Sample size is a crucial aspect of research design, determining the minimum number of participants required to detect a meaningful effect while ensuring statistical validity and ethical feasibility. An inadequate sample size may lead to unreliable results, while an excessively large sample increases costs and complexity. Key factors influencing sample size include P-value, power, confidence interval, margin of error, effect size, and variability. The P-value and alpha (α) set the threshold for statistical significance, with lower alpha requiring a larger sample. Power (1 - β) represents the probability of detecting a true effect, typically set at ≥80%, meaning a 20% chance of missing a real effect (Type II error). A narrow confidence interval (CI) ensures a more precise estimate, whereas higher variability in data necessitates a larger sample to maintain accuracy. Different statistical formulas are used to estimate sample size depending on study design. For continuous variables, sample size is calculated based on the mean difference and standard deviation between groups. For proportions, calculations consider the expected event rate in study groups. Case-control and cohort studies use different approaches to compare qualitative and quantitative variables, while animal studies apply the resource equation method to determine an optimal number of subjects. Additionally, researchers adjust for potential dropouts and confounding factors, following the 10% rule, which recommends increasing the sample size by 10% per confounder. Software tools such as G*Power, OpenEpi, and nQuery simplify these calculations, ensuring that the study remains statistically sound while being cost-effective and ethically appropriate.

1. Dr Shreedhar Angadi 1
Calculating Sample Size And Power
Dr. Shreedhar Angadi
Junior Resident 3
Department of Pharmacology & Therapeutics
King George’s Medical University, Lucknow, U.P., India
E-mail: drshreedhar.kgmu@gmail.com
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Content
Introduction
Parameters required for sample size calculation
Formulas for sample size calculation
Software-based sample size calculation
References
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Learning Objectives
1.Understand sample size and power importance
2.Identify parameters influencing calculations
3.Apply formulas for sample size estimation
4.Explore software tools for calculations
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Introduction
• Sample Size is the number of participants included in a study to represent a
population
• It determines the minimum number of participants required to detect a clinically
relevant treatment effect
• It Ensures validity, reliability, and ethical balance
• Too Small: Invalid results, poor population coverage
• Too Large: Unnecessary cost, ethical concerns, false significance
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PARAMETERS REQUIRED FOR SAMPLE SIZE CALCULATION
1.P value ( alpha)
2. Power
3. Confidence interval
4. Margin of error
5. Precision
6. Effect size
7. Variability
8. Two-tailed/One-tailed
test
9. Event rate in population
10. Dropout rate
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1.P value
• P value: The P-value is the calculated probability of obtaining results as extreme
as those observed in your study, assuming the null hypothesis is true
• Alpha (α): Pre-set threshold for statistical significance
• Common α Value: 0.05 (5% risk of Type I error)
• P value and α : P value < α: Reject null hypothesis
• α and Sample Size : Lower alpha requires larger sample size
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2.Power
• It is the probability that a statistical test will correctly reject the null hypothesis
when it is false
• It is the probability of detecting a true effect
• Power = 1 - Type II error (β)
• Standard: Power ≥ 80% (β ≤ 20%)
• Influencing Factors : Sample size, Effect size, Variability
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3.Confidence Interval
• It is a range within which the true value of the population parameter lies
• Example: BP reduction= 10 mmHg (95% CI: 8–12mmHg)
• Confidence Level:The percentage (or probability) that the confidence interval
contains the true population parameter across many samples
Confidence level = 1 - α
• Factors Affecting CI: Sample Size, Confidence Level , Variability
• CI and Precision:Narrow CI = More precise estimate
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4.Margin of Error (MOE)
• It represents the range within which we expect the true population parameter to
lie, based on our sample data
• It indicates how much the sample estimate is expected to differ from the true
population value due to random sampling
• Expression: ± Deviation from population mean
• Example : BP reduction: 10 mmHg ± 2 mmHg
• CI and MOE :MOE is half the width of the confidence interval
• Factors Affecting MOE : Sample Size, Confidence Level ,Variability
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5.PRECISION
• It refers to how consistently an estimate or measurement can be reproduced
• It indicates the degree of variability or consistency in repeated measurements or
estimates. In other words, it is how close repeated results are to each other
• Example : Blood pressure readings like 120 mmHg,121 mmHg,119 mmHg
• Precision vs. Accuracy
• Factors Affecting Precision : Sample size, Measurement tools , Consistency of data
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6.Effect Size (ES)
• It describes the magnitude of the difference or relationship between two groups,
treatments, or variables
• It helps in understanding how big the effect is rather than just whether the effect
exists (which is what p-value indicates)
• Types:
• Cohen’s d : Compares 2 groups(e.g., treatment vs placebo)
• Pearson’s r : Measures the strength of a linear relationship between 2 variables
• Odds Ratio : Quantifies the odds of an event in one group compared to other
• Example : Drug A reduces BP by 10 mmHg (mean difference) with a SD of 15 mmHg
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7.Variability
• It refers to how much the data points in a dataset differ from the mean or central
value
• It is a measure of the spread or dispersion in the data
• Example: FBS -Group 1: 90, 91, 92, 89, 88 mg/dL and Group 2: 70, 110, 85, 120, 95
mg/dL
• Variability and Power
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Type Description
Range Difference between maximum and minimum values
Variance Average of squared deviations, measures overall spread
Standard Deviation (SD) Square root of variance, same units as data, measures
spread

