By determination of sample size, we mean to select the appropriate number of observations/ persons/ subjects from a large group to use in a sample. A sample with an appropriate number of observations and a sample with an appropriate size so that the results are statistically valid and accurate estimate the population parameters.

## Table of Contents

### Importance of Determining the Sample Size

Determination of sample size is important as appropriate sample size usually saves time, costs, and labor involved in studying the members of the population. It also helps to select a representative sample of objects/subjects if an appropriate sampling technique is used for the selection of objects/subjects.

Therefore, it is important to remember that, a good sample size depends on the contexts and goals of the research being done. On the other hand, a good sample size results in reliable statistical estimates and represents the population under study accurately. In general, large sample sizes are considered better as they reduce the likelihood of sampling error. However, the larger the sample larger the time, cost, and labor required to collect the sample. The sample size directly affects the accuracy and reliability of your findings.

The margin of error will decrease by drawing a larger sample, for a given confidence level say $c$, standard deviation $\sigma$.

### Determination of Sample Size and Sample Size Formula

One can determine the sample size if the maximum allowable error and level of confidence are known. If population standard deviation can be estimated, then the necessary sample size can be determined by simplifying the error formula for $n$.

The maximum allowable error is: $E=Z \frac{\sigma}{\sqrt{n}}$

By multiplying both sides with $\sqrt{n}$, we have

$E\sqrt{n} = Z \sigma$

Dividing both sides by $E$, we obtain $\sqrt{n} = \frac{Z\sigma}{E}$

Finally, squaring both sides, we get the sample size formula:

$$n=\left(\frac{Z\sigma}{E}\right)^2$$

### Example: Determining Sample Size

Suppose, we are interested in finding the average weight of Pakistani men, and we want to be 95% confident that our estimate falls within $\pm2$lbs, of the actual mean. Let’s suppose that according to the previous studies, the population standard deviation (or estimated standard deviation) is $\sigma = 18.4$lbs. We are interested in determining sample size from the above-given information.

According to the given information

$\alpha = 0.05, Z_\alpha = 1.96$, the desired maximum error is $E=2$, and the estimated $\sigma = 18.4$. Therefore,

$$n=\left(\frac{Z\sigma}{E}\right)^2 = \left( \frac{1.96 \times 18.4}{2}\right)^2 \approx 325.15$$

The appropriate sample size for the above scenario should be 326 men for the given desired level of accuracy.

### Summary

Note that If the population under study is highly diverse (heterogeneous population), a larger sample size may be necessary to ensure adequate representation of different subgroups. The type of study (e.g., survey, experiment) and the research questions can also influence the appropriate sample size. Similarly, the Practical Constraints: Factors such as budget, time, and accessibility can limit the feasible sample size.