# Sample Size Determination and Power of a Study

Sample Size Determination and Power of a Study: Sample size determination is a major step in the design of a research study. It is very important to have a proper sample size for study or justification of the problem. The size of the sample depends upon the precision, the researcher desires in estimating the population parameter at a particular confidence level. There is no specific rule that can be used to determine sample size. The sample size should be small enough to avoid unnecessary expenses and large enough to avoid intolerable sampling errors. Sample size which fulfills the requirements of efficiency representativeness, reliability, and flexibility is called an optimum sample size. The sample size required to reject or accept a study hypothesis is determined by the power of the test.

The sample size is based on the following parameters:

• Size of the Population: If the total population to be studied is very large then, the sample size is also more.
• Nature of Population: The population may be homogeneous or heterogeneous. If the population is homogeneous, a small sample may serve the purpose but if the population is heterogeneous, a large sample is required to serve the purpose.
• Nature of Study: In some investigations, the items need to be intensively and continuously studied. In such cases, the sample size should be small. Studies which are not likely to be repeated and are quite extensive then the sample size may be large.
• Type of Sampling: Sampling methods play an important role in determining the size of a sample. In simple random sampling require a larger sample but properly drawn stratified sampling plan, the small sample gives better results.
• Desired Accuracy or Confidence Level: If the sample size is larger then the degree of accuracy is more. The researcher has to think of the level at which he will be confident that his sample is representative. The 95% confidence level is chosen to mean that one anticipates that there is a 95% chance that the sample and the population will look alike and a 5% chance that it will not.
• Sampling Error or Desired Risk Level: If the sample size is more than the minimum sample error. The study of parents who want to send their children to English medium private schools or government schools. If the average annual family income. of parents in the area is 10,00,000 then, the investigator should make sure that his sample’s average income is close to 10,00,000.
• Purpose of Study: Sample size depends on the study whether it is descriptive, exploratory, or explanatory. Qualitative or quantitative studies may be considered for sampling size.
• Availability of Funds: The size of the sample also depends upon the availability of funds for the research process. Financial sources should be kept in view while determining the size of a sample.

It is a very difficult task for the researcher to determine the size of the sample. In an experimental study, it is essential to equate the control and experimental groups, but in a survey, the study sample should be representative of the population. Therefore, the size of the sample is an important parameter for representativeness. The precision of data is determined primarily by the size of the sample, rather than by the percentage of the population represented in the sample.

Several formulae have been devised for determining the sample size depending upon the availability of information.

Where,

• n = Sample size
• z = Specific level of confidence or desired degree of precision
• σ= Standard deviation of the population
• d = Difference between the population mean and sample mean

The specific level of confidence (z) at 1% level of significance or 99% confidence level the value of z is 2.58 and 5% level of significance or 95% confidence level the value of z is 1.96. If the standard deviation of the population is 12, the population mean is 36, the sample mean is 30, and the confidence level of 99% then the sample size is calculated as follows:

• The confidence level of 99% i.e. z = 2.58.
• Standard deviation of population i.e. sigma = 12
• Difference between population mean (36) and sample mean (30) i.e. d = 36-30 = 6.

In studies where the plan is to estimate the proportion of successes in a dichotomous outcome variable (yes/no) in a single population, the formula used for determining sample size is:

Where,

• n = Sample size
• Z = Value from the standard normal distribution reflecting the confidence level
• E = the Desired margin of error.
• p = Proportion of successes in the population.

The equation to determine the sample size for determining p seems to require knowledge of p. The range of p is 0 to 1, and therefore the range of p(1 – p) is 0 to 1. The value of p that maximizes p(1 – p) is p = 0.5 . Consequently, if there is no information available to approximate p, then p = 0.5 can be used to generate the most conservative, or largest, sample size.

In studies where the plan is to estimate the difference in means between two independent populations, the formula for determining the sample sizes required in each comparison group is given below:

Where,

• ni = Sample size required in each group (i = 1, 2)
• Z = Value from the standard normal distribution reflecting the confidence level
• E = Desired margin of error.
• σ = Standard deviation of the outcome variable.

In studies where the plan is to estimate the difference in proportions between two independent populations (ie, to estimate the risk difference), the formula for determining the sample sizes required in each comparison group is:

Where,

• ni = Sample size required in each group (i = 1, 2)
• Z = Value from the standard normal distribution reflecting the confidence level (e.g., Z= 1.96 for 95%)
• E = Desired margin of error.
• p₁ and p₂ = The proportions of successes in each comparison group.

The difference between two groups in a study will be explored in terms of the estimate of effect, appropriate confidence interval, and P-value. The confidence interval indicates the likely range of values for the true effect in a population while P-value determines how likely it is that the observed effect in the sample is due to chance. A related quantity is the statistical power of the study and it is the probability of detecting a predefined clinical significance. High power ideal study means that the study has a high chance of detecting a difference between groups if it exists. The ideal power for any study is considered to be 80%. For example, if the study has 80% power, it has an 80% chance of detecting an effect that exists Generally, 90% power or more is recommended to calculate sample size to achieve the real effect of the experiment.

Statistical power is generally calculated with major two objectives given as follows:

• Power can be calculated before data collection based on information from previous studies to decide the sample size needed for the current study.
• It can also be calculated after data analysis.

The second situation occurs when the result turns out to be non-significant. In this case, statistical power is calculated to verify whether the non-significance result is due to a lack of relationship between the groups or due to a lack of statistical power.

Power is the probability that the statistical test results in rejection of Ho when a specified alternative is true. The ‘stronger the power, the better the chance that the null hypothesis will be rejected when, in fact, Ho, is false. The larger the power, the more sensitive is the test. Power is defined as 1 – β. The larger the error, the weaker is the power.

Different parameters can affect the power of a test such as a sample size (n). the significance level of the test (α), “true value of the tested parameter, etc. The power of a study or statistical test is the probability that it correctly rejects the null hypothesis (Ho) when the null hypothesis is false (i.e. the probability of not committing a Type II error or β).

Statistical power is positively correlated with the sample size as a larger sample size gives greater power. However, researchers should be clear to find a difference between statistical difference and scientific difference. A larger sample size enables statisticians to find smaller differences statistically significant, the difference found may not be scientifically meaningful.

`Make sure you also check our other amazing Article on : Research Hypothesis`
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