This document provides an overview of statistical tests of significance used to analyze data and determine whether observed differences could reasonably be due to chance. It defines key terms like population, sample, parameters, statistics, and hypotheses. It then describes several common tests including z-tests, t-tests, F-tests, chi-square tests, and ANOVA. For each test, it outlines the assumptions, calculation steps, and how to interpret the results to evaluate the null hypothesis. The goal of these tests is to determine if an observed difference is statistically significant or could reasonably be expected due to random chance alone.

1. Test of Significance
Vikash R Keshri
Moderator: Mr. M. S. Bharambe

2. Outline
• Introduction:
• Important Terminologies.
• Test of Significance:
– Z test.
– t test.
– F test.
– Chi Square test.
– Fisher’s Exact test.
– Significant test for correlation Coefficient.
– One Way Analysis of Variance (ANOVA).
• Conclusion:

3. Introduction:
• All scientists work look for the answer to following
questions:
– How probable the difference between the observed and
expected results by chance only ?
– Is the difference statistically significant?

4. Important Terminologies:
• Population & Sample:
Population is any infinite collection of elements i.e.
individual, items, observations etc.
A part or subset of population. But The Basic problem of
the sample is generalization.
• Parameters & Statistic:
A parameter is a constant describing a population.
Statistic is quantity describing the sample i.e. a function
of observation.

5. • Normal Distribution:

6. Sampling Distribution:
• The distribution of the value of statistics which would
arise from all possible samples are called sampling
distribution.

7. Standard Error (SE):
• The standard deviation of sampling distribution is called
as the Standard Error. It provides the estimate that how far
from the true value the estimated value is likely to be.

8. Confidence Limits:
Confidence Limit is range within which all the Possible
sample mean will lie.
 A population mean ± 1 Std. Error limit correspond to
68.27 percent of sample mean value.
 A population mean ± 1.96 Std. Error correspond to
95.0% of the sample mean values.
 Population mean ± 2.58 stand. Error corresponds to 99
% sample mean values.
 Population mean ± 3.29 correspond to 99.9% of the
sample mean value.
• Interval is confidence interval.

9. • Hypothesis:
A statistical Hypothesis is a statement about the parameter
(forms of population).
i.e. x1 = x2 or x = µ or p1 = p2 or p = P
• Null Hypothesis (H0):
It is hypothesis of no difference between two outcome
variables.
• Alternative Hypothesis (H1):
There is difference between the two variables under study.
• Hypotheses are always about parameters of populations,
never about statistic from samples.
• Test of Significance:
Testing the null hypothesis.

10. Type 1 and Type 2 Error:
Null Hypothesis
Test Result
True False
Significant
Accepting Hi
Rejecting Ho
Type 1 Error No error
Power (1- β)
Not significant
Accepting Ho
Rejecting Hi
No Error Type 2 Error

11. Parametric Vs. Non – Parametric test;
Parametric test
• Based on assumptions that
data follow normal
distribution or normal
family of distribution.
• Estimate parameter of
underlying normal
distribution.
• Significance of difference
known
Non parametric test
• Variable under study don’t
follow normal distribution
or any other distribution
of normal family.
• Association can be
estimated.

12. P – Value:
• P value provides significant departure or some degree of evidence
against null hypothesis.
• P value derived from statistical tests depend on the size and direction
of the effect.
• P < 0.05 = significant = 1.96 Std. Error = 95% Confidence Interval.
• P < 0.01 or p < .001 = highly significant = 99% and 99.9% Confidence
Interval.
• The Non Significant departure doesn’t provide the positive evidence in
favour of hypothesis.
• Dependent on Sample Size.
• If P > alpha, calculate the power
– If power < 80% - The difference could not be detected; repeat the
study with deficit number of study subjects.
– If power ≥ 80 % - The difference between groups is not statistically
significant.

13. One Sided ( One tailed) Vs. Two Sided (two
tailed) :
• Two Sided test:
Significantly large departure from Null Hypothesis in
either direction will be judged by significance.
• One Sided Test:
Is used we are interested in measuring the departure in
only one particular direction.
• A one sided test at level P is same as two sided test at level
2P.
• Example: test to compare population mean of two group A
and B
– Alternate Hypothesis mean of A > mean of B. – One tailed test.
– Alternate Hypothesis Mean of B > mean of A > mean of B. – two
tailed test.

