Research calculators
Unlimited free calculations for effect size, sample size, power, intervals and reliability. Free, rate-limited DOI citation count lookups.
Effect size
Formula and supported method
Independent d=(mean1−mean2)/pooled SD; g=Jd, with exact gamma correction J. Paired dz uses SD of differences; dav uses sqrt((SD1²+SD2²)/2). OR=ad/bc with log-Wald CI; Fisher r interval=tanh(atanh(r) ± z/sqrt(n−3)).
Choose the standardizer explicitly. Positive effects refer to group 1 minus group 2. Pearson uses complete independent pairs. A zero odds-ratio cell triggers 0.5 added to all four cells. d/g confidence intervals are unsupported.
Worked example: Means 10 and 8, SDs 4 and 5, sizes 20 and 30 give d≈0.431959 and g≈0.425169.
Method reference · Implementation 2026-09-14
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How to use research calculators
- Choose a calculator and supported method, or load its example.
- Enter values with the labelled units and matrix orientation.
- Calculate and review the estimate, intermediate quantities and assumptions.
- Copy or download the text or CSV result with its assumptions before leaving; the page does not retain it.
Methods
Effect size
Independent d=(mean1−mean2)/pooled SD; g=Jd, with exact gamma correction J. Paired dz uses SD of differences; dav uses sqrt((SD1²+SD2²)/2). OR=ad/bc with log-Wald CI; Fisher r interval=tanh(atanh(r) ± z/sqrt(n−3)).
Choose the standardizer explicitly. Positive effects refer to group 1 minus group 2. Pearson uses complete independent pairs. A zero odds-ratio cell triggers 0.5 added to all four cells. d/g confidence intervals are unsupported.
Example: Means 10 and 8, SDs 4 and 5, sizes 20 and 30 give d≈0.431959 and g≈0.425169.
Sample size
Independent shift=d/sqrt(1/n1+1/n2); paired/one-sample shift=d sqrt(n). Power=Φ(shift−zcrit)+Φ(−shift−zcrit) for two-sided; solve numerically, ceil each group, then ceil(n/(1−attrition)). Binary effect is Cohen h=2 asin sqrt(p1)−2 asin sqrt(p2).
Normal approximation, NOT noncentral t or exact binomial. Independent groups use common planning SD; paired use SD of differences. No cluster, survival, regression or noninferiority designs.
Example: d=.5, alpha=.05, power=.8, equal allocation, two-sided: 63+63 analyzable; 10% attrition: 70+70 recruited.
Power analysis
Φ(shift−zcrit)+Φ(−shift−zcrit) two-sided, Φ(shift−zcrit) one-sided. Independent effective n=1/(1/n1+1/n2); paired/one-sample effective n=n.
Normal approximation throughout; one-sided effect assumed in the prespecified alternative direction. Enter analyzable counts; attrition only adds a recruitment target. No t-based small-sample claim.
Example: d=.5 with 63+63 observations at two-sided alpha=.05 gives about 80% power.
Citation count lookup
Enter a DOI to retrieve OpenAlex and Crossref citation counts separately. Coverage differs, so counts are not summed; unavailable values are not zero.
Example: Look up DOI 10.1038/s41586-020-2649-2 and retain both dated provider counts.
Article fee calculator
Subtotal=APC×(1−discount/100)+colour+page+submission; total=subtotal×(1+tax/100). Full waiver replaces APC with0. Optional converted total=total×user exchange multiplier.
All charges use one entered currency. Discount and waiver apply only to APC and are mutually exclusive. Optional charges remain payable; tax applies to the post-discount subtotal. No live exchange rate, publisher quote or tax advice.
Example: APC1000, colour100, submission50, discount10%, tax20% →1260. Manual exchange .9 →1134 target currency units.
Confidence interval
Mean: mean ± tcrit SD/sqrt(n). Difference: (mean1−mean2) ± Welch tcrit sqrt(SD1²/n1+SD2²/n2), with Satterthwaite df. Wilson: center=(p+z²/2n)/(1+z²/n); half-width=z sqrt(p(1−p)/n+z²/4n²)/(1+z²/n).
Independent observations, t sampling model for means, independent binomial trials for Wilson. No paired difference interval or exact binomial interval. Confidence is a fraction; output retains the measurement units.
Example: Mean10, SD2, n25 at 95%: [9.17444,10.82556]. Wilson for4/10: [0.16818,0.68733].
p-value
z and t: left CDF, right survival, or two-sided 2×survival(|statistic|). Chi-square and F use upper-tail survival only; F requires numerator and denominator df.
