Table of contents

1. Defination
2. Key points
3. Factors affecting
4. Examples
5. Impact of population on sample size
6. Formulas when population won't follow Normal Distribution
7. Summary

What is sample size ?

Sample size refers to the number of individual observations or data points collected from a population for a specific study or analysis.

The key points for related to sample size

Representativeness: The sample should be representative of the population from which it is drawn. A larger and more diverse sample generally provides more accurate and generalizable results.

Precision and Confidence: The sample size affects the precision of estimates and the level of confidence in the study's results. A larger sample size often leads to more precise estimates and higher confidence levels.

Statistical Power: The statistical power of a study is the probability of detecting a true effect if it exists. A larger sample size generally increases the statistical power, allowing researchers to detect smaller effects.

Resource Constraints: The available resources, including time, budget, and manpower, may limit the sample size. Researchers often need to balance the desire for a larger sample with practical constraints.

Type of Study: The required sample size may vary based on the type of study (e.g., observational, experimental, survey) and the analysis techniques used.

Heterogeneity: If the population is highly heterogeneous, a larger sample size may be needed to capture the variability within the population.

Factors Influencing Sample Size Calculation:

Acceptable level of significance:
Researchers typically aim for a certain level of confidence (e.g., 95% confidence level) in their study results. The higher the confidence level, the larger the required sample size.
Researchers often grapple with the concept of the desired level of significance, a critical parameter that shapes the interpretation of study results. This level, commonly denoted as α (alpha), influences the balance between making Type I errors (false positives) and the acceptance of a null hypothesis when it is true.

Margin of Error:
The margin of error (or precision) is the acceptable range within which the true population parameter is expected to fall. A smaller margin of error necessitates a larger sample size.
The margin of error is a statistical measure that provides an estimate of the amount of random sampling error in a survey's results. It is commonly used in the context of polling and survey research to quantify the uncertainty or variability associated with the sample data when making inferences about the entire population.

Margin of error = Z * σ / n^0.5
Z - Z-score corresponding to the desired level of confidence (e.g., 1.96 for a 95% confidence level)
In most statistical tables or using statistical software, you would look up the Z-score for a cumulative probability of 0.975. However, if you don't have access to these resources, you can use a standard normal distribution calculator or the inverse of the standard normal cumulative distribution function (invNorm) if you are using a calculator.
σ - the population standard deviation (or an estimate if the population standard deviation is unknown)
n - the sample size.

A confidence level, in statistics, represents the degree of certainty or reliability associated with a statistical inference or estimate. It is often expressed as a percentage and indicates the likelihood that the true value of a parameter falls within a specified range.
Let's say you conduct a survey to estimate the average height of adults in a city. After analyzing the data, you might find that the 95% confidence interval for the average height is 165 cm to 175 cm. This implies that if you were to conduct the same survey multiple times and calculate a confidence interval each time, you would expect about 95% of those intervals to contain the true average height of the population. The confidence level helps quantify the reliability of your estimate and provides a measure of uncertainty in statistical analysis.

Population Variability:
The extent of variability within the population affects the required sample size. More variability often demands a larger sample to ensure representation.
There are different statistical measures used to quantify population variability, and some of the common ones include - range, variance, standard deviation, coefficient of variation, interquartile range.

Effect Size:
The effect size represents the magnitude of the difference or relationship under investigation. Smaller effect sizes require larger sample sizes to detect significance.
It provides an assessment of the practical or substantive significance of an observed effect, independent of sample size. Effect size is particularly useful when comparing groups or analyzing the impact of an intervention, as it helps researchers evaluate the real-world importance of their findings.
Cohen's d: Cohen's d is widely used in the context of comparing means between two groups. It is calculated by taking the difference between the means and dividing it by the pooled standard deviation. A larger Cohen's d indicates a larger effect size.
Eta-squared (η²) and Partial Eta-squared (η²p):
These are measures of effect size for analysis of variance (ANOVA) and are used to assess the proportion of variance in the dependent variable explained by the independent variable(s). Eta-squared considers total variance, while partial eta-squared controls for the effects of other variables.
Hedges' g: Hedges' g is a variation of Cohen's d that corrects for bias, especially in small sample sizes. It is often preferred when dealing with meta-analyses.

Calculation

To determine Z-score from given confidence level

1. Find out the confidence level, denoted as C, where C is a number in between 0 and 100
2. Find out alpha = 1 - C/100
3. Use Z-table or calculator to find Z-score

The basic formula for sample size calculation in estimating a population mean (assuming a normal distribution) is:
n = (Z^2 × σ^2)/E^2
n is the required sample size.
Z is the Z-score corresponding to the desired level of confidence. For example, for a 95% confidence level, Z might be 1.96.
σ is the estimated population standard deviation.
E is the desired margin of error.

