Calculator Inputs
Example Data Table
| Case | Expected r | Alpha | Power | Dropout | Approximate n |
|---|---|---|---|---|---|
| Weak field relation | 0.20 | 0.05 | 0.80 | 10% | 213 |
| Moderate sensor link | 0.30 | 0.05 | 0.80 | 10% | 94 |
| Strong lab response | 0.50 | 0.05 | 0.90 | 5% | 41 |
Formula Used
The calculator uses Fisher transformation for Pearson correlation planning. For detecting an expected correlation, it applies:
n = ((Z alpha + Z power) / FisherZ(r))² + 3
For two-tailed tests, alpha is divided by two. For one-tailed tests, alpha is used directly. FisherZ(r) equals 0.5 × ln((1 + r) / (1 - r)). The result is then adjusted for design effect, finite population, dropout, and a minimum sample rule.
How to Use This Calculator
Enter the expected Pearson r from prior research, pilot data, or a practical physics target. Choose alpha and power. Use two-tailed testing when the direction may be positive or negative. Use one-tailed testing only when a justified direction exists. Add dropout if observations may fail quality checks. Use design effect for clustered readings.
Article: R Sample Size Planning in Physics
Why Sample Size Matters
Physics experiments often compare two measured quantities. A researcher may study voltage and temperature. Another may compare force and displacement. The Pearson r value describes the strength and direction of a linear relation. A sample size plan helps decide how many observations are needed before collecting data.
Using Expected Correlation
The expected r should come from a pilot test, earlier lab work, or a realistic physical model. Small r values need larger samples. Strong r values need fewer observations. This happens because weak relations are harder to detect with confidence. The calculator changes r into Fisher z before applying the sample size equation.
Power and Alpha
Power is the chance of detecting the planned relation when it is truly present. Many studies use 0.80 or 0.90 power. Alpha is the allowed chance of a false positive result. A common alpha is 0.05. Lower alpha values increase the required sample size.
Precision Planning
Some experiments do not only ask whether r exists. They ask how tightly r should be estimated. The precision method uses a target confidence interval half width. A smaller half width means a more precise estimate. That also means more observations are required.
Adjustments for Real Work
Real physics data may include failed sensors, missing readings, rejected trials, or clustered measurements. The dropout field increases the sample count for expected data loss. The design effect handles grouped observations. The finite population field reduces the sample when the available population is limited and known.
Interpreting the Output
The final answer is rounded upward. This protects the planned study from underestimation. Use the table details to review the base sample size, corrected sample size, and final adjusted value. Export the result for lab notebooks, proposals, reports, or internal review files.
FAQs
1. What does r mean here?
It means Pearson correlation coefficient. It measures the linear relation between two numeric variables, such as pressure and volume or voltage and temperature.
2. Can r be negative?
Yes. A negative r means one variable tends to fall as the other rises. The calculator uses the absolute strength for sample size.
3. What power should I use?
Many studies use 0.80 as a minimum. Use 0.90 when stronger protection against missed effects is needed.
4. What alpha is common?
Alpha 0.05 is common. A smaller alpha, such as 0.01, requires more observations because the test becomes stricter.
5. Should I choose one-tailed or two-tailed?
Use two-tailed when either direction matters. Use one-tailed only when the direction is justified before data collection.
6. What is design effect?
Design effect adjusts for grouped or clustered observations. Values above 1 increase sample size to protect reliability.
7. What does dropout mean?
Dropout is the expected percentage of unusable observations. It may include sensor errors, missing trials, or rejected readings.
8. Is this result exact?
It is a planning estimate based on Fisher transformation. Confirm final methods with your study protocol and statistical review process.