This slide contains method of analysis of data( mean, median, mode, range. t-test, ANOVA etc..

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