The manual methods of calculating the sample size of quantitative research

By Riya Jain and Priya Chetty on July 5, 2019

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 = [z­*p*(1-p)]/ e2
e = Margin of error (desired level of precision or percentage of permitted error) 
z = z-score value at the confidence level of the study (derived from z-value table) 
p = estimated proportion of the population who has study attributes 

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, in case of small population, the sample size computed from the above formula could be modified by using the below equation

Modified Sample size = N*SS / (N + SS – 1)  
N = Population size
SS = Computed Sample Size

Alternative formula that can be used for sample size calculation using standard deviation is stated below:

SS= [(z2σ2)/E]2
SS = Sample size
z = z-score
σ= Standard deviation 
E = Margin of error 

Calculating the sample size

Use the formula;

SS or Sample size = [z­*p*(1-p)] / e2

Suppose the total number of population considered for the study is 500. The proportion of people who possess the attribute of the study is 50% and the permitted error (margin of error) while computing the sample size is 5%. Confidence level of the study is 95% thus depicting the z-score value as 1.96.

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

X = [(1.96)2 * 0.50 * (1-0.50)] / (0.05)2
X = [3.8416 * 0.50 * 0.50] / 0.0025
X = 0.9604 / 0.0025 = 384.16

Therefore, the sample size is:

SS = 500 * 384.16 / (384.16 + 500 - 1)
SS = 192080 / 883.16
SS = 217.49.  

Thus, 217 sample size is the best fit for the study.

Again, using the alternative formula and considering the standard deviation is 0.6 at 95% confidence level;

SS = [(1.962(0.6)2)/0.05]2
SS = [(3.8416 * 0.36)/0.05]2  (follow the BODMAS rule)
SS = [1.382976 * 100/5]2  (reverting the decimals)
SS = [138.2976/5]2 
SS = (27.65952)2  
SS = 765.049047 

Following the above stated population size, the computed sample size is modified i.e.

SS = 500 *  765.049047 / ( 765.049047 + 500 - 1)
SS = 382524.524 / 1264.04905
SS = 302.618418

Thus, 303 sample size is the 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.
Riya Jain


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