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Feature One-Tailed Test Two-Tailed Test
Test Direction
Tests for effect in one direction
(greater or smaller)
Tests for effect in both
directions (greater or
smaller)
Critical Region
Only on one side of the distribution
(either left or right)
Critical regions on both
sides of the distribution
Hypothesis Null hypothesis is tested against a
specific direction (e.g., > or <)
Null hypothesis is tested
for deviations in both
directions (e.g., ≠)
Type of Research
Used when you have a specific
direction in mind for the effect
Used when the effect could
go in either direction
Significance Level
Split across one tail (α = 0.05 in one
tail)
Split across two tails (α =
0.025 in each tail )
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8.Two-Tailed vs One-Tailed Test

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9.Event Rate
• It is the proportion of individuals who experience a specific event in the total
population or in a particular group.
• Formula:
• Example:50 out of 100 patients experience side effects : ER=50%
• Role of ER in sample size calculation : Lower ER requires a larger sample size to
detect a difference
• ER in different types of studies:
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Cross-Sectional Studies Represents the prevalence of a condition in the population.
Clinical Trials To refer to Proportion of participants experiencing an adverse
event, disease progression, or treatment response.

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10.Dropout Rate
It is the percentage of participants who fail to complete the study due to various
reasons, such as side effects, lack of compliance, or personal reasons
Formula:
Importance : High dropout rate = larger sample size needed.
Adjusted Sample Size:
Impact on Study Design:
Reduces statistical power
Bias in results
Increased Cost and Time
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Rule of Thumb : Critical Z-Scores for Confidence Levels
Confidence
Level (%)
Critical Zα​Score-
(Two-Tailed)
Critical Zα/2-Score
(One-Tailed) Application
90% 1.645 1.28
Exploratory studies or less
stringent precision
requirements.
95% 1.96 1.645
Most common in research for
balance of precision and
certainty.
99% 2.576 2.33
Critical studies where high
certainty is required.
99.9% 3.291 3.09
Rarely used; for extremely
high precision requirements.
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FORMULAS FOR SAMPLE SIZE CALCULATION
For Single
Group
Mean
1
For
Comparing
Two Means
2
For Single
Group
Proportions
3
For
Comparing
Two
Proportions
4
For Case Control
Studies
5
For Cohort
Studies
6
For Animal
Studies
7
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Formula for Sample Size:
Example:
A junior resident is conducting a thesis study to evaluate the effect of a new
antidiabetic drug on the HbA1c levels of patients with Type 2 Diabetes Mellitus (T2DM).
In a pilot study, the drug resulted in a mean HbA1c reduction of 2%, with a standard
deviation (SD) of 4%. The resident sets the alpha level at 5% for a two-tailed test.
Sample Size:16
If the possible dropout is 20%, then the adjusted sample (Nadj): ? 20
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1.For Single Group Mean
Z alpha-1.96,SD:4,d=2

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2.For Comparing Two Means
Formula for Sample Size:
Example: A researcher is conducting a randomized placebo-controlled trial to
assess a new drug's effect on hemoglobin levels in iron-deficiency anemia patients.
A pilot study showed a 2 g/dL increase in hemoglobin with a standard deviation of
4 g/dL. The study uses a 5% alpha level (Zα/2 = 1.96), 80% power (Zβ = 0.84), and a
1:1 allocation ratio.
Sample Size:62 participants(Each group=31)
If 1:2 allocation rate, then (N)=? 93 Participants (Placebo 31+Treatment 62)
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r=1,Z beta=0.84,SD=4,d=2

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3.For Single Group Proportions
Formula for Sample Size:
Example : A researcher is evaluating the efficacy of a new antibiotic in preventing
postoperative staphylococcal infections at the incision site. Data shows a prevalence of
such infection is 70%. The researcher aims to detect a 10% reduction in the infection
rate, considering this a significant outcome. Researcher fixed the alpha level at 5% (for
two-tailed) and the study is powered at 80% .
Sample Size:9 Participants
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p(proportion of events in a population)=70%=0.7
q(Proportion of non events) =1-p=(1-0.7)=0.3
d=70%-10%=60%=0.6

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4.Comparing Two Proportions
Formula for Sample Size:
Example :A researcher is conducting a study to compare the effectiveness of a new
antibiotic against standard treatment for preventing postoperative staphylococcal
infections at the incision site . Literature review shows that 15% of patients receiving
standard treatment develop infections, while previous pilot study shows 5% of patients
receiving the new antibiotic develop infections.
Sample Size:80
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P1=0.05, p2=0.15, p=(p1+p2/2)=0.1,
d=(p1-p2)=10%=0.1

