Effect Sizes Template (Statistics, Choice, Formulas, CIs, R and Python)

Effect Sizes: [Your study question or course]

A working reference and practice sheet for effect sizes: how to choose one for your design, how to compute it by hand and in R and Python, how to convert between effect size families, how to put a confidence interval around it, and how to interpret it without leaning on "small, medium, large" labels. It is for statistics students preparing for an exam, master's and PhD students reporting thesis results, and analysts who need to say how big an effect is, not only whether it exists. It speeds up learning with a decision diagram, a fully solved example, a half solved example, an exercise, self-test questions and a separate answer key. The worked examples use fictional data; replace them with your own. Source basis: standard effect size formulas (Cohen 1988; Hedges 1981; Hedges and Olkin 1985; Lakens 2013) written in original notation and words.

Document control

This table records the course or project the sheet supports, the data set behind the numbers and the software that produced them. Rows marked Example: show how to fill a field.

What an effect size is and why it matters

A hypothesis test answers a narrow question: is the observed pattern surprising if there were no effect at all? An effect size answers the question your reader actually cares about: how big is the effect? A p-value mixes the size of the effect with the sample size n, so a trivial effect becomes "significant" with enough data and a large effect can miss significance in a small study. The effect size separates the two. It is the number that goes into a meta-analysis, the number a power analysis needs, and the number a decision maker uses to judge whether a result is worth acting on.

Effect sizes come in two kinds. Unstandardized effect sizes keep the original units: "students who revised with flashcards scored 5.4 points higher", "the new yeast raised loaves by 0.7 cm". When the units are meaningful, these are usually the best thing to report, because anyone can judge whether 5.4 exam points matters. Standardized effect sizes divide by a measure of variability or express the effect as a share of variance: Cohen's d, r, η², Cramér's V. They let you compare studies that used different scales and they feed power calculations. Report the unstandardized effect whenever the scale has meaning, and add a standardized one when you need comparability.

Every effect size is an estimate from a sample, so it has sampling error. A d of 1.10 from 20 students is compatible with true effects from small to very large. That is why every effect size you report should carry a 95% confidence interval (CI).

Choosing an effect size

The right effect size follows from three facts about your study: the type of outcome (continuous, binary, counts in categories), the number of groups or predictors, and whether the measurements are independent or paired. The diagram below walks through those decisions; the table that follows gives the default choice and a common alternative.

Example: the Lakemoor revision study compares exam scores (continuous) between two independent groups of students, so the default is Hedges' g, reported with the raw mean difference.

The data used in the examples

All examples use small fictional data sets typed into the code, so nothing needs to be downloaded.

The revision study scores are: flashcards 72, 75, 81, 66, 79, 70, 74, 84, 77, 71 (mean 74.9, SD 5.466); re-reading 65, 70, 74, 68, 72, 63, 77, 71, 66, 69 (mean 69.5, SD 4.249). The chart below shows every score so you can see the overlap that a single d value hides.

Formulas: mean differences

Cohen's d says how many standard deviations apart two group means are. It divides the difference in means by the pooled standard deviation, which is a weighted average of the two group SDs.

Hedges' g removes the small upward bias of d. The exact correction factor uses the gamma function; the approximation below is accurate to three decimals once the degrees of freedom df are above about 10.

Paired designs have two different "d" values and you must say which one you report.

• d_z standardizes the mean of the differences by the SD of the differences. It is the effect size the paired t-test actually tests, so it is the right input for a paired power analysis, and it equals the paired t divided by the square root of n.
• d_av standardizes the mean difference by the average of the two raw SDs. It is on the same scale as an independent-groups d, so use it when you want to compare a within-subject result with between-subject studies.

When the two measurements are strongly correlated, s_D is small and d_z is much larger than d_av. With roughly equal SDs the link is $d_z \approx d_{av} / \sqrt{2(1 - r)}$, where r is the correlation between the two measurements.

Formulas: variance explained, association and regression

Eta squared (η²) is the share of the total variation that lies between groups. Partial η² divides the effect's sum of squares by itself plus the error sum of squares only, leaving out other effects in the model. In a one-way ANOVA there are no other effects, so η² and partial η² are identical. Omega squared (ω²) is a less biased estimate of the population share; it is always smaller than η² and turns negative when F < 1 (report it as 0).

