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1. Calculate the sample size needed given these factors:one-tailed t-test with two independent groups of equal sizesmall effect size (see Piasta, S.B., & Justice, L.M., 2010)alpha =.05beta = .2Assume that the result is a sample size beyond what you can obtain. Use the compromise function to compute alpha and beta for a sample half the size. Indicate the resulting alpha and beta. Present an argument that your study is worth doing with the smaller sample (include peer-reviewed journal articles as needed to support your response). 2. Calculate the sample size needed given these factors:ANOVA (fixed effects, omnibus, one-way)small effect sizealpha =.05beta = .23 groupsAssume that the result is a sample size beyond what you can obtain. Use the compromise function to compute alpha and beta for a sample approximately half the size. Give your rationale for your selected beta/alpha ratio. Indicate the resulting alpha and beta. Give an argument that your study is worth doing with the smaller sample.3. In a few sentences, describe two designs that can address your research question. The designs must involve two different statistical analyses. For each design, specify and justify each of the four factors and calculate the estimated sample size you’ll need. Give reasons for any parameters you need to specify for G*Power.Support your paper with a minimum of 5 resources. In addition to these specified resources, other appropriate scholarly resources, including older articles, may be included.Length: 5-7 pages not including title and reference pagesReferences: Minimum of 5 scholarly resources.The first two charts in the word document will pertain to the first question and the last two charts will pertain to the second questions. Please let me know if there are any questions.


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Encyclopedia of Research Design
Cohen’s d Statistic
Contributors: Shayne B. Piasta & Laura M. Justice
Edited by: Neil J. Salkind
Book Title: Encyclopedia of Research Design
Chapter Title: “Cohen’s d Statistic”
Pub. Date: 2010
Access Date: October 26, 2017
Publishing Company: SAGE Publications, Inc.
City: Thousand Oaks
Print ISBN: 9781412961271
Online ISBN: 9781412961288
Print pages: 181-185
©2010 SAGE Publications, Inc.. All Rights Reserved.
This PDF has been generated from SAGE Knowledge. Please note that the pagination of
the online version will vary from the pagination of the print book.
Copyright © 2010 by SAGE Publications, Inc.
SAGE Reference
Cohen’s d statistic is a type of effect size. An effect size is a specific numerical nonzero value
used to represent the extent to which a null hypothesis is false. As an effect size, Cohen’s d is
typically used to represent the magnitude of differences between two (or more) groups on a
given variable, with larger values representing a greater differentiation between the two
groups on that variable. When comparing means in a scientific study, the reporting of an
effect size such as Cohen’s d is considered complementary to the reporting of results from a
test of statistical significance. Whereas the test of statistical significance is used to suggest
whether a null hypothesis is true (no difference exists between Populations A and B for a
specific phenomenon) or false (a difference exists between Populations A and B for a specific
phenomenon), the calculation of an effect size estimate is used to represent the degree of
difference between the two populations in those instances for which the null hypothesis was
deemed false. In cases for which the null hypothesis is false (i.e., rejected), the results of a
test of statistical significance imply that reliable differences exist between two populations on
the phenomenon of interest, but test outcomes do not provide any value regarding the extent
of that difference. The calculation of Cohen’s d and its interpretation provide a way to estimate
the actual size of observed differences between two groups, namely, whether the differences
are small, medium, or large.
Calculation of Cohen’s d Statistic
Cohen’s d statistic is typically used to estimate between-subjects effects for grouped data,
consistent with an analysis of variance framework. Often, it is employed within experimental
contexts to estimate the differential impact of the experimental manipulation across conditions
on the dependent variable of interest. The dependent variable must represent continuous
data; other effect size measures (e.g., Pearson family of correlation coefficients, odds ratios)
are appropriate for non-continuous data.
General Formulas
Cohen’s d statistic represents the standardized mean differences between groups. Similar to
other means of standardization such as z scoring, the effect size is expressed in standard
score units. In general, Cohen’s d is defined as
where d represents the effect size, μ1 and μ2 represent the two population means, and σ∊
represents the pooled within-group population standard deviation. In practice, these
population parameters are typically unknown and estimated by means of sample statistics:
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The population means are replaced with sample means (
,j) and the population standard deviation is replaced with Sp, the pooled standard deviation
from the sample. The pooled standard deviation is derived by weighing the variance around
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each sample mean by the respective sample size.
Calculation of the Pooled Standard Deviation
Although computation of the difference in sample means is straightforward in Equation 2, the
pooled standard deviation may be calculated in a number of ways. Consistent with the
traditional definition of a standard deviation, this statistic may be computed as
where nj represents the sample sizes for j groups and s2j represents the variance (i.e.,
squared standard deviation) of the / samples. Often, however, the pooled sample standard
deviation is corrected for bias in its estimation of the corresponding population parameter, σ∊.