Select the distribution used by the original test. Positive finite df required. No two-sided F or chi-square shortcut. A p-value is not a posterior probability or a measure of effect importance.
Example: z=1.96 gives two-sided p≈.0499958; t=2 with df10 gives p≈.0733880.
Adjusted R²
Adjusted R²=1−(1−R²)(n−1)/(n−k−1).
OLS with an intercept; k excludes the intercept, n>k+1. Ordinary R² must be between0 and1; negative adjusted R² is preserved. Intercept-free and weighted-model conventions unsupported.
Example: R²=.1, n10, k3 gives adjusted R²=−.35.
Cronbach's alpha
Raw alpha=k/(k−1)×(1−sum(item variances)/variance(total score)). Standardized alpha=k rbar/(1+(k−1)rbar).
Item matrix rows=subjects. Listwise deletion removes any incomplete subject. Explicit reverse flags negate selected columns; an additive scale constant does not affect covariance. Alpha measures internal consistency, not validity; negative estimates are possible. No polychoric estimates or CI.
Example: Five items with mean inter-item correlation .3 give standardized alpha=.681818.
ICC
MSB=subject, MSJ=rater, MSE=residual, MSW=within-subject mean squares. ICC1=(MSB−MSW)/(MSB+(k−1)MSW); ICC2=(MSB−MSE)/(MSB+(k−1)MSE+k(MSJ−MSE)/n); ICC3=(MSB−MSE)/(MSB+(k−1)MSE). Average-measure forms use k·ICC/(1+(k−1)ICC).
Input rows=raters, columns=subjects. Choose one-way random agreement, two-way random absolute agreement, or two-way mixed consistency, single or average measures. Complete balanced design only; no confidence bounds or missing-value handling. No default model is selected.
Example: Load the3-rater,6-subject example and explicitly choose ICC(2,1) for absolute agreement; compare ICC(3,1) only if consistency is the intended estimand.
Cohen's kappa
κ=(weighted observed agreement−weighted expected agreement)/(1−weighted expected agreement). Weights: identity for unweighted; 1−|i−j|/(K−1) linear; 1−((i−j)/(K−1))² quadratic.
Exactly two raters; square contingency table in the same category order on both axes. Choose weights explicitly. Ordered-category distances are meaningful only with an ordinal scale. No Fleiss kappa or CI; expected agreement1 is undefined.
Example: Load the3-category count table, then choose unweighted, linear or quadratic to compare the explicitly different agreement definitions.
NNT / NNH
ARR=control adverse-event risk−treatment adverse-event risk. Point NNT/NNH=ceil(1/|ARR|). Wald ARR CI=ARR ± z sqrt(pt(1−pt)/nt+pc(1−pc)/nc); Altman interval inverts endpoints separately on each side of zero.
Common follow-up horizon and adverse-event definition required. Positive ARR means benefit (NNT); negative means harm (NNH). Zero ARR is undefined. A crossing CI is disjoint benefit/harm ranges to infinity, not one finite range. Wald coverage can be poor for sparse/boundary risks.
Example: 10/100 treatment versus20/100 control gives ARR=.1 and NNT10; the95% Wald ARR interval is about[.002,.198], yielding a wide benefit interval. A comparison of10/100 versus11/100 crosses zero.
FDR
Sort m p-values; adjusted p at rank i=min over j≥i of (m p(j)/j), capped1. BY additionally multiplies by the harmonic sum1+1/2+…+1/m. Restore original order and reject adjusted≤q.
Supply the entire prespecified testing family. BH assumes independence or suitable positive dependence; BY allows arbitrary dependence. Ties remain equal. This controls false discovery rate, not family-wise error; no selective subset adjustment.
Example: [.01,.04,.03,.002,.04] → BH [.025,.04,.04,.01,.04], all rejected at q=.05.
z and t scores
z=(observation−population mean)/population SD. t=(sample mean−null mean)/(sample SD/sqrt(n)), df=n−1.
Known population SD for z; estimated sample SD and independent observations for one-sample t. SD must be positive. The score is signed and dimensionless. A score alone does not determine p without the distribution and tail.
Example: Observation12, population mean10, SD2 → z1. Sample mean12, null10, SD2,n25 → t5,df24.
Choose the statistic that matches the question
Start with the research question, the unit of observation and the quantity you want to estimate. Comparing two unrelated groups is a different design from measuring the same people twice. A sample mean needs a measure of uncertainty appropriate to an estimated standard deviation; a proportion needs an interval that respects binomial sampling. Selecting a familiar calculator without identifying those distinctions can produce a precise number for the wrong problem. The tabs name the available families; the method selector inside each form identifies the supported design. The mobile calculator selector changes the same selection as the desktop tabs.
The collection includes effect size, sample size, power analysis, confidence intervals, p-values, adjusted R², Cronbach alpha, ICC, Cohen kappa, NNT and NNH, false discovery rate adjustment, and z and t scores. Bibliographic tools provide DOI citation counts. The article fee calculator adds user-supplied publication charges and an optional manual exchange multiplier. All numeric calculations run in your browser. DOI citation count lookups require the network and query the named providers through a bounded server route. No calculator uses an AI model to invent missing measurements.
Read the method before interpreting the output
Effect-size methods keep the sign of the supplied contrast. Independent Cohen d divides group 1 minus group 2 by the pooled sample standard deviation; Hedges g applies a small-sample correction. Paired dz instead uses the standard deviation of individual differences. These numbers can differ even when the mean difference is identical. Odds ratios use an explicitly labelled event table. Correlations describe linear association between complete pairs and cannot establish causation. Confidence intervals quantify sampling uncertainty under their stated assumptions; they are not a probability that a fixed parameter lies inside this particular interval.
Planning calculations use a normal approximation throughout, including continuous outcomes. They do not use noncentral t distributions and should not replace small-sample design software. Binary planning uses the arcsine-standardized difference between probabilities. Reliability calculations also require deliberate choices: raw alpha uses item covariances after listwise deletion, standardized alpha uses the mean inter-item correlation, ICC depends on the rater model and agreement definition, and weighted kappa requires ordered categories. Neither a high consistency coefficient nor a small p-value establishes validity, importance or a sound design. Report the estimate with the design and assumptions.
Keep a reproducible calculation record
Use Load example to inspect a realistic input before entering your own values. Check whether each probability is requested as a fraction or each percentage as a percentage: 0.05 is significance alpha, whereas 10 is ten percent attrition. Keep means and standard deviations on the same measurement scale. Enter integer participant counts and the entire p-value family for FDR adjustment. Matrices have explicitly labelled orientations; swapping subjects and raters changes the ICC analysis. Blank or NA cells are accepted only where the form describes a missing-data policy. Other missing data must be handled outside this calculator.
After calculating, review intermediate quantities such as the pooled SD, degrees of freedom, analyzable counts and recruitment totals. Copy or download the text and export CSV when you need a spreadsheet record. Results include method assumptions and input values; browser reloads do not preserve them. For external metrics, retain the selected OpenAlex identifier, provider name, update date where supplied and retrieval timestamp. Citation databases have different coverage, so their counts should remain separate. Journal Impact Factor and licensed quartiles are not reconstructed from OpenAlex. A missing APC price is unknown, not a promise of free publication. These tools support research preparation; a statistical reviewer must confirm any consequential study design.
Frequently asked questions
Which calculators work offline?
Every numeric calculator works without a provider request. DOI citation counts require OpenAlex or Crossref access.
What does the example demonstrate?
The initial example compares means 10 and 8 with SDs 4 and 5 and sample sizes 20 and 30, returning signed Cohen d and exact-correction Hedges g.
Can I use this for any study design?
No. Cluster, survival, regression-power, noninferiority, sequential and complex repeated-measure designs are unsupported. The selected method names the implemented assumptions.
Are calculations free and unlimited?
Yes. Local calculations have no quota and require no login. External lookups are rate limited to protect provider access; waiting and retrying never requires signup.
Does AnswerThis save or receive my result?
No. Signup opens the AnswerThis app; it does not save or transfer this result. Copy or download your calculation before leaving the page.
How can I reproduce a result?
Keep the supplied inputs, selected method, units, assumptions and implementation date in the text or CSV export. Numerical fixtures are checked against SciPy 1.15.3, statsmodels 0.14.4 and pingouin 0.5.5.
What if a value is missing or invalid?
Required fields show a specific error. Zero-variance and invalid-degree-of-freedom cases are refused. Missing provider metadata is labelled unknown or not reported, and a failed lookup is never replaced by invented data.
Sources and related research tools
Method version: 14 September 2026. Numeric reference validation: SciPy 1.15.3, statsmodels 0.14.4, pingouin 0.5.5. DOI citation counts use OpenAlex and Crossref. Retrieval time is included with every live result.
SciPy statistical distributions; statsmodels statistical methods; OpenAlex access and limits; OpenAlex metric definitions.