Pearson r is itself an effect size: the strength of a straight-line relationship, from -1 to 1. Its confidence interval uses Fisher's z transformation, because the sampling distribution of r is skewed when r is far from zero.

Categorical outcomes. In a 2 x 2 table with cells a, b (first row) and c, d (second row), φ is the Pearson correlation between the two binary variables, Cramér's V extends it to larger tables, the odds ratio compares the odds of the outcome, and the risk ratio compares the probabilities.

Confidence intervals for OR and RR are built on the log scale and exponentiated, so they are not symmetric around the estimate.

Regression. R² is the share of outcome variance explained by the model. Cohen's f² turns a share of variance into a signal-to-noise ratio and is the input for regression power analysis. The most useful version is the f² for a set of predictors B added to a model that already contains A.

Converting between effect sizes

You often need an effect size that a paper did not report, or need to put two studies on one scale. The conversions below are exact for the stated design or are standard approximations (Borenstein et al. 2009). They assume the design is what the formula says: using an independent-groups conversion on a paired t gives a wrong answer.

Two warnings. First, d and r measure different things: r depends on how the sample is split between groups, so the same d gives a smaller r when the groups are unequal. Second, the OR to d conversion assumes an underlying logistic distribution; treat it as a rough bridge, not an identity.

Confidence intervals for effect sizes

An effect size without an interval invites over-interpretation. The CI shows the range of population effects that are compatible with your data at the 95% level. Common methods:

• Noncentral t inversion for d. The observed t follows a noncentral t distribution whose noncentrality parameter is proportional to the true d. Find the two noncentrality values that put the observed t at the 97.5th and 2.5th percentiles, then multiply by $\sqrt{1/n_1 + 1/n_2}$. This is the method in the Example A code (Steiger and Fouladi 1997; Cumming and Finch 2001).
• Large-sample standard error for d. $SE_d \approx \sqrt{(n_1 + n_2)/(n_1 n_2) + d^2 / (2(n_1 + n_2))}$ and CI = d ± 1.96 SE. For the Lakemoor data this gives [0.16, 2.04], close to the exact [0.14, 2.04]. It is less accurate in small samples and for large d.
• Fisher z for r. Transform, build a symmetric interval, transform back (see the formula above).
• Log scale for OR and RR. Build the interval for ln(OR) or ln(RR) and exponentiate.
• Noncentral F for η² and ω². Invert the noncentral F distribution in the same way as for d. Packages such as effectsize (R) automate this; check which interval they return, because conventions differ: recent effectsize versions return a one-sided interval (upper limit fixed at 1) by default, and some authors use a two-sided 90% interval so that it matches an F test at α = 0.05. State the type you report.
• Bootstrap. Resample the data, recompute the effect size, take percentile or BCa limits (Efron 1979). Useful when no formula exists; set and report the seed.

For Hedges' g, multiply both limits of the d interval by J. A wide interval is a finding in itself: it says the study cannot pin down the size of the effect, which belongs in your limitations section.

Interpreting magnitude: why Cohen's benchmarks are a last resort

Cohen (1988) offered rough conventions (d of 0.2, 0.5, 0.8; r of .10, .30, .50; η² of .01, .06, .14; f² of .02, .15, .35) and explicitly described them as a fallback for when nothing better is known. They were based on his reading of behavioural science research in the 1960s, not on any principle. Using them as fixed thresholds causes three problems. They ignore the field: a d of 0.2 on mortality is enormous, while a d of 0.8 on a reaction-time task can be routine. They ignore the cost and benefit of acting: a cheap intervention with a small effect may be worth more than an expensive one with a large effect. And they ignore uncertainty: calling d = 1.10 "large" hides that the 95% CI runs from 0.14 to 2.04, which includes effects Cohen would call small.