14. STEPS :
• Defining the research question.
• Null Hypothesis (H0) - there is no difference between the
group.
• Alternative hypothesis (H1) – there is some difference
between the groups.
• Selecting appropriate test.
• Calculation of test criteria (c).
• Deciding the acceptable level of significance (α). Usually
0.05 (5%).
• Compare the test criteria with theoretical value at α.
• Accepting Null Hypothesis or Alternative Hypothesis.
• Inference.

15. Common concerns:
• Sample mean and Population mean
• Two or more sample mean.
• Sample Proportion (percentage) vs. Population proportion
(percentages).
• Two or more Sample Proportion (percentages).
• Sample Correlation Coefficient vs. population correlation
coefficient.
• Two sample correlation coefficient.

16. Why test of significance?
• Testing SAMPLE and commenting on POPULATION.
• Two different SAMPLES (group means) from same or different
POPULATIONS (from which the samples were drawn)?
• Is the difference obtained TRUE or by chance alone?
• Will another set of samples be also different?
• Significance Testing - Deals with answer to above Questions.

17. Standard Normal Deviate (Z) test
• Assumptions:
 Samples are selected randomly.
 Quantitative data.
 Variable follow normal distribution in the population.
 Sufficiently large sample Size.

18. The steps:
• To find out the problem and question to be answered.
• Statement of Null (H0)
• Alternative Hypothesis (H1).
• Calculation of standard Error.
• Calculation of Critical ratio.
• Fixation of level of significance. (α) critical level of significance.
• Comparison of calculated critical ratio with the theoretical
value.
• Drawing the inference.

19. Comparison of Means of Two Samples:
• Zc = x1 – x2 / SE (x1 –x2).
• SE of (x1 – x2) = √ [ (SE1
2 + SE2
2)]
• SE of (x1 – x2) = [SD1
2 /n1 + SD2
2/ n2] ½
• Example: We have to compare and infer from the given
data that the arm circumference of Indian and American
children.
Details American Indian
No. of Subjects 625 625
Mean 20.5 15.5
Standard Deviation 5.0 5.4

20. Interpreting Z value:
• Area under curve:
 Z 0.05, = 1.96
 Z0.001 = 2.56
 Z0.01 = 3.29
• If Calculated Z value (Zc ) > Z 0.05, Z0.01, Z0.001
• Null hypothesis is rejected
• Alternate Hypothesis is accepted.

21. Comparing Sample Mean with Population
Mean:
• Z = difference between sample and population mean / SE
of sample mean.
• SE of sample mean= sample std. deviation / square root of
n
• Example:
If the Mean weight of population Follow normal
distribution. Do the mean weight of 17.8 kg. of 100
children with std. deviation of 1.25 Kg. different from the
population mean wt. of 20 kg.

22. Difference between two sample Proportions:
• Difference in proportion / SE (Difference in proportion)
• Z = p1 – p2 / [PQ (1/n1 + 1/n2)]1/2
• Here p1 = Proportion of sample 1
p2 = Proportion of sample 2
• P = p1 n1 + p2n2 / n1 + n2 and Q = 1- P
• Example:
Given table provides data for Prevalence of Overweight and Obesity
among Indians and USA. can we conclude that the Prevalence of
Overweight and Obesity among Indians and USA is same?
Details India USA
Sample Size 500 500
Prevalence of overweight or
obesity
p1 = 28.0 p2 = 30.0
Proportion 0.28 0.30

23. Comparison of Sample Proportion with
Population Proportion:
• Zc = Difference between sample proportion and
population proportion / SE of Difference between sample
proportion and population proportion.
• Zc = p – P / [PQ (1/n)] ½
• p= Sample proportion , P = Population Proportion and Q
= 1-P. , n = Sample Size.
• Example:
In school health survey the prevalence of nutritional
dwarfism among the school age children in class 10 is 18.3.
Sample size studied was 250. Does it confirm that 20% of
school age of children is nutritional dwarf?

24. Variance Ratio test (F – test).
• Developed by Fisher and Snedecor.
• Comparison of Variance between two groups (or Sample).
• Involves the distribution of F.
• Applied If the
 SD 1
2 and SD 2
2 of two sample is known.
 SD 1
2 > SD 2
2 than
 SD 1
2 / SD 2
2 follows the F distribution at n1 -1 and n2 – 1
Degree of Freedom.
• F = SD 1
2 / SD 2
2
• Example:
SD1
2 of 25 males’ adults for height is 5.0. SD 1
2 for 25 females
is 9.0. Can we conclude that the variance in height is same
in both male and female adults?