Equation 4 denotes this correction of bias in the sample statistic (with the resulting effect size
often referred to as Hedge’s g):
When simply computing the pooled standard deviation across two groups, this formula may
be reexpressed in a more common format. This formula is suitable for data analyzed with a
two-way analysis of variance, such as a treatment-control contrast:
The formula may be further reduced to the average of the sample variances when sample
sizes are equal:
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in the case of two groups.
Other means of specifying the denominator for Equation 2 are varied. Some formulas use the
average standard deviation across groups. This procedure disregards differences in sample
size in cases of unequal n when one is weighing sample variances and may or may not
correct for sample bias in estimation of the population standard deviation. Further formulas
employ the standard deviation of the control or comparison condition (an effect size referred
to as Glass’s Δ). This method is particularly suited when the introduction of treatment or other
experimental manipulation leads to large changes in group variance. Finally, more complex
formulas are appropriate when calculating Cohen’s d from data involving cluster randomized
or nested research designs. The complication partially arises because of the three available
variance statistics from which the pooled standard deviation may be computed: the withincluster variance, the between-cluster variance, or the total variance (combined between- and
within-cluster variance). Researchers must select the variance statistic appropriate for the
inferences they wish to draw.
Expansion beyond Two-Group Comparisons: Contrasts and Repeated Measures
Cohen’s d always reflects the standardized difference between two means. The means,
however, are not restricted to comparisons of two independent groups. Cohen’s d may also be
calculated in multigroup designs when a specific contrast is of interest. For example, the
average effect across two alternative treatments may be compared with a control. The value of
the contrast becomes the numerator as specified in Equation 2, and the pooled standard
deviation is expanded to include all j groups specified in the contrast (Equation 4).
A similar extension of Equations 2 and 4 may be applied to repeated measures analyses. The
difference between two repeated measures is divided by the pooled standard deviation across
the j repeated measures. The same formula may also be applied to simple contrasts within
repeated measures designs, as well as interaction contrasts in mixed (between- and withinsubjects factors) or split-plot designs. Note, however, that the simple application of the pooled
standard deviation formula does not take into account the correlation between repeated
measures. Researchers disagree as to whether these correlations ought to contribute to effect
size computation; one method of determining Cohen’s d while accounting for the correlated
nature of repeated measures involves computing d from a paired t test.
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Additional Means of Calculation
Beyond the formulas presented above, Cohen’s d may be derived from other statistics,
including the Pearson family of correlation coefficients (r), t tests, and F tests. Derivations
from r are particularly useful, allowing for translation among various effect size indices.
Derivations from other statistics are often necessary when raw data to compute Cohen’s d are
unavailable, such as when conducting a meta-analysis of published data. When d is derived
as in Equation 3, the following formulas apply:
Note that Equation 10 applies only for F tests with 1 degree of freedom (df) in the numerator;
further formulas apply when df> 1.
When d is derived as in Equation 4, the following formulas ought to be used:
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Again, Equation 13 applies only to instances in which the numerator df= 1.
These formulas must be corrected for the correlation (r) between dependent variables in
repeated measures designs. For example, Equation 12 is corrected as follows:
Finally, conversions between effect sizes computed with Equations 3 and 4 may be easily
Variance and Confidence Intervals
The estimated variance of Cohen’s d depends on how the statistic was originally computed.
When sample bias in the estimation of the population pooled standard deviation remains
uncorrected (Equation 3), the variance is computed in the following manner:
A simplified formula is employed when sample bias is corrected as in Equation 4:
Once calculated, the effect size variance may be used to compute a confidence interval (CI)
for the statistic to determine statistical significance:
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The z in the formula corresponds to the z-score value on the normal distribution
corresponding to the desired probability level (e.g., 1.96 for a 95% CI). Variances and CIs may
also be obtained through bootstrapping methods.
Cohen’s d, as a measure of effect size, describes the overlap in the distributions of the
compared samples on the dependent variable of interest. If the two distributions overlap
completely, one would expect no mean difference between them
. To the extent that the distributions do not overlap, the difference ought to be greater than
zero (assuming
Cohen’s d may be interpreted in terms of both statistical significance and magnitude, with the
latter the more common interpretation. Effect sizes are statistically significant when the
computed CI does not contain zero. This implies less than perfect overlap between the
distributions of the two groups compared. Moreover, the significance testing implies that this
difference from zero is reliable, or not due to chance (excepting Type I errors). While
significance testing of effect sizes is often undertaken, however, interpretation based solely on
statistical significance is not recommended. Statistical significance is reliant not only on the
size of the effect but also on the size of the sample. Thus, even large effects may be deemed
unreliable when insufficient sample sizes are utilized.
Interpretation of Cohen’s d based on the magnitude is more common than interpretation
based on statistical significance of the result. The magnitude of Cohen’s d indicates the extent
of nonoverlap between two distributions, or the disparity of the mean difference from zero.
Larger numeric values of Cohen’s d indicate larger effects or greater differences between the
two means. Values may be positive or negative, although the sign merely indicates whether
the first or second mean in the numerator was of greater magnitude (see Equation 2).
