23-02-2024

Statistics:
Determination of
Sample Size
Dr. Nilesh Fichadiya
M.D.(Community Medicine), D.P.H.

Calculation of Sample Size

• What should be the sample size?
 To get the correct results

• To small samples  Study is not valid

• To large sample  Laborious, costly & time
consuming

• We need optimum sample size, which gives reliable

results

Sample and population

• If sufficiently large and unbiased sample is taken
from defined population  Inference drawn from
sample can be applied to whole population

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Sample and population

• If sample is representative of population, sample
statistics Mean (X ), SD and proportion(p) will
NOT differ significantly from population parameters
µ,  and P

Characteristics of Representative
Sample
1. Precision: Sample Size

2. Unbiased character: Technique of sample

selection

Precision
• Also known as reproducibility or reliability

• Defined as Ability of an instrument/test/sample to

provide the same or a very similar result with
repeated measurements of the same factor

• In case of sample, it depends on Sample size

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Precision

Precision =
n= sample size
s = Standard deviation, SD

Sample size depends on:

1. Study Objective and Study design:
 To estimate a value : Descriptive study

 To test hypothesis : Analytical study or experimental

study

 Whether we have one or two set of data

 Whether the data is paired or unpaired

Sample Size and Study Design

• In analytical study, there is ALWAYS comparison
 So we have cases and controls or comparison group
 Here we need to multiply sample size with (r+1)/r where
r= ratio of control to case

• In interventional study:
 We have intervention and control group
 We have to double the sample size

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Sample Size and Study Design

• In before-after study,
 We have one group
 We need not double the sample size

• In cluster sampling,
 We have to multiply the sample size with design effect
(discussed in previous topic)

Sample size depends on:

2. The degree of type I error (denoted as α)
 Type I error (α) is also known as p value or
level of significance
 1- α is confidence level (taken as 95% or 99%)

3. The degree of type II error (denoted as β)

 (1 – β) is “power” of the study (taken as 80% or 90%)

 and  errors
• It is the probability of the difference occurring by
chance and not in reality (Type I error) but we
conclude that the difference is real

• So, 1- (Confidence level) is probability the

difference is NOT occurring by chance

• Prior to starting a study, we set an acceptable value

for this “p.” which can be p<0.05 or p<0.01
(accordingly confidence level is 95% or 99%)

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 and  errors
•  error is the probability that the study will miss a
true difference
 i.e. there is a real difference but we miss to conclude
that

• So, (1 – β) or the “power” of the study is

probability that if there is a difference  we will
detect it correctly

 and  errors
• As a rule:
• For estimating a value  only  error is considered
• For testing hypothesis  both  and  errors are
considered

• If  error is considered in the formula of sample

size, we get a higher sample size

Sample size depends on:

4. Standard deviation (SD) or variance(s) in the
population

5. The proportion of people experiencing (p) and

people not experiencing (1-p or q) the attribute

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Sample size depends on:

6. The level of acceptable or allowable error (E)
 Can be taken as 5% or 10% or 20%

7. The degree of difference or effect size expected

between the two set of data (d)
 We can estimate the effect size based on previously
reported or preclinical studies
 If the effect size is large  the sample size is less
 If the effect size is small  the sample size is large

Sample size is larger if:

• Confidence level desired is large
• Power is large
• Variance (or SD) is large
• Proportion of people with attribute is small
• Margin of error (allowable error) is small

Various Formulae for Calculation of

Sample Size:
• To estimate the value by descriptive cross-sectional
study
 Quantitative variable (mean):

𝑍 𝑠
𝑁=
𝐸
where,
𝑍  = Value of Std. normal deviate for value of ,
s = standard deviation,
E = allowable error

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Value of Z1-/2 & Z1- for two

tailed and one tailed studies

Exercise
• Mean hemoglobin level of girls students in the
colleges is estimated to be 11.5 gm% with SD of 1.5
gm%. Calculate sample size for a study of
Haemoglobin estimation of physiotherapy colleges
of Saurashtra region with allowable error of 0.2 at
5% significance level

• 𝑁=

• N = (1.96)2(1.5)2
(0.2)2

= 3.84 x 2.25
0.04
= 216

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Sample size calculation: For

Quantitative data:
• Exercise:
• Mean pulse rate of a population is believed to be
72/minute with Standard Deviation of 8. Calculate
minimum sample size to verify this if allowable
error is 1 at 5% significance level.