25. t – test:
• Prof. W.S. Gosset. ( pen name of student.)
• Difference b/w Normal and t Distribution:
• Very Small Sample size don’t follow the normal
distribution.
• They follow the t distribution.
• Bell shaped vs. symmetrical.

26. Prerequisite:
Unpaired data:
– Sample size is small (Usually < 30)
– Population variance is not known.
– Two separate group of samples drawn from two separate
population group.
– These two groups can be control and cases also.
Paired data:
– Applied only when each individual gives a pair of data.
i.e. study of accuracy of two instruments or study on
weight of one individual on two different occasion.

27. Assumptions:
 Samples are randomly selected.
 Quantitative data.
 Variable under study follow normal
distribution family.
 Sample variances are mostly same in both
group.
 Sample size is small (usually < 30).

28. Unpaired t test:
• Mean of two independent samples.
• Example:
• Mean value of birth weight with std. deviation is given
below by socio- economic status.
• Small randomly selected sample size. Variance is mostly
the same, so t test can be applied.
Details HSES LSES
Sample size 15 10
Mean Birth weight 2.91 2.26
Standard deviation 0.27 0.22

29. Steps:
• State Null hypothesis (H0): X1 = X2
• Alternative Hypothesis (H1): H0 is not true.
• Test criteria t = mean difference between two samples / SE
(mean difference between two samples)
• t = x1 – x2 / SE (x1 – x2).
• SE (x1 – x2) = SD [1/n2 + 1/n2]1/2 SD = [(n1-1)SD 1
2 + (n2 -1)
SD2
2 / n1 + n2 -2]
• Calculate df = (n1 – 1) + (n2-1) = n1+ n2 -2.
• Compare of calculated t value with its table value at t0.05,
t0.01 , t0.001 at n1+ n2 -2 df.
• Inference: if calculated value is > or equal to theoretical
value Null Hypothesis rejected.

30. Difference between sample mean and
population mean:
• t = [x – u ] / SE
• t = [x – u ] / SD/ n1/2
• Degree of freedom: n -1
• Example:
– mean Hb. Level of 25 school children is 10.6 gm% with
SD of 1.15 gm. / dl. Is it significantly different from
mean value of 11.0 gm%.

31. For difference between two small sample
Proportion:
• t = p1 – p2 / [PQ (1/n1 + 1/n2)]1/2
• P = p1 n1 + p2n2 / n1 + n2 Q = 1- P
• df = n1+ n2 -2.
• Example:
Proportion of infant with frequent diarrhea by type of
feeding habits is given. Is there significant difference
between the incidence of frequent diarrhea among EBF
babies and not EBF babies.
Details Exclusive breast fed Not EBF
Sample size 30 30
Percentage of infants
with diarrhea
10.0 80.0
Proportion 0.10 0.80

32. Paired t test:
• Pre-requisite:
– When each individual is providing a pair of result.
– When the pair of results are correlated.
• t = mean d – 0 /SE (d)
• t = mean d / SD/ (n)1/2
• SE = SD / (n)1/2 = [SD2 / n ] 1/2
• SD2 = Σ (d - mean d)2 / n-1
• Σ (d - mean d)2 = Σ d2 – (Σ d)2/n

33. Example:
The fat fold at triceps was recorded on 12 children before and at the
end of commencement of feeding programme. Is there any
significant change in the fat fold at triceps at the end of the
programme?
Child no. Triceps before
X1
Triceps after
X2
Difference (d)
X2 – X1
d2
1 6 8 2 4
2 8 8 0 0
3 8 10 -2 4
4 6 7 1 1
5 5 6 1 1
6 9 10 1 1
7 6 9 3 9
8 7 8 1 1
9 6 5 -1 1
10 6 7 1 1
11 4 4 0 0
12 8 6 2 4
Σ d = 9 Σ d2=27

34. • t = mean d – 0 /SE (d) = mean d / SD/ (n)1/2
• Σ (d - mean d)2 = Σ d2 – (Σ d)2/n = 27 – 81/12 = 27 – 6.75 =
20.25
• SD2 = Σ (d - mean d)2 / n-1 = 20.25 / 11 = 1.84
• SE = SD / (n)1/2 = [SD2 / n ] 1/2 = [1.84 / 12]1/2 = [0.1533]1/2 =
0.3917
• t = 0.75 / 0.3917 = 1.92
• df = n -1 = 11
• calculated t value is < t0.05 at 11 df. Difference is not
statistically significant.

