This is an excellent and common challenge in mixed methods research! Determining sample sizes for both quantitative rigor and qualitative depth requires different approaches.

Let's break down each strand:

Part 1: Quantitative Sample Size using Power Analysis

Power analysis is the gold standard for determining the minimum sample size needed for a quantitative study to detect a statistically significant effect, if one truly exists, with a specified probability.

To help you calculate this, I need more information about your specific quantitative research questions and planned statistical analyses. However, I can give you a step-by-step guide and examples.

Key Components of a Power Analysis:

• Alpha ($\alpha$): Significance Level


• This is the probability of making a Type I error (false positive – rejecting a true null hypothesis).


• Commonly set at 0.05 (meaning a 5% chance of a false positive).


• Power (1 - $\beta$): Statistical Power


• This is the probability of correctly rejecting a false null hypothesis (i.e., detecting an effect when there is one).


• Commonly set at 0.80 (meaning an 80% chance of detecting an effect if it exists). Higher power (e.g., 0.90) requires larger samples.


• Effect Size (ES): Magnitude of the Effect


• This is the most crucial and often trickiest component to estimate. It describes the strength or magnitude of the relationship between variables or the difference between groups.


• Effect sizes vary depending on the statistical test (e.g., Cohen's d for t-tests, Pearson's r for correlations, f or $\eta^2$ for ANOVA, etc.).


• How to estimate Effect Size:


• Literature Review: Look for similar studies and their reported effect sizes. This is the best method.


• Pilot Study: Conduct a small preliminary study to estimate the effect size.


• Theoretical/Clinical Significance: What is the smallest effect size that would be practically meaningful or clinically relevant in your field?


• Cohen's Conventions (as a last resort): Jacob Cohen provided general guidelines for "small," "medium," and "large" effect sizes for various tests. Use these cautiously, as what's "small" in one field might be "large" in another.


• Small: An effect barely detectable or considered trivial.


• Medium: An effect that is typically noticeable to the naked eye.


• Large: An effect that is strikingly obvious.


• Intended Statistical Test:


• The type of statistical analysis you plan to use (e.g., t-test, ANOVA, regression, chi-square, correlation) directly impacts the power calculation. Each test has specific formulas for power.

Tools for Power Analysis:

• **G*Power (Free Software):** Highly recommended and widely used. Download it for Windows or Mac: http://www.gpower.hhu.de/


• R Packages: pwr package.


• Online Calculators: Many exist, but ensure they are reputable and specify their underlying assumptions.


• Commercial Software: SAS, SPSS (with specific modules), Stata, etc.

Step-by-Step Guide Using G*Power (General Example)

Let's assume you're using G*Power.

• **Open G*Power.**


• Select "Test family": This depends on your broad type of analysis (e.g., t-tests, F-tests, Chi-square, Z-tests, Exact).


• Select "Statistical test": Choose the specific test (e.g., "Means: Difference between two independent means (t-test)," "ANOVA: Fixed effects, omnibus, one-way," "Linear multiple regression: Fixed model, R² deviation from zero").


• Select "Type of Power Analysis": Usually "A priori: Compute required sample size - given $\alpha$, power, and effect size."


• Enter Parameters:


• "Effect size d" (or f, r, $\eta^2$ depending on test): Enter your estimated effect size. This is the critical input!


• "$\alpha$ err prob": Enter 0.05.


• "Power (1-$\beta$ err prob)": Enter 0.80.


• Other specific parameters:


• For t-tests: "Allocation ratio N2/N1" (usually 1 if groups are equal).


• For ANOVA: "Number of groups," "Number of predictors."


• For Regression: "Number of predictors."


• Click "Calculate" and then "Calculate and transfer to main window" if applicable.


• Interpret the Output: G*Power will provide the "Total sample size" (N).

**To help me calculate for your study, please provide the following for your quantitative strand:**

• What is your primary quantitative research question?


• Example: "Is there a significant difference in job satisfaction scores between employees who receive flexible work arrangements and those who do not?"


• Example: "What is the relationship between hours spent studying and exam performance?"


• Example: "Do different teaching methods (Method A, Method B, Method C) lead to different student achievement scores?"


• What are your key variables (independent and dependent)?


• Example: Independent Variable = Flexible Work (yes/no), Dependent Variable = Job Satisfaction (continuous scale).


• What specific statistical test(s) do you intend to use to answer that question?