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A . For Qualitative Variables (Proportions)
Formula:
A researcher is conducting a case-control study to investigate the association
between smoking (risk factor) and lung cancer. Previous studies show that the
proportion of smoking exposure in the lung cancer (case) group is 0.4 and in the
control (non-cancer) group is 0.2. The study is set with a 5% alpha level, 80%
power, and equal numbers of participants in both groups.
Sample Size (N) = 82 (41 cases, 41 controls)
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5.For Case-Control Studies
P1=0.4, p2=0.2, p=0.3 , d=0.2,

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B . For Quantitative Variables
Formula:
Example : A researcher is studying the association between cognitive decline
(MMSE scores) and Alzheimer’s disease. The expected difference in MMSE scores
between the Alzheimer’s group and healthy controls is 4 points, with a standard
deviation of 6 points. Using a 1:1 case-control ratio, 95% confidence (Z = 1.96), and
80% power (Z = 0.84), the required sample size for each group is?
Sample size: 35 participants (35 cases and 35 controls)
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r=1, s=6, d=4

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Formula :
Example: A researcher is conducting a cohort study to evaluate the impact of regular
30-minute walking on cardiovascular mortality. According to previous literature, the
proportion of cardiovascular mortality is 20% among those who do regular walking
and 40% for those who do not walk regularly. The study is set with a 5% alpha level,
80% power, and equal numbers in both groups.
• Sample Size: 92 participants (46 per group)
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6.Sample Size for Cohort Studies
p1=proportion of events in non exposed group=0.4
P2=proportion of events in exposed group=0.2
P=p1+p2/2 = 0.3
m=ratio of exposed to unexposed participants=1
d=p1-p2=0.4-0.2=0.2

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• Formula:
• Guidelines for E:
• Optimum Size: 10≤E≤20
• If E<10: Add more animals
• If E>20: Reduce sample size
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7. For Animal studies :Resource equation method

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Example Calculation:
1.Initial Case:
Groups: 4 (positive control, negative control, low-dose, high-dose)
Animals per group: 6
Total animals: 4×6=24
E=24−4=20 (Appropriate sample size)
2.Adjusted Case:
Animals per group: 7
Total animals: 4×7=28
E=28−4=24 (Too large, reduce sample size)
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The 10% Rule
• Initial Sample size calculations assume a simple relationship (exposure →
outcome) i.e., no confounders are considered
• Confounders can distort results if not adjusted
• The 10% Rule: “ Increase the sample size 10% for each confounder added”
• Ensures study accuracy and power
• Example:
Initial Sample Size: 100 participants
Confounders: Age, gender, smoking status (3)
Adjusted Sample Size: 133 participants
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Software Type Software Name Link/Description
Free Software
1.G*Power http://www.gpower.hhu.de
2.OpenEpi OpenEpi Menu
3.R Packages https://cran.r-project.org/web/packages/pwr
Paid Software
a. PASS (Power Analysis
and Sample Size
Software)
https://www.ncss.com/software/pass
b. nQuery
How to use nQuery
- Calculate sample size and optimize your tri
als
c.SPSS (Sample Power) Power Analysis - IBM Documentation
d.STATA (power) https://www.stata.com/features/power-and-
sample-size/
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Software-Based Sample Size Calculation

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Summary
• Sample Size: Balances validity, reliability, and ethics.
• Power: Ensures detection of true effects.
• Key Parameters: P value,Power, CI, MOE, ES, Variability.
• Adjustments: Account for dropouts and variability.
• Tools: G*Power, OpenEpi, and nQuery streamline calculations.
• Outcome: Accurate, ethical, cost-effective research design
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REFERENCES
• Mehta T. Basic Course in Biomedical Research Handbook. 1st ed. Chennai: Notion Press; 2021.
• Gupta KK, Attri JP, Singh A, Kaur H, Kaur G. Basic concepts for sample size calculation: critical step for
any clinical trials! Saudi J Anaesth. 2016;10:328-31.
• Hazra A, Gogtay N. Biostatistics series module 5: Determining sample size. Indian J Dermatol.
2016;61:496-504.
• Charan J, Biswas T. How to calculate sample size for different study designs in medical research?
Indian J Psychol Med. 2013;35:121-6.
• Bujang MA. A step-by-step process on sample size determination for medical research. Malays J
Med Sci. 2021;28:15-27.
• Das S, Mitra K, Mandal M. Sample size calculation: basic principles. Indian J Anaesth. 2016;60:652-6.
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THANK YOU
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•What is precision in research?
•How does the effect size influence sample size?
•Differentiate between a one-tailed and a two-tailed test.
•What parameters are needed to calculate sample size?
•List some software tools used for sample size estimation.
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Questions