35. Chi Square (Ϫ2) test:
Underlying theory:
If the two variables are not associated the value of observed
and expected frequencies should be close to each to each
other and any discrepancies should be due to
randomization only.
• Non-parametric test.
• Statistical significance for bivariate tabular analysis.
• Evaluate differences between experimental or observed
data and expected or hypothetical data.

36. Ϫ2 Assumptions:
1. Quantitative data.
2. One or more categories.
3. Independent observations.
4. Adequate sample size.
5. Simple random sample.
6. Data in frequency form.

37. Contingency table:
• A frequency table where sample classified in to two
different attributes.
• A contingency table may be 2 x 2 table or r x c table.
• Marginal total = (a + b) or (a + c) or (c + d) or (b +d)
• Grand total = N = a + b + c + d
• Expected value (E) = R X C / N
where R = row total, C = Column total and N = Grand total.
Disease Smoker Non – smoker Total
Cancer 6 a 4 b 10 (a + b)
No cancer 94 c 96 d 190 (c + d)
100 (a+c) 100 (b+d) 200 ( a +b +c +d)

38. • Calculation:
= (O – E) 2 / E
• Degree of freedom: df = (r-1) (c-1)
• for 2x2 table:
Ϫ2 = (ad – bc)2 N / (a+b) (b+d) (c+d) (a+c) with 1 df

39. • In given example calculation of expected value:
Ea = 10 x100 / 200 = 5 O – Ea = 1 (O – Ea)2= 1
Eb = 10 x100 / 200 = 5 O –Eb = -1 (O-Ea)2 = 1
Ec = 190 x 100 /200 = 95 O- Ec = 95 -96 = 1
(O-Ec)2 = 1
Ed = 190 x 100 /200 = 95 O- Ed = -1
(O-Ed)2 = 1
•
• Ϫ2 = 4 at 1 df
• Calculated value Ϫ2 < Ϫ2 at 0.05 for 1 df. The difference is
statistically significant

40. Yates's continuity correction:
• Described by F. Yates.
• When the value in a 2x2 table is fairly small , correction for
continuity is required.
• No precise rule for situation in which the Yates correction
needs to be applied.
• Generally it is applied if the grand total is < 100 or a
Expected value is < 5 in any cell.
• Ϫ2 = [(ad – bc) –N/2]2 N / (a+b) (b+d) (c+d) (a+c)

41. Exact Probability test or Fisher’s Exact test:
Cochran’s Criteria:
• Recommended by W. G. Cochran in 1954.
• Fisher’s Exact test should be used if:
– If n < 20
– 40 < n >20 and smallest expected value is less than 5.
– For contingency table more than 1 df the criteria states
that if Expected value < 5 in more than 20% of cells.
• What if the observed value is 0 in one cell?
– Chi square can still be applied if it fulfills the above
criteria of expected value.

42. Fisher’s Exact test…….
• Devised by Fisher, Yates and Irwin.
• Example:
Survival rate after two different types of treatments:
• Is the difference in survival statistically significant?
• No. of tables possible with marginal total is 4 = lowest
total marginal +1.
Survived Died Total
Treatment A 3 1 4 r1
Treatment B 2 2 4 r2
5 s1 3 s2 8 n

43. Table 1 Survived Died Total
Treatment
A
4 0 4 r1
Treatment
B
1 3 4 r2
5 s1 3 s2 8 n
Table 2 Survived Died Total
Treatment
A
3 1 4 r1
Treatment
B
2 2 4 r2
5 s1 3 s2 8 n
Table 3 Survived Died Total
Treatmen
t A
2 2 4 r1
Treatmen
t B
3 1 4 r2
5 s1 3 s2 8 n
Table 4 Survived Died Total
Treatmen
t A
1 3 4 r1
Treatmen
t B
4 0 4 r2
5 s1 3 s2 8 n

44. • Exact probability P value
=
• The P value for each table is 0.O71, 0.429, 0.429 and 0.071.
• Table 2 is similar to the test table.
• Final P value:
• Conventional Approach:
P = P of observed set + extreme value
= O.429 +0.071 = 0.5
• Mid P approach given by Armitage and Berry:
P = 0.5 X observed P + Extreme value
= 0.2145 + 0.071 = 0.286
• Exact probability is essentially One sided.
• For two sided test double the P value.