Better ways to interpret, in order of preference:

1. Raw units with a practical anchor. "5.4 exam points, about half a grade band at Lakemoor."
2. Comparison with prior evidence. "Larger than the median effect of revision techniques in [prior review]."
3. A smallest effect size of interest (SESOI) fixed before the study, then ask whether the CI excludes it.
4. Distribution-based restatements. With d = 1.10, about 86% of the flashcard group score above the re-reading mean (the normal cumulative probability at 1.10), assuming normal distributions with equal SDs.
5. Cohen's labels, flagged as conventions, only when none of the above is available (Funder and Ozer 2019).

Worked example A (fully solved): two independent groups

Question. At Lakemoor University (fictional), did students who revised with flashcards score higher on the statistics exam than students who re-read their notes, and by how much?

Design and choice. Two independent groups, continuous outcome, so Hedges' g with the raw mean difference. The test is the Welch t-test (the default for two independent groups); the effect size uses the pooled SD as in the formula above.

By hand from the summary statistics.

1. Mean difference: 74.9 - 69.5 = 5.4 points.
2. Pooled SD: $s_p = \sqrt{(9 \cdot 5.466^2 + 9 \cdot 4.249^2)/18} = \sqrt{(268.9 + 162.5)/18} = \sqrt{23.97} = 4.896$.
3. Cohen's d: 5.4 / 4.896 = 1.103.
4. Correction: J = 1 - 3/(4 · 20 - 9) = 1 - 3/71 = 0.958, so g = 0.958 · 1.103 = 1.056.
5. CI for d by noncentral t inversion: [0.14, 2.04]; multiply by J for g: [0.14, 1.95].

R code

flash  <- c(72, 75, 81, 66, 79, 70, 74, 84, 77, 71)
reread <- c(65, 70, 74, 68, 72, 63, 77, 71, 66, 69)
n1 <- length(flash); n2 <- length(reread)
sp <- sqrt(((n1 - 1) * var(flash) + (n2 - 1) * var(reread)) / (n1 + n2 - 2))
d  <- (mean(flash) - mean(reread)) / sp
J  <- 1 - 3 / (4 * (n1 + n2) - 9)          # small-sample correction
g  <- J * d
# CI for d by inverting the noncentral t distribution
t_obs <- unname(t.test(flash, reread, var.equal = TRUE)$statistic)
df <- n1 + n2 - 2; k <- sqrt(1 / n1 + 1 / n2)
ncp_at <- function(prob) suppressWarnings(
  uniroot(function(ncp) pt(t_obs, df, ncp) - prob, c(-20, 20))$root)
ci <- c(ncp_at(0.975), ncp_at(0.025)) * k
welch <- t.test(flash, reread)              # Welch is the R default
round(c(sp = sp, d = d, J = J, g = g, t_pooled = t_obs,
        ci_low = ci[1], ci_high = ci[2], ci_g_low = J * ci[1], ci_g_high = J * ci[2]), 3)
round(c(t_welch = unname(welch$statistic), df_welch = unname(welch$parameter),
        p_welch = welch$p.value), 3)

       sp         d         J         g  t_pooled    ci_low   ci_high  ci_g_low
    4.896     1.103     0.958     1.056     2.466     0.143     2.037     0.137
ci_g_high
    1.951
 t_welch df_welch  p_welch
   2.466   16.968    0.025

Python code

import numpy as np
from scipy import stats, optimize

flash = np.array([72, 75, 81, 66, 79, 70, 74, 84, 77, 71])
reread = np.array([65, 70, 74, 68, 72, 63, 77, 71, 66, 69])
n1, n2 = len(flash), len(reread)
sp = np.sqrt(((n1 - 1) * flash.var(ddof=1) + (n2 - 1) * reread.var(ddof=1))
             / (n1 + n2 - 2))
d = (flash.mean() - reread.mean()) / sp
J = 1 - 3 / (4 * (n1 + n2) - 9)
t_obs = stats.ttest_ind(flash, reread).statistic       # scipy default: pooled
df, k = n1 + n2 - 2, np.sqrt(1 / n1 + 1 / n2)
ncp_at = lambda prob: optimize.brentq(
    lambda nc: stats.nct.cdf(t_obs, df, nc) - prob, -20, 20)
lo, hi = ncp_at(0.975) * k, ncp_at(0.025) * k
welch = stats.ttest_ind(flash, reread, equal_var=False)
print(f"sp={sp:.3f} d={d:.3f} J={J:.3f} g={J * d:.3f} t_pooled={t_obs:.3f}")
print(f"95% CI d [{lo:.2f}, {hi:.2f}]  g [{J * lo:.2f}, {J * hi:.2f}]")
print(f"Welch t={welch.statistic:.3f} df={welch.df:.2f} p={welch.pvalue:.3f}")

sp=4.896 d=1.103 J=0.958 g=1.056 t_pooled=2.466
95% CI d [0.14, 2.04]  g [0.14, 1.95]
Welch t=2.466 df=16.97 p=0.025

R and Python defaults differ. t.test() in R runs the Welch test unless you pass var.equal = TRUE; scipy.stats.ttest_ind() runs the pooled test unless you pass equal_var=False. With equal group sizes the two t values are identical (2.466 here) but the df differ (18 pooled, 16.97 Welch), so the p-values differ slightly. Always state which test you ran.

Interpretation. Flashcard students scored 5.4 points higher on average. Standardized, that is g = 1.06, but the 95% CI [0.14, 1.95] is wide: the data are compatible with anything from a small to a very large advantage. With only 10 students per group, the study shows the direction convincingly but not the size.

Report sentence. Students who revised with flashcards scored higher (M = 74.9, SD = 5.47) than students who re-read their notes (M = 69.5, SD = 4.25), a difference of 5.4 points, Welch t(16.97) = 2.47, p = .025, Hedges' g = 1.06, 95% CI [0.14, 1.95].

Worked example B (half solved): paired design

Question. Eight Lakemoor students took a test anxiety questionnaire before and after a study skills workshop. How large was the reduction?

Steps:

1. Mean difference $\bar{D}$ = 24 / 8 = [ ].
2. SD of the differences $s_D$ = 1.852 (given).
3. d_z = $\bar{D}$ / $s_D$ = [ ].
4. SD before = 4.140, SD after = 2.828, so the average SD = [ ].
5. d_av = $\bar{D}$ / average SD = [ ].
6. Check: paired t = d_z · √8 = [ ] (the software gives 4.583).
7. The correlation between before and after scores is 0.927. In one sentence, explain why d_z is about twice d_av: [ ].
8. Which value goes into a power analysis for a repeat of this paired study, and which one would you use to compare with a between-groups study of the same workshop? [ ]

Exercise C (unsolved): a 2 x 2 table

Lakemoor recorded whether 80 students attended optional revision sessions and whether they passed the exam.

Compute: (a) the risk ratio of passing with its 95% CI; (b) the odds ratio with its 95% CI (log method); (c) φ; (d) one sentence saying which of RR and OR you would put in a report for students and why. The Pearson χ²(1) without continuity correction is 5.698.

Code for every family

The scripts below compute the effect sizes for Example B, the bakery ANOVA, the study hours correlation, Exercise C and the regression. Both print the same numbers.

# 1. Paired: anxiety score before and after a workshop (8 students)
pre  <- c(24, 30, 27, 22, 35, 29, 26, 31)
post <- c(21, 26, 25, 22, 30, 24, 25, 27)
dif <- pre - post
d_z  <- mean(dif) / sd(dif)
d_av <- mean(dif) / ((sd(pre) + sd(post)) / 2)
# 2. One-way ANOVA: Brightwell Bakery loaf height (cm) by yeast strain
height <- c(9.8, 10.4, 10.1, 9.6, 10.0, 10.3, 10.6, 11.0, 10.8,
            10.5, 11.2, 10.7, 10.2, 10.9, 10.4, 10.0, 10.6, 10.5)
yeast <- factor(rep(c("A", "B", "C"), each = 6))
tab <- summary(aov(height ~ yeast))[[1]]
ssb <- tab$`Sum Sq`[1]; ssw <- tab$`Sum Sq`[2]
dfb <- tab$Df[1]; dfw <- tab$Df[2]; msw <- tab$`Mean Sq`[2]; Fv <- tab$`F value`[1]
eta2   <- ssb / (ssb + ssw)
omega2 <- (ssb - dfb * msw) / (ssb + ssw + msw)
peta2  <- Fv * dfb / (Fv * dfb + dfw)          # from F and df only
# 3. Correlation: hours studied and exam score (12 students)
hours <- c(2, 4, 5, 3, 8, 6, 7, 1, 9, 5, 6, 4)
score <- c(62, 60, 74, 66, 72, 64, 80, 58, 77, 61, 70, 69)
ct <- cor.test(hours, score)                    # Fisher z CI
# 4. 2 x 2 table: attended revision sessions x passed (80 students)
pass <- matrix(c(32, 8, 22, 18), nrow = 2, byrow = TRUE)
chi <- chisq.test(pass, correct = FALSE)
V  <- sqrt(unname(chi$statistic) / (sum(pass) * (min(dim(pass)) - 1)))
OR <- (32 * 18) / (8 * 22); se_lor <- sqrt(sum(1 / pass))
RR <- (32 / 40) / (22 / 40); se_lrr <- sqrt(1/32 - 1/40 + 1/22 - 1/40)
# 5. Regression: score on hours and sleep
sleep <- c(6, 7, 6, 8, 7, 5, 8, 6, 7, 7, 8, 5)
full <- summary(lm(score ~ hours + sleep)); red <- summary(lm(score ~ hours))
f2_model <- full$r.squared / (1 - full$r.squared)
f2_sleep <- (full$r.squared - red$r.squared) / (1 - full$r.squared)
round(c(d_z = d_z, d_av = d_av, r_pre_post = cor(pre, post),
        eta2 = eta2, omega2 = omega2, partial_eta2 = peta2, F = Fv,
        r = unname(ct$estimate), r_low = ct$conf.int[1], r_high = ct$conf.int[2],
        chi2 = unname(chi$statistic), p_chi2 = chi$p.value, V = V,
        OR = OR, OR_low = exp(log(OR) - 1.96 * se_lor), OR_high = exp(log(OR) + 1.96 * se_lor),
        RR = RR, RR_low = exp(log(RR) - 1.96 * se_lrr), RR_high = exp(log(RR) + 1.96 * se_lrr),
        R2 = full$r.squared, adj_R2 = full$adj.r.squared,
        f2_model = f2_model, f2_sleep = f2_sleep), 3)

         d_z         d_av   r_pre_post         eta2       omega2 partial_eta2
       1.620        0.861        0.927        0.578        0.508        0.578
           F            r        r_low       r_high         chi2       p_chi2
      10.285        0.753        0.314        0.926        5.698        0.017
           V           OR       OR_low      OR_high           RR       RR_low
       0.267        3.273        1.211        8.844        1.455        1.056
     RR_high           R2       adj_R2     f2_model     f2_sleep
       2.004        0.579        0.486        1.376        0.030

import numpy as np
import pandas as pd
from scipy import stats
import statsmodels.formula.api as smf
from statsmodels.stats.anova import anova_lm

# 1. Paired
pre = np.array([24, 30, 27, 22, 35, 29, 26, 31])
post = np.array([21, 26, 25, 22, 30, 24, 25, 27])
dif = pre - post
d_z = dif.mean() / dif.std(ddof=1)
d_av = dif.mean() / ((pre.std(ddof=1) + post.std(ddof=1)) / 2)
# 2. One-way ANOVA
bake = pd.DataFrame({
    "height": [9.8, 10.4, 10.1, 9.6, 10.0, 10.3, 10.6, 11.0, 10.8,
               10.5, 11.2, 10.7, 10.2, 10.9, 10.4, 10.0, 10.6, 10.5],
    "yeast": ["A"] * 6 + ["B"] * 6 + ["C"] * 6})
tab = anova_lm(smf.ols("height ~ C(yeast)", data=bake).fit())
ssb, ssw = tab["sum_sq"].iloc[:2]
dfb, msw = tab["df"].iloc[0], tab["mean_sq"].iloc[1]
eta2 = ssb / (ssb + ssw)
omega2 = (ssb - dfb * msw) / (ssb + ssw + msw)
# 3. Correlation with Fisher z CI
hours = np.array([2, 4, 5, 3, 8, 6, 7, 1, 9, 5, 6, 4])
score = np.array([62, 60, 74, 66, 72, 64, 80, 58, 77, 61, 70, 69])
res_r = stats.pearsonr(hours, score)
r_ci = res_r.confidence_interval()               # Fisher z by default
# 4. 2 x 2 table
table = np.array([[32, 8], [22, 18]])
chi2, p, dof, _ = stats.chi2_contingency(table, correction=False)
V = np.sqrt(chi2 / (table.sum() * (min(table.shape) - 1)))
OR = stats.contingency.odds_ratio(table, kind="sample").statistic
or_ci = np.exp(np.log(OR) + np.array([-1.96, 1.96]) * np.sqrt((1 / table).sum()))
rr = stats.contingency.relative_risk(32, 40, 22, 40)
rr_ci = rr.confidence_interval()
# 5. Regression
dat = pd.DataFrame({"score": score, "hours": hours,
                    "sleep": [6, 7, 6, 8, 7, 5, 8, 6, 7, 7, 8, 5]})
full = smf.ols("score ~ hours + sleep", data=dat).fit()
red = smf.ols("score ~ hours", data=dat).fit()
print(f"d_z={d_z:.3f} d_av={d_av:.3f}")
print(f"eta2={eta2:.3f} omega2={omega2:.3f}")
print(f"r={res_r.statistic:.3f} [{r_ci.low:.3f}, {r_ci.high:.3f}]")
print(f"chi2={chi2:.3f} p={p:.3f} V={V:.3f} OR={OR:.2f} [{or_ci[0]:.2f}, {or_ci[1]:.2f}]"
      f" RR={rr.relative_risk:.2f} [{rr_ci.low:.2f}, {rr_ci.high:.2f}]")
print(f"R2={full.rsquared:.3f} f2_model={full.rsquared / (1 - full.rsquared):.3f} "
      f"f2_sleep={(full.rsquared - red.rsquared) / (1 - full.rsquared):.3f}")

d_z=1.620 d_av=0.861
eta2=0.578 omega2=0.508
r=0.753 [0.314, 0.926]
chi2=5.698 p=0.017 V=0.267 OR=3.27 [1.21, 8.84] RR=1.45 [1.06, 2.00]
R2=0.579 f2_model=1.376 f2_sleep=0.030

Notice three teaching points in this output. The bakery ω² (0.508) is smaller than η² (0.578), as it always is. The regression explains 57.9% of the variance overall, yet adding sleep to a model that already has hours adds almost nothing (f² = 0.030): the model-level effect size and the effect size of one predictor answer different questions. And chisq.test() in R applies a continuity correction to 2 x 2 tables by default, as does scipy.stats.chi2_contingency(); both scripts switch it off with correct = FALSE and correction=False so that V matches the uncorrected χ².

Reading software output

Reporting template and checklist

Fill-in sentence:

[Group 1] scored [higher / lower] (M = [ ], SD = [ ]) than [group 2] (M = [ ], SD = [ ]), a difference of [raw difference and units], [test] t([df]) = [ ], p = [ ], [Hedges' g / d_z / d_av] = [ ], 95% CI [ , ].

Checklist:

• The effect size matches the design (independent, paired, factorial, categorical).
• For paired designs, the text says whether d is d_z or d_av.
• The raw (unstandardized) effect is reported when the units have meaning.
• Every effect size has a 95% CI and the CI method is stated.
• Hedges' g (not d) is used for small samples, or the choice is explained.
• Partial η² is labelled as partial; ω² is preferred when samples are small.
• Interpretation compares with a practical anchor or prior evidence before any Cohen label.
• R and Python (or two tools) give the same numbers, or the difference is explained.

Common mistakes and misconceptions

Self-test questions

1. A study reports t(38) = 2.10 for two independent groups of 20. What is Cohen's d?
2. Why is Hedges' g always slightly smaller in absolute value than Cohen's d, and when does the difference stop mattering?
3. A one-way ANOVA has η² = 0.12. Is partial η² larger, smaller or the same? Is ω² larger, smaller or the same?
4. In a 2 x 2 table the pass rates are 90% and 75%. Without computing, will the odds ratio be closer to or further from 1 than the risk ratio?
5. A colleague writes "R² = .58, a large effect". What two things would you ask them to add?
6. Which effect size do you put into a power analysis for a paired t-test, d_z or d_av, and why?

My notes and next steps

Sources and further reading

Based on the following sources, cited by name; all text, notation and examples here are original.

• Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.
• Hedges, L. V. (1981). Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128.
• Hedges, L. V., and Olkin, I. (1985). Statistical Methods for Meta-Analysis. Academic Press.
• Olejnik, S., and Algina, J. (2003). Generalized eta and omega squared statistics: measures of effect size for some common research designs. Psychological Methods, 8(4), 434–447.
• Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, 863.
• Borenstein, M., Hedges, L. V., Higgins, J. P. T., and Rothstein, H. R. (2009). Introduction to Meta-Analysis. Wiley.
• Steiger, J. H., and Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical models. In L. L. Harlow, S. A. Mulaik, and J. H. Steiger (Eds.), What If There Were No Significance Tests? Lawrence Erlbaum Associates.
• Cumming, G., and Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61(4), 532–574.
• Fisher, R. A. (1921). On the "probable error" of a coefficient of correlation deduced from a small sample. Metron, 1(4), 3–32.
• Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7(1), 1–26.
• Funder, D. C., and Ozer, D. J. (2019). Evaluating effect size in psychological research: sense and nonsense. Advances in Methods and Practices in Psychological Science, 2(2), 156–168.
• Wasserstein, R. L., and Lazar, N. A. (2016). The ASA statement on p-values: context, process, and purpose. The American Statistician, 70(2), 129–133.
• Further reading (by name only): NIST/SEMATECH e-Handbook of Statistical Methods (public domain); de Smith, M. J. (2024). Statistical Analysis Handbook (2024 edition, StatsRef); OpenIntro Statistics.

Answer key

Example B. (1) $\bar{D}$ = 3.0. (3) d_z = 3.0 / 1.852 = 1.620. (4) Average SD = (4.140 + 2.828) / 2 = 3.484. (5) d_av = 3.0 / 3.484 = 0.861. (6) 1.620 · √8 = 1.620 · 2.828 = 4.58. (7) Because before and after scores are highly correlated (r = 0.927), the SD of the differences (1.852) is much smaller than the raw SDs, so dividing by it gives a larger number. (8) d_z goes into the paired power analysis because it is what the paired t-test detects; d_av is the one to compare with a between-groups study.

Exercise C. (a) RR = (32/40) / (22/40) = 0.80 / 0.55 = 1.45; SE of ln RR = √(1/32 - 1/40 + 1/22 - 1/40) = 0.163; 95% CI [1.06, 2.00]. (b) OR = (32 · 18) / (8 · 22) = 576 / 176 = 3.27; SE of ln OR = √(1/32 + 1/8 + 1/22 + 1/18) = 0.507; 95% CI [1.21, 8.84]. (c) φ = (32 · 18 - 8 · 22) / √(40 · 40 · 54 · 26) = 400 / 1498.8 = 0.267, which equals √(5.698 / 80). (d) The risk ratio: "attenders were 1.45 times as likely to pass (80% vs 55%)" is easy to understand, while the OR of 3.27 would be misread as "three times as likely" because passing is common.

Self-test.

1. d = 2.10 · √(1/20 + 1/20) = 2.10 · 0.316 = 0.66.
2. J is below 1 because d overestimates the population effect in small samples. The shrinkage is about 3/(4 df), so it is about 4% at df = 18 and under 2% once df is above about 50, where it rarely changes the interpretation.
3. The same: in a one-way design there is only one effect, so partial η² = η². ω² is smaller.
4. Further from 1. The OR exaggerates relative to the RR when the outcome is common: here RR = 0.90/0.75 = 1.20 and OR = (0.90/0.10) / (0.75/0.25) = 3.0.
5. A confidence interval (or at least adjusted R² and n), and an anchor for "large": compared with what prior models or practical need? Also whether any single predictor matters (f² for the predictor of interest).
6. d_z, because the paired t statistic is d_z · √n. Using d_av would misstate the power: it underestimates power when the before-after correlation is above 0.5 (as here, r = 0.927, where d_z > d_av) and overestimates it when the correlation is below 0.5.

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