On what basis is sample size in numbers determined? any formula or calculation exists

 Yes, ✅ sample size determination is based on statistical principles and can be calculated using formulas, especially in quantitative research. The appropriate sample size depends on:

๐Ÿ”น Factors That Determine Sample Size

Factor Description
Population size (N) Total number of individuals in the population
Margin of error (E) Acceptable difference between sample result and population parameter (e.g., ±5%)
Confidence level (Z) Degree of certainty (e.g., 95% = Z value of 1.96; 99% = Z value of 2.576)
Standard deviation (ฯƒ or p) Variability in the population; estimated proportion if binary variable (e.g., 0.5 is max variability)
Power of the test (1 - ฮฒ) Probability of correctly rejecting a false null hypothesis (usually set at 80% or 90%)

๐Ÿ”น Common Formula for Sample Size (for Large Populations)

If population is large or infinite (N > 10,000), and you're estimating a proportion, the sample size is calculated as:

n=Z2⋅p⋅(1−p)E2n = \frac{{Z^2 \cdot p \cdot (1 - p)}}{E^2}

Where:

  • nn = required sample size

  • ZZ = Z-score for desired confidence level

  • pp = estimated proportion (use 0.5 if unknown)

  • EE = margin of error

✅ Example:

Suppose you want to conduct a survey with:

  • 95% confidence level (Z = 1.96)

  • 5% margin of error (E = 0.05)

  • Estimated proportion (p) = 0.5 (maximum variability)

n=(1.96)2⋅0.5⋅(1−0.5)(0.05)2=3.8416⋅0.250.0025=0.96040.0025=384.16n = \frac{{(1.96)^2 \cdot 0.5 \cdot (1 - 0.5)}}{(0.05)^2} = \frac{{3.8416 \cdot 0.25}}{0.0025} = \frac{0.9604}{0.0025} = 384.16

✅ Sample size = 385 respondents

๐Ÿ”น Sample Size Formula for Finite Population

If population size (N) is known and not very large (say < 10,000), adjust the above formula:

n′=n1+n−1Nn' = \frac{n}{1 + \frac{n - 1}{N}}

Where:

  • nn = sample size from the large population formula

  • NN = actual population size

  • n′n' = adjusted sample size

๐Ÿ”ธ Example:

Let N=1000N = 1000 and n=385n = 385

n′=3851+3841000=3851.384≈278n' = \frac{385}{1 + \frac{384}{1000}} = \frac{385}{1.384} \approx 278

✅ Adjusted sample size = 278 respondents

๐Ÿ”น Tools for Sample Size Calculation

  • Online calculators (e.g., Raosoft, SurveyMonkey)

  • Statistical software (e.g., SPSS, G*Power, R, STATA)

  • Excel-based formulas (can be built using the above)

๐Ÿ”น Summary Table

Confidence Level Z-score
90% 1.645
95% 1.96
99% 2.576


Comments

Post a Comment

Popular posts from this blog

Research Methodology vs Research methods

The terms research method and research methodology are often used interchangeably, but they refer to distinct concepts in research. Here's a clear comparison to help differentiate them:  1. Research Method Aspect Description Definition The techniques or procedures used to gather and analyze data related to a research question or hypothesis. Focus “What” tools or techniques are used in the research. Examples Surveys, interviews, observations, experiments, case studies, content analysis, etc. Nature Practical and procedural – refers to actions or steps taken during the study. Purpose To collect data or evidence to address the research problem.  2. Research Methodology Aspect Description Definition The philosophical framework and justification behind the choice and use of research meth...
Read more

Types of variables in Finance Research

 In finance research, variables represent different financial aspects, behaviors, or indicators that are measured, analyzed, and used to test hypotheses. Here are the main types of variables commonly used in finance research: 1. Dependent and Independent Variables Dependent Variables : These are the outcomes or responses that researchers are trying to explain or predict. They depend on other variables. Examples : Stock returns, profitability, firm value, credit risk. Independent Variables : Also known as predictor variables, these are factors that are hypothesized to affect or influence the dependent variable. Examples : Interest rates, inflation, firm size, leverage ratio. 2. Control Variables Control variables are used to account for additional factors that might influence the dependent variable but aren’t the main focus of the study. They help isolate the effect of the independent variables on the dependent variable. Examples : Firm age, industry type, market capitalization, GDP...
Read more

Explain sum of squares.

 Certainly! The Sum of Squares (SS) is a fundamental concept in statistics, especially in regression and ANOVA . It measures variability —that is, how much values differ from a mean or predicted value. ๐Ÿ” What Is Sum of Squares? Sum of Squares quantifies the total variation in a dataset by summing the squared deviations from a reference value (usually the mean or predicted value). The general form is: SS = ∑ ( X i − Reference Value ) 2 \text{SS} = \sum (X_i - \text{Reference Value})^2 Why squared? To avoid cancellation of positive and negative deviations To give more weight to larger deviations ๐ŸŽฏ Types of Sum of Squares in ANOVA/Regression 1. Total Sum of Squares (SST) Measures the total variation in the dependent variable Y Y Y Compares each actual value to the grand mean Y ห‰ \bar{Y} Y ห‰ S S T = ∑ ( Y i − Y ห‰ ) 2 SST = \sum (Y_i - \bar{Y})^2 2. Explained Sum of Squares (SSR) — also called Regression SS or Between-Group SS Measures the vari...
Read more