Various Formulae for Calculation of

Sample Size:
• To estimate the value by descriptive cross-
sectional study
 Qualitative variable (proportion):

𝑍 𝑝𝑞 where,
𝑁= p = positive character,
𝐸
q = negative character = 1 – p, or
q = 100 – p in percentage as p+q =
100%,
E = allowable error of p, usually 10%
or 20% of p

Example
Incidence rate in the last SARS CoV epidemic was
found to be 50/1000 (5%) of the population exposed.
What should be the size of sample to find
incidence rate SARS CoV in the current epidemic if
allowable error is 10% and 20%?

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p = 5%,
q = p-100=100-5=95%
L = 0.5 (at 10% of p)
L = 1 (at 20% of p)

• Sample size calculation at 10% allowable error

n= 4pq = 4 x 5 x 95 = 7600
E2 0.5 x 0.5
• Sample size calculation at 20% allowable error
n= 4pq = 4 x 5 x 95 = 1900
E 2 1x1

Sample size calculation: For Qualitative

data:

• Exercise:
Prevalence rate of Musculoskeletal disorders were
found to be 40% in earlier studies.
Calculate the size of sample required to find the
prevalence rate of Musculoskeletal disorders in your
area if allowable error is 10% or 20%.

Various Formulae for Calculation of

Sample Size:
• To establish association by case control study or
independent sample cohort study:
 Quantitative Data (sample size for each group):

𝑁 =2 𝑍 +𝑍 2 𝑠 ×𝑠

---------------------------------------
𝑑

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Value of Z1- for different

Value of Z1-/2 & Z1- for two power of studies
tailed and one tailed studies

Example:
• An investigator wants to conduct a study to find out
whether there is any difference in effect of pollution on
lung function by studying force expiratory volume (FEV)
between traffic police and general population.
• From the previous study it is known that S.D.(s) of FEV are
3.5 l/min and 5 l/min among traffic police and general
population respectively.
• How many subjects from traffic police and general
population required for testing the null hypothesis that
there is no difference in FEV between traffic police and
general population.
• The investigator wishes to be 90% confident of detecting a
difference of 2.5 l/min or more in either direction at 5%
level of significance?

𝑁 =2 𝑍 +𝑍 2 𝑠 ×𝑠

---------------------------------------
𝑑
=2 (1.96 + 1.28)2 (3.5 x 5)
(2.5)2
= 2 (10.49 x 17.5)
6.25
= 58.74  59 from each population

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Various Formulae for Calculation of

Sample Size:
• To establish association by case control study or
independent sample cohort study:
 Qualitative Data (sample size for each group):

𝑁 = 𝑍 +𝑍 2 𝑝1𝑞1 + 𝑝2𝑞2

----------------------------------------------
𝑑

Example
• An investigator is interested to study whether the lower
back pain among Bank officers in Ahmedabad is more
or equal to the rest of the population?
• Previous studies reported that the prevalence of Lower
back pain among Bank officers in Ahmedabad is 55%
whereas it is 45% among general population.
• The investigator wishes to be 80% confident of
detecting a difference of 10% or more in either
direction at 1% level of significance.
• How many Bank Officers and members of General
population should be included in this study ?

𝑁= 𝑍 +𝑍 2 𝑝1𝑞1 + 𝑝2𝑞2

----------------------------------------------
𝑑
= (2.57 + 0.84)2 [(0.45 x 0.55) + (0.55 x 0.45)]
(0.1)2
= (3.41)2 (0.25 + 0.25) = 581
0.01

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Various Formulae for Calculation of

Sample Size:
• To identify significant difference between two
groups by intervention study
 Quantitative Data (sample size for each group):

𝑁 =2 𝑍 +𝑍 2 𝑠
For Unpaired Data,
------------------------- • Take pooled ‘s’
𝑑 • Calculated sample
size is for each
group

Calculate Pooled s or p

• Pooled 𝑠 =

• Pooled 𝑝 =

Various Formulae for Calculation of

Sample Size:
• To identify significant difference between two
groups by intervention study
 Qualitative Data (sample size for each group):

𝑁 =2 𝑍 +𝑍 2 𝑝𝑞
For Unpaired Data,
-------------------------------- • Take pooled ‘p’ & ‘q’
𝑑 • Calculated sample
size is for each
group

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Bias in Sampling
• Bias: A result that differs from the true values

• Examples of Bias

• Types:
1. Selection Bias
2. Information Bias
3. Measurement Bias
4. Bias due to confounding

Bias in Sampling
• Selection Bias
 Selection of subjects
 Non-response
 Loss to follow up

• Information Bias
 Quality and extent of information obtained from
different subject
 Recall bias

Bias in Sampling
• Measurement Bias
 Misclassification or mis-diagnosis of subjects
 Unequal diagnostic work-up in different groups
 Measurement error

• Bias due to confounding

 A factor that is associated with the risk factor and
disease outcome

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Thank You

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