• Example: Independent Samples t-test


• Example: Pearson Correlation


• Example: One-way ANOVA


• Example: Multiple Linear Regression


• Example: Chi-square test of independence


• What is your best estimate of the effect size you expect or would be meaningful to detect?


• This is the most crucial piece of information.


• Can you cite an effect size (e.g., Cohen's d = 0.5, r = 0.3, f = 0.2, $\eta^2$ = 0.06) from similar studies?


• If not, would you consider a "small," "medium," or "large" effect size based on Cohen's conventions, and for which test?


• For t-tests (Cohen's d): Small = 0.2, Medium = 0.5, Large = 0.8


• For ANOVA (Cohen's f): Small = 0.1, Medium = 0.25, Large = 0.4


• For Correlation (Pearson's r): Small = 0.1, Medium = 0.3, Large = 0.5


• For Regression (f²): Small = 0.02, Medium = 0.15, Large = 0.35


• Are there multiple groups? If so, how many, and how will participants be allocated?

Quantitative Sample Size Examples (using common scenarios)

Scenario 1: Comparing Two Independent Groups (e.g., Treatment vs. Control)

• Research Question: Is there a difference in post-test scores between a group receiving a new intervention and a control group?


• Statistical Test: Independent Samples t-test


• Assumptions: $\alpha = 0.05$, Power = 0.80, Equal group sizes.


• Effect Size: Let's assume you want to detect a medium effect size (Cohen's d = 0.5) based on previous literature.


• **G*Power Input:**


• Test family: t-tests


• Statistical test: Means: Difference between two independent means (two groups)


• Type of Power Analysis: A priori


• Effect size d: 0.5


• $\alpha$ err prob: 0.05


• Power (1-$\beta$ err prob): 0.80


• Allocation ratio N2/N1: 1


• **Result (G*Power): Total Sample Size = 128** (64 per group)

• Research Question: Do three different teaching methods (A, B, C) result in different student achievement scores?


• Statistical Test: One-way ANOVA


• Assumptions: $\alpha = 0.05$, Power = 0.80, 3 equal groups.


• Effect Size: Let's assume you want to detect a medium effect size (Cohen's f = 0.25).


• **G*Power Input:**


• Test family: F tests


• Statistical test: ANOVA: Fixed effects, omnibus, one-way


• Type of Power Analysis: A priori


• Effect size f: 0.25


• $\alpha$ err prob: 0.05


• Power (1-$\beta$ err prob): 0.80


• Number of groups: 3


• **Result (G*Power): Total Sample Size = 159** (53 per group)

• Research Question: How much variance in job performance is explained by experience, education level, and personality trait X?


• Statistical Test: Multiple Linear Regression (testing R² deviation from zero)


• Assumptions: $\alpha = 0.05$, Power = 0.80, 3 predictors.


• Effect Size: You want to detect a medium effect size (f² = 0.15), which corresponds to a change in R² of 0.15 (meaning 15% of variance explained).


• **G*Power Input:**


• Test family: F tests


• Statistical test: Linear multiple regression: Fixed model, R² deviation from zero


• Type of Power Analysis: A priori


• Effect size f²: 0.15


• $\alpha$ err prob: 0.05


• Power (1-$\beta$ err prob): 0.80


• Number of predictors: 3


• **Result (G*Power): Total Sample Size = 68**

Part 2: Qualitative Sample Size

Qualitative sample size determination is fundamentally different from quantitative. It's not about statistical power or generalizability to a larger population in the same way. Instead, it's about achieving saturation or information power.

Key Principles for Qualitative Sample Size:

• Saturation:


• The point at which no new themes, categories, or insights emerge from additional data collection. You keep interviewing/collecting data until you hear the same things repeatedly, and new data offers diminishing returns.


• This is often determined iteratively during data collection and analysis. You might start with a smaller sample and add participants until saturation is reached.


• Information Power (Malterud et al., 2016):


• This concept suggests that the more information a sample holds, the fewer participants are needed. Factors that increase information power:


• Broader Study Aim: Less specific, need more participants.


• Specific Study Aim: More focused, can use fewer participants.


• Dense Sample Specificity: Participants are very relevant to the research question (e.g., all have a rare experience).


• Strong Theoretical Foundation: Using an established theory to guide the study.


• High Quality of Dialogue/Data: Rich, detailed interviews.


