no.
of observation
25/9= 2.78
- Total sum of square= ∑(X-X̅)²
= 13.6
- Sum of square with in the group=
∑(Xᴀ-X̅ᴀ)² + ∑(Xʙ-Xʙ̅)² + ∑(Xᴄ-X̅ᴄ)² = 13.37

31. - Sum of square between the group =
total S.S. - S.S. with in the group
13.6-13.37= 0.23
4. Calculate variance-
between the group= S.S. between group/ d.f.
between group
- .23/2 = 0.12
with in the group= S.S. with in the group/ d.f.
with in the group
- 13.34/6= 2.22

32. 5. F-value-
variance between the group/ variance
with in the group
0.12/2.22= 0.5
RESULT- 0.5< 5.14
we fail to reject null hypothesis. Hence
there is no significant between these
three groups.

33. REFRENCES
Mendham J, Denny RC, Barnes JD, Thomas M,
Sivasanker B. Vogel’s textbook of quantitative
chemical analysis. 6th ed. Delhi: Pearson Education
Ltd; 2000: 110-133.
Patel GC, Jani GK. Basic biostatistics for pharmacy.
2nd ed. Ahemdabad: Atul Parkashan; 2007-2008.
Manikandan S. Measures of central tendency:
Median and mode. J Pharmacol Pharmacother.
2011: 2(3): 214-215.

34. KNOWLEDGE NOT SHARE, IS WASTED.
- CLAN JACOBS