45. Significance test for Correlation Coefficients:
• Sample correlation coefficient (r) and Population with
correlation coefficient (r = 0 in population).
• Is the sample correlation coefficient r is from the
population with correlation coefficient o?
• Valid if at least one variable follow normal distribution.
• Null hypothesis H0 p = 0. Sample correlation coefficient is
zero).
• Std Error of r = [(1-r2)/ n-2] 1/2
• For small sample test:
t = r – 0 / SE (r) = r / SE ( r) at n-2 df.

46. Example:
• Correlation coefficient between intake of calories and
protein in adults is 0.8652. The sample size studied was 12.
Is this r value statistically significant?
• First calculate SE(r ) = [ 1-(0.8652)2/ 10]1/2 = 0.1585
• t = r – 0 / SE (r)
t = 0. 8652 / 0.1585 = 5.458
• df = n -2 = 10
• t value is > t value at 0.001 for 10 df.
• so the r value is highly significant.

47. Two independent correlation coefficient.
• r1 and r2 are two independent correlation coefficient based
on n1 and n2 sample size.
• First z transformation: (also known as Fisher’s Z
transformation).
Z1 = ½ log 1+r1 / 1-r2 and Z2 = ½ log 1+r2 / 1-r1
• For small sample t test is used:
t = Z1 - Z2 / [1/ n1 -3 + 1/n2-3]1/2 at n1 + n2 – 6 df.
• For large sample test of significance:
Z = Z1 - Z2 / [1/ n1 -3 + 1/n2-3]1/2
• Z value follow normal distribution.

48. • Example:
Correlation coefficient between protein and calorie intakes
calculated from two samples of 1200 and 1600 are 0.8912 and
0.8482 respectively. Do the two estimates differs significantly?
n1 = 1200 n2 = 1600 r 1 = 0.8912 and r2 = 0.8482
• then Z1 = 1.4276 and Z2 = 1.2496 from fisher’s table
• Z = Z1 - Z2 / [1/ n1 -3 + 1/n2-3]1/2 = 4.659
• Z calculated > Z at 0.001 level.
• The difference in correlation between two sample is highly
significant.

49. Effect of Sample Size:
• If sample size is 12 and 16.
• Data given:
n1 = 12 n2 = 16 and r 1 = 0.8912 , r2 = 0.8482
• Z1 = 1.4276 and Z2 = 1.2496 from fisher’s table
• t = Z1 - Z2 / [1/ n1 -3 + 1/n2-3]1/2
• t = 0.41.
• Df = n1 + n2 – 6 = 22
• Calculated t < t 0.05
• So P > 0.05.
• No difference between correlation Coefficient.

50. Conclusion: Significance of test of Significance ?
 Strength of association?
 Result is meaningful in practical sense ?
 Result fails the test of significance doesn’t mean there is no
relationship between two variables.
 Significance only relates to probability of result being commonly or
rarely by chance.
 The results are statistically significant but no clinical or biochemical
significance.
• Assumption for test of significance:
– Group to be equal in all respect other than the factor under study.
– Random selection of the patient for each group.
• Factors where significance test is not full proof:
– Small Sample size.
– Matching

51. Selecting Appropriate test:
 Goal of Analysis
 Type of Data
 Distribution of data
 No. of Groups
 Design of Study

52. Selecting Appropriate test:

53. Selecting Appropriate test ……

54. References:
• Rao VK. Biostatistics: A manual of statistical method for use in health
nutrition and anthropometry. 2nd ed. New Delhi: Jaypee Brothers;
2007.
• Armitage P, Berry G. Statistical Method in Medical Research. 3rd ed.
London: Oxford Blackwell scientific publication; 1994
• Swinskow TV, Campbell MJ. Statistics at Square One. 10th ed. London:
BMJ Books; 2002.
• Bland M. An Introduction to Medical Statistics. 3rd ed. New York:
Oxford University Press; 200.
• Moye LA. Statistical Reasoning in Medicine: The Intuitive P Value
Primer. 1st ed. New York: Springer- Verlag. 2000.
• Mahajan BK. Methods in Biostatistics. 7th ed. New Delhi: Jaypee
Brothers; 2010.