Appendix E — Statistical Calculators

This appendix provides interactive calculators for common statistical computations used in movement science research. These tools help you perform power analyses and test for normality assumptions.

E.1 Power Analysis Calculator

This calculator computes statistical power for various hypothesis tests. Power is the probability of correctly rejecting the null hypothesis when it is false. Select your test type and enter the required parameters.

Statistical Power Calculator

What do you want to calculate? Calculate Power (given sample size) Calculate Sample Size (given desired power) Select Test Type: Two-Sample (Independent) t-Test One-Sample t-Test Paired (Dependent) t-Test One-Way ANOVA Correlation (Pearson r)

Effect Size (Cohen's d):

Expected Correlation (r):

Effect Size (Cohen's f):

Sample Size per Group (n):

Total Sample Size (N):

Number of Groups (k):

Desired Power (0-1):

Alpha Level (α):

Test Type: Two-tailed One-tailed

TipChoosing Parameter Values

Effect Size (Cohen’s d or f): Small = 0.2, Medium = 0.5, Large = 0.8 for d. Use pilot data or prior research when available. Medium effects (d = 0.5) are common starting points for planning.

Desired Power: Convention is 0.80 (80% chance of detecting real effects). Higher-stakes research may use 0.90. Lower power (<0.70) increases risk of missing true effects.

Alpha Level (α): Standard is 0.05 for most research. Use 0.01 for more conservative tests or when multiple comparisons are involved. Lower alpha reduces Type I error but requires larger samples.

Correlation (r): Small = 0.1, Medium = 0.3, Large = 0.5 (Cohen’s benchmarks for behavioral sciences).

Tip🎮 Use this calculator to explore — not just to plan

The power calculator above is a great tool for building intuition about how statistical power works. Try changing one parameter at a time while holding the others constant and see what happens: What happens to power when you double the sample size? How does a small effect size (d = 0.2) compare to a large one (d = 0.8) in terms of the sample size needed to achieve 80% power? What does it take to push power from .70 to .90?

This kind of exploratory “what-if” thinking is exactly how researchers develop a feel for the relationships among sample size, effect size, alpha, and power — and it is far more instructive than simply plugging in numbers to get a required N.

For formal research planning (grant proposals, IRB submissions, publishable studies), this calculator is a useful starting point, but you should confirm your final estimates with dedicated software that uses exact probability distributions:

• G*Power — free, purpose-built power analysis software covering a wide range of tests. This is the gold standard for most movement science researchers and is widely cited in published methods sections.
• SPSS Statistics 31+ — beginning with version 31, SPSS includes a built-in power analysis module accessible directly from the menus (Analyze → Power Analysis). This allows you to estimate required sample size or achieved power for t-tests, ANOVA, correlation, and several other procedures without leaving the software you already use for your analyses.
• R packages — pwr, WebPower, and simr (for simulation-based power analysis with mixed models) are free and highly flexible options for more advanced designs.

The goal is not to pick one tool exclusively — many researchers use the in-book calculator to explore, G*Power or SPSS to confirm, and then report the method they used.

E.2 Z-Test for Skewness

This calculator tests whether the skewness of your sample differs significantly from zero (perfect symmetry). Enter the skewness value, standard error, and sample size from your SPSS output.

Z-Test for Skewness

Sample Skewness:

Standard Error of Skewness:

Sample Size (n):

E.3 Z-Test for Kurtosis

This calculator tests whether the kurtosis of your sample differs significantly from 3 (mesokurtic, normal distribution). Enter the kurtosis value, standard error, and sample size from your SPSS output. If SPSS reports excess kurtosis, add 3 before entering.

Z-Test for Kurtosis

Sample Kurtosis (raw, not excess):

Standard Error of Kurtosis:

Sample Size (n):

E.4 Confidence Interval Calculator

This calculator computes confidence intervals for a population mean. A confidence interval provides a range of values that likely contains the true population parameter. For example, a 95% CI means that if you repeated your study many times, about 95% of the intervals would contain the true population mean.

When to use: After collecting sample data when you want to estimate the population mean with a specified level of confidence. Larger samples produce narrower (more precise) intervals.

Confidence Interval for Mean

Sample Mean (x̄):

Sample Standard Deviation (s):

Sample Size (n):

Confidence Level: 90% 95% 99%

E.5 Effect Size Calculator

Calculate Cohen’s d from sample statistics. Effect size quantifies the magnitude of the difference between groups, independent of sample size. Unlike p-values, effect sizes tell you how big the difference is, not just whether it exists.

When to use: After finding a significant difference between two groups, to understand the practical importance of that difference. Essential for power analysis planning and meta-analyses. Cohen’s benchmarks: d = 0.2 (small), 0.5 (medium), 0.8 (large).

Cohen's d Calculator

Group 1 Mean (M₁):

Group 1 SD (SD₁):

Group 1 Sample Size (n₁):

Group 2 Mean (M₂):

Group 2 SD (SD₂):

Group 2 Sample Size (n₂):

E.6 Z-Score and Percentile Converter

Convert between z-scores and percentiles for the standard normal distribution. Z-scores indicate how many standard deviations a value is from the mean, while percentiles show the percentage of scores below a given value.

When to use: To interpret standardized scores, find critical values for hypothesis tests, or determine what proportion of a normal distribution falls above/below a certain value. For example, z = 1.96 corresponds to the 97.5th percentile (used for 95% CIs).

Z-Score ↔ Percentile Converter

Z-Score to Percentile

Z-Score:

---

Percentile to Z-Score

Percentile (0-100):

E.7 Standard Error Calculator

Calculate standard error for various statistics. Standard error measures the precision of a sample statistic—smaller values indicate more precise estimates. It reflects how much your sample statistic (mean, proportion, or difference) would vary across repeated samples.

When to use: To understand sampling variability, construct confidence intervals, or calculate test statistics. SE decreases as sample size increases (more data = more precision). Choose the appropriate type based on your research question.

Standard Error Calculator

What do you want to calculate? Standard Error of the Mean (SEM) Standard Error of a Proportion (SEp) Standard Error of Difference Between Means

Standard Deviation (s):

Sample Size (n):

Sample Proportion (p):

Sample Size (n):

SD Group 1 (s₁):

n₁:

SD Group 2 (s₂):

n₂:

NoteNote on Calculator Use

These calculators use normal-approximation formulas and provide results comparable to professional software for educational purposes and preliminary planning. However, for high-stakes research (grant proposals, pilot study design, publication-quality analyses), use dedicated power analysis software:

• G*Power: Free, comprehensive power analysis tool with exact calculations for a wide range of statistical tests
• SPSS Statistics 31+: Now includes a dedicated power analysis module with advanced options
• R packages: pwr, WebPower, or simr for simulation-based power analysis

For kurtosis inputs, enter raw kurtosis (normal = 3); if you have excess kurtosis, add 3 before entering.