For estimating population proportions, the formula is slightly different and involves the use of the estimated population proportion Again,

n = Z^2 * p * (1 - p) / E^2

n is the required sample size, Z is the Z-score corresponding to the desired level of confidence, p is the estimated population proportion, and E is the desired margin of error.

Let's go through a couple of examples to illustrate the calculation of sample size using the formulas mentioned.

Estimating a Population Mean
Suppose you are conducting a study to estimate the average income of a population with a 95% confidence level and a margin of error of $500. You have an estimate of the population standard deviation (σ) as $5,000.
n = (1.96^2 × 5000^2)/500^2
n ≈ 384.16
Since the sample size must be a whole number, you round up to the nearest whole number. Therefore, the estimated sample size (n) is 385.

Estimating a Population Proportion
Imagine you want to determine the proportion of customers who are satisfied with a product, and you want to estimate this proportion with a 90% confidence level and a margin of error of 0.05. If you don't have an estimate of the population proportion, you might use 0.5 (which gives the maximum sample size for a given margin of error).
n = 1.645^2 x 0.5 x (1 - 0.5) / 0.05^2
n ≈ 267.78

The Impact of population on sample size

The impact of population size on sample size is most noticeable when dealing with finite populations, where the number of individuals or elements in the entire population is relatively small. In such cases, adjustments may be made to the formula for calculating sample size to account for the finite population size. This is known as the finite population correction (FPC).
The standard formula for calculating the sample size without considering finite population correction is:
n = Z^2 x p x (1 - p) / E^2

n is the required sample size,
Z is the Z-score corresponding to the desired confidence level,
p is the estimated proportion of the population with a certain characteristic, E is the desired margin of error.

When dealing with a finite population, the formula is adjusted with the finite population correction term:
n_f = n x N / (n + N - 1)
n_f is the finite population corrected sample size,
n is the sample size calculated without considering finite population correction,
N is the total population size.

The rationale behind this correction is that as the population size becomes a larger proportion of the sample size, the variability within the population becomes more accurately represented in the sample, and therefore, a smaller sample size may be sufficient.

When Population Doesn't follow Normal Distribution

When the population does not follow a normal distribution, and you want to estimate a population parameter (such as the mean or proportion) with a sample, you may use different methods and formulas based on the characteristics of the distribution.

Unknown Population Standard Deviation (for estimating the mean):
When the population standard deviation (σ) is unknown, and the sample size is sufficiently large the sample mean can be approximately modeled as a normal distribution due to the Central Limit Theorem. In this case, you can use the t-distribution for confidence intervals and hypothesis testing. The formula for the confidence interval is:

Average(X) +/- t x (s/square_root(n))
+/- indicates either + or -
t is the critical value from the t-distribution based on the desired confidence level and degrees of freedom,
s is the sample standard deviation,
n is the sample size.

Unknown Population Proportion (for estimating proportions):
When estimating a population proportion (p) and the sample size is sufficiently large, you can use the normal distribution. The formula for the confidence interval is:

p +/- z x square_root(p*(1-p)/n)
z is the critical value from the standard normal distribution based on the desired confidence level,
n is the sample size.

Summary

When calculating sample size for a study, there are several key takeaways to consider:

1. Confidence Level:
   The confidence level reflects the level of certainty you want in your estimate. Common confidence levels include 90%, 95%, and 99%.
   As the confidence level increases, the required sample size also increases because a higher level of confidence requires a wider interval.

2. Margin of Error:
   The margin of error represents the acceptable range of deviation from the estimated parameter (e.g., mean or proportion).
   A smaller margin of error requires a larger sample size to increase precision.

3. Standard Deviation:
   The population standard deviation is a measure of variability in the population. If known, it can be used to determine the required sample size.
   If standard deviation is unknown, researchers often use a larger sample size or conduct a pilot study to estimate it.

4. Z-Score:
   The Z-score is associated with the chosen confidence level and is used to determine the critical value for constructing the confidence interval.
   Higher confidence levels require larger Z-scores and, consequently, larger sample sizes.

5. Population Proportion:
   When estimating a population proportion, the estimated proportion is used in the sample size formula.
   If p is unknown, researchers may use 0.5 to obtain the maximum required sample size.

6. Type of Distribution:
   The sample size calculations often assume a normal distribution or rely on large sample approximations, especially when estimating population means.

7. Practical Considerations:
   Practical constraints such as time, budget, and resources may influence the feasibility of obtaining a specific sample size.
   Larger sample sizes generally provide more precise estimates but may be more resource-intensive.

8. Consulting with Statisticians:
   It's advisable to consult with statisticians or use specialized software for accurate sample size calculations, as they can account for specific study designs and considerations.

9. Adjustments:
   In some cases, adjustments to the sample size calculation may be necessary based on the study design, anticipated non-response rates, or other factors.

Remember that sample size calculations are a critical aspect of study design, and careful consideration of these factors is essential to ensure the reliability and validity of study results. Additionally, flexibility and adaptability may be needed as practical circumstances evolve during the research process.