• Detailed Analysis Strategy: In-depth, thematic analysis.

Factors Influencing Qualitative Sample Size:

• Methodology:


• Phenomenology: Often smaller (5-25) to deeply explore lived experiences.


• Grounded Theory: Larger (20-30+) to develop a robust theory, requiring continuous comparison and saturation.


• Ethnography: Often fewer formal interviews, but prolonged engagement and observation with a community.


• Case Study: Can be a single case or a few cases, but each case is deeply explored.


• Scope of the Study:


• A very narrow, focused question with a homogeneous group will require fewer participants than a broad, exploratory question with diverse participants.


• Nature of the Topic:


• Sensitive or complex topics might require fewer participants but deeper engagement, as participants might provide rich data quickly.


• Diversity/Homogeneity of Participants:


• If you're studying a very specific, homogeneous group, you might reach saturation with fewer participants.


• If your aim is to capture a wide range of perspectives from a diverse group, you'll need more.


• Data Collection Method:


• Individual Interviews: Typically require more participants than focus groups to cover similar ground.


• Focus Groups: Fewer groups, but each group offers multiple perspectives. Typically 3-6 focus groups for a topic.


• Analytic Strategy:


• Highly detailed, intensive analyses (e.g., discourse analysis) might produce a lot of data from a small number of participants, whereas broader thematic analysis might benefit from more participants to identify recurring patterns.

Common Recommendations/Ranges:

• Individual In-Depth Interviews:


• General thematic analysis: 10-15 (Guest et al., 2006 suggests 12 is often sufficient for basic themes)


• Phenomenological studies: 5-25 (Creswell, 2013 suggests 5-8)


• Grounded theory: 20-30+ (until theoretical saturation)


• Focus Groups:


• Typically 3-6 focus groups, each with 6-10 participants.


• Case Studies:


• 1-5 cases, with each case deeply investigated.

Justifying Your Qualitative Sample Size:

In your proposal, you should explain:

• Your sampling strategy: (e.g., purposeful/purposive sampling, snowball, convenience).


• Your target range: "We anticipate interviewing approximately 10-15 participants, based on similar studies and the need to achieve thematic saturation."


• Your criteria for saturation: "Data collection will continue until no new themes or categories emerge from the interviews, and existing themes are well-developed."


• How you will monitor saturation: "Analysis will be iterative, with transcripts being reviewed after each interview, allowing for adjustments to the interview guide and identification of preliminary themes."


• Reference relevant literature: Cite authors like Guest, Creswell, Mason, or Malterud who discuss qualitative sample size.

Mixed Methods Considerations

Since this is a mixed methods study, you also need to think about how the two strands relate.

• Relationship between Samples:


• Same Participants: Do the same individuals participate in both the quantitative and qualitative strands? (e.g., survey followed by in-depth interviews with a subset of survey respondents).


• Parallel but Separate: Are the samples distinct but drawn from the same broader population? (e.g., a large survey of students, and separate interviews with a different group of students).


• Nested: Is one sample a subset of the other? (e.g., a few participants from the quantitative study are selected for the qualitative phase).


• Implications for Integration:


• Concurrent Designs: If data is collected concurrently, you'll need to plan both sample sizes simultaneously.


• Sequential Designs (e.g., Explanatory: QUAN $\rightarrow$ QUAL): The quantitative sample size is determined first by power analysis. The qualitative sample is then often a smaller, purposive subset of the quantitative participants, chosen to help explain the quantitative results.


• Sequential Designs (e.g., Exploratory: QUAL $\rightarrow$ QUAN): The qualitative sample size is determined by saturation, and its findings then inform the development of quantitative instruments or hypotheses, which then dictates a new quantitative sample size calculation.


• Overall Justification:


• Your mixed methods proposal should explain why two different sample sizes are appropriate, given the different aims of each strand and how they contribute to answering your overall research questions.

What I Need From You to Help Further:

To give you more specific guidance and help with the quantitative calculation, please provide the details requested under **"To help me calculate for your study"** for your quantitative strand.

For your qualitative strand, tell me:

• What is your primary qualitative research question?


• What is your qualitative methodology? (e.g., phenomenology, grounded theory, thematic analysis, case study)


• What data collection methods will you use? (e.g., individual in-depth interviews, focus groups, observations, document analysis)


• What are the characteristics of your target qualitative participants? (e.g., homogeneous, diverse, specific experience)