Students frequently ask about sample size in research methods, but the decision depends on various factors and there is no definitive answer. Decisions are often influenced by time and cost constraints, making them a compromise between precision, precision, and other considerations.

#### 1.Absolute and Relative Sample Size in Research

- Absolute sample size is more important than relative size.
- A national probability sample of 1,000 individuals in the UK has the same validity as a sample of 1,000 individuals in the USA, despite a larger population.
- As sample size increases, the precision of a sample decreases, reducing the 95% confidence interval.
- The less sampling error one is prepared to tolerate, the larger a sample will need to be.
- Fowler (1993) warns against relying on a single estimate of a variable in decision-making about sample size.
- Most survey research generates multiple estimates, not a single 'desired level of precision'.
- The notion of using a desired level of precision as a factor in a decision about sample size is not realistic.

#### 2.Time and Cost in Sampling Size

- Larger sample sizes lead to greater precision due to less sampling error.
- Gains in precision are noticeable up to a sample size of around 1,000.
- Sharp increases in precision become less announced after a certain point, often in the region of 1,000.
- A slowing-down in the extent to which precision increases and the extent to which the mean sample error of the mean declines.
- Considerations of sampling size are profoundly affected by matters of time and cost, as striving for smaller increments of precision becomes increasingly uneconomical.

#### 3. Non-Response in Survey Research

- Non-response in survey surveys is a significant issue, with a 20% rate of non-response.
- To ensure 450 employees are interviewed, it may be advisable to sample 540–550 individuals, with approximately 90 being non-respondents.
- Non-response rates are declining in many countries, indicating a growing tendency towards people refusing to participate in survey research.
- Business Week and a report from the Market Research Society’s Research and Development Committee in Britain have reported concerns about declining response rates.
- T. W. Smith (1995) suggests no consistent evidence of such a decline, and it's difficult to separate general trends from variables like research subject matter, respondent type, and effort level.
- Strategies to improve responses to survey instruments like structured interviews and postal questionnaire.

#### 4. Population Heterogeneity and Sample Size

- Heterogeneity of population affects sample size.
- Highly heterogeneous samples, like entire countries or cities, show high variation.
- Relatively homogeneous samples, like company members or occupations, show less variation.
- Greater population heterogeneity necessitates larger sample sizes.

#### 5. Understanding the Type of Analysis

- Researchers should consider the type of analysis they plan to undertake.
- A contingency table is a useful tool for illustrating the relationship between two variables.
- For instance, a contingency table could show the variation in employee skill development and learning among the 100 largest UK companies.
- The table would also reflect differences between companies representing the seventeen main SIC code sections.
- However, the initial criterion of selecting the 100 largest companies may not represent all SIC sections, leading to empty cells.
- To overcome this, the sample would need to reflect a wider range of public and private organizational activity.

**Some principles that influence sample size include:**

- The greater the dispersion or variance within the population, the larger the sample must be to provide estimation precision.
- The greater the desired precision of the estimate, the larger the sample must be.
- The narrower or smaller the error range, the larger the sample must be.
- The higher the confidence level in the estimate, the larger the sample must be.
- The greater the number of subgroups of interest within a sample, the greater the sample size must be, as each subgroup must meet minimum sample size requirements.

**Population**: Population refers to a collection of items that share similar traits. It consists of a number of observations from which we hope to draw conclusions. A sample is the subset of the population.**Margin of Error**: The margin of error is a statistic that describes how much random sampling error there is in experiment results. It establishes how far your sample mean can deviate from the population mean, either higher or lower. The margin of error is frequently stated as a percentage.**Confidence Level**: The projected likelihood that a population estimate falls within a certain margin of error is known as the confidence level. In other words, it expresses your level of confidence that the real mean is within your margin of error. The three most popular levels of confidence are 90%, 95%, and 99%.**Degree of Variability**: The degree of variability refers to how much the sample measurements vary from the population measure. The sample size should be increased as variation increases.**Sampling Method:**The sample size depends on the sampling method used. For example, in simple random sampling, the sample size is typically determined by the desired level of precision and the size of the population. In stratified sampling, the sample size may be determined by the desired level of precision within each stratum, and the proportion of the population in each stratum. In cluster sampling, the sample size may be determined by the desired level of precision within each cluster, and the number of clusters selected. In general, larger sample sizes are needed for more precise estimates and for populations with greater variability

## Sample Size Calculation Formula

The size of the sample is very important for getting accurate, statistically significant results and running your study successfully.

- If your sample is too small, you risk including an excessive number (within the sample size) of people who are anomalies and outliers. You don't get a true representation of the entire population because these influence the results.
- The study becomes complicated, expensive, and time-consuming to conduct if the sample is too large, and even though the results are more accurate, the advantages do not exceed the disadvantages.

In order to find the optimal sample size, we use two types of standard formulas:

### For Small Sample Size

The T distribution notion is utilized in place of the normal distribution when there is a lower sample size. This distribution is specifically used when the sample size is less than 30. In this test, if the population variance is unknown and the sample size is small, we use the t statistic to test the null hypothesis using both one-tailed and two-tailed tests. Another name for it is an adjusted sample size.

where,

A is the adjusted sample size,

n is the sample size,

P is the population size.

### For Large Sample Size

For infinite population size, the formula is expressed in terms of z-value and error margin.

Sample Size (n) =

Where,

n is the sample size,

Z is the z-value,

p is the proportion of population (generally taken as 0.5),

m is the margin of error.

## Solved Examples of Sample Size

Here are some solved examples on sample size for you to practice.

**Solved Example 1:** Calculate the sample if the z value is 3.2 and the margin of error is 3.2%.

**Solution: **We have,

z = 3.2

m = 3.2% = 0.032

p = 0.5

Using the formula we have,

N = 2500

**Solved Example 2: **Calculate the z-value if the sample size is 3000 and the margin of error is 2.5%.

**Solution: **We have,

N = 295

m = 2.5% = 0.025

p = 0.5

Using the formula we have,

Thus,

N = 7.5

**Solved Example 3:** Calculate the adjusted sample size for a sample size of 400 and a population of 60000.

**Solution: **We have,

n = 400

P = 60000

Using the formula we have,

A = 397.42

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