The manual methods of calculating a sample size

Sample size calculation is the process used by social science researchers that helps to access the number of people needed to gather information from the people or collected data (Das et. al., 2016). Sample size simply indicates the number of people associated in a study. The sample size does not remain limited to quantitative study or clinical study where people participate but also qualitative studies whereby interviews and focus groups remain constructed. The sample size is dependent on the sampling plan of the study.

The margin of error

The margin of error or confidence interval is the amount of error that can be tolerated. Lower margin of error requires a larger sample size. The smaller the margin of error, the closer it gets in achieving the answer at a given confidence level. It remains expressed as a percentage and represents how often the true percentage of the population who would pick an answer lies within the confidence interval. The confidence interval shows how much significance remain expected from the survey results to reflect the views of the overall population.

Confidence level

The confidence level remains expressed as a percentage and represents how often the true percentage of the population pick an answer lies within the confidence interval. The 95% confidence level means you can be 95% certain; the 99% confidence level means you can be 99% certain. Most researchers use the 95% confidence level. Higher confidence level requires a larger sample size.

Population size

Population size is the total number of people expected to be added for the study.

Response distribution

Response distribution used as the percentage of the population needed to determine a general level of accuracy. It helps in assessing the expected sample size or predicted population in the study. For instance, choose 1000 respondents and response distribution of 50, then there is a 50% chance that all the 1000 people will participate in the study.

Types of formula

SS or Sample size = N*X / (X + N – 1) and X = z­*p*(1-p) / e2
N = population size
e = Margin of error (percentage in decimal form)
z = z-score (achieved from the Confidence level and z-score table)
p = significance value at 95% (0.05)

The following table presents the list of z-values while conducting the sample size calculation based on the confidence level of choice.

Desired confidence levelz-score

However, there is another formula that can be used for sample size calculation using standard deviation. The formula is;

N= z2σ2 / E2
N = Sample size
z = z-score
σ= Standard deviation
E = Margin of error

Calculating the sample size

Use the formula;

SS or Sample size = N*X / (X + N – 1) and X = z­*p*(1-p) / e2

So, the study has a 95% confidence level (0.05) and a 5% margin of error with an expected sample size of 800 and a confidence interval of 1.96.

The expected target sample remains considered on the basis of the size of the sample or the targetted segment of participants and from the limitations of previous works of literature.

X = (1.96)2 * 0.05 * (1-0.05) / (0.05)2
X = 3.841 * 0.05 * 0.95 / 0.0025
X = 3.4665 / 0.0025 = 1386.6

Therefore, the sample size is:

SS = 800 * 1386.6 / (1386.6 + 800 - 1)
SS = 1109280 / 2185.6

SS = 507.54 or 508 sample size is the best fit for the study.

Again, using the second formula considering the standard deviation is 5 at 95% confidence level;

N = 1.96252 / 0.052
N = 3.841 * 25 / 0.0025 (follow the BODMAS rule)
N = 96.025 * 100 / 25 (reverting the decimals)
N = 9602.5 / 25 N = 384.1 or 384 sample size will be best fit for the study.

Consider using the first formula when the required standard deviation of the sample size not known or not considered for the statistical study.

This is how the sample size for a primary quantitative study is considered and calculated. Include sample size calculation as the most important part of an empirical study and include in the methodology section of any thesis or dissertation where primary study remains considered.


  • Das, S., Mitra, K. and Mandal, M., 2016. Sample size calculation: Basic principles. Indian journal of anaesthesia60(9), p.652.

Avishek Majumder

Research Analyst at Project Guru
Avishek is a Master in Biotechnology and has previously worked with Lifecell International Private Limited. Apart from data analysis and biological research, he loves photography and reading. He loves to play football and basketball in his spare time with an avid interest in adventure and nature. He was also a member of the Scouts in his school and has attended Military training.
Avishek Majumder


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