Typically, researchers choose to subtract the smaller mean from the larger, resulting in a
positive effect size. As a standardized measure of effect, the numeric value of Cohen’s d is
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interpreted in standard deviation units. Thus, an effect size of d =0.5 indicates that two group
means are separated by one-half standard deviation or that one group shows a one-half
standard deviation advantage over the other.
The magnitude of effect sizes is often described nominally as well as numerically. Jacob
Cohen defined effects as small (d=0.2), medium (d= 0.5), or large (d=0.8). These rules of
thumb were derived after surveying the behavioral sciences literature, which included studies
in various disciplines involving diverse populations, interventions or content under study, and
research designs. Cohen, in proposing these benchmarks in a 1988 text, explicitly noted that
they are arbitrary and thus ought not be viewed as absolute. However, as occurred with use of
.05 as an absolute criterion for establishing statistical significance, Cohen’s benchmarks are
oftentimes interpreted as absolutes, and as a result, they have been criticized in recent years
as outdated, atheoretical, and inherently nonmeaningful. These criticisms are especially
prevalent in applied fields in which medium-to-large effects prove difficult to obtain and
smaller effects are often of great importance. The small effect of d=0.07, for instance, was
sufficient for physicians to begin recommending aspirin as an effective method of preventing
heart attacks. Similar small effects are often celebrated in intervention and educational
research, in which effect sizes of d= 0.3 to d= 0.4 are the norm. In these fields, the practical
importance of reliable effects is often weighed more heavily than simple magnitude, as may
be the case when adoption of a relatively simple educational approach (e.g., discussing vs.
not discussing novel vocabulary words when reading storybooks to children) results in effect
sizes of d= 0.25 (consistent with increases of one-fourth of a standard deviation unit on a
standardized measure of vocabulary knowledge).
Critics of Cohen’s benchmarks assert that such practical or substantive significance is an
important consideration beyond the magnitude and statistical significance of effects.
Interpretation of effect sizes requires an understanding of the context in which the effects are
derived, including the particular manipulation, population, and dependent measure(s) under
study. Various alternatives to Cohen’s rules of thumb have been proposed. These include
comparisons with effects sizes based on (a) normative data concerning the typical growth,
change, or differences between groups prior to experimental manipulation; (b) those obtained
in similar studies and available in the previous literature; (c) the gain necessary to attain an a
priori criterion; and (d) cost–benefit analyses.
Cohen’s d in Meta-Analyses
Cohen’s d, as a measure of effect size, is often used in individual studies to report and
interpret the magnitude of between-group differences. It is also a common tool used in metaanalyses to aggregate effects across different studies, particularly in meta-analyses involving
study of between-group differences, such as treatment studies. A meta-analysis is a statistical
synthesis of results from independent research studies (selected for inclusion based on a set
of predefined commonalities), and the unit of analysis in the meta-analysis is the data used
for the independent hypothesis test, including sample means and standard deviations,
extracted from each of the independent studies. The statistical analyses used in the metaanalysis typically involve (a) calculating the Cohen’s d effect size (standardized mean
difference) on data available within each independent study on the target variable(s) of
interest and (b) combining these individual summary values to create pooled estimates by
means of any one of a variety of approaches (e.g., Rebecca DerSimonian and Nan Laird’s
random effects model, which takes into account variations among studies on certain
parameters). Therefore, the methods of the meta-analysis may rely on use of Cohen’s d as a
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way to extract and combine data from individual studies. In such meta-analyses, the reporting
of results involves providing average d values (and CIs) as aggregated across studies.
In meta-analyses of treatment outcomes in the social and behavioral sciences, for instance,
effect estimates may compare outcomes attributable to a given treatment (Treatment X) as
extracted from and pooled across multiple studies in relation to an alternative treatment
(Treatment Y) for Outcome Z using Cohen’s d (e.g., d =0.21, CI = 0.06, 1.03). It is important to
note that the meaningful-ness of this result, in that Treatment X is, on average, associated
with an improvement of about one-fifth of a standard deviation unit for Outcome Z relative to
Treatment Y, must be interpreted in reference to many factors to determine the actual
significance of this outcome. Researchers must, at the least, consider whether the one-fifth of
a standard deviation unit improvement in the outcome attributable to Treatment X has any
practical significance.
effect size
standard deviations
statistical significance
repeated measures
significance testing
sampling bias
Shayne B.Piasta, and Laura M.Justice
See also
Analysis of Variance (ANOVA)
Effect Size, Measures of
Mean Comparisons
Statistical Power Analysis for the Behavioral Sciences
Further Readings
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (
2nd ed.
). Mahwah, NJ: Lawrence Erlbaum.
Cooper, H., & Hedges, L. V. (1994). The handbook of research synthesis. New York: Russell
Sage Foundation.
Hedges, L. V.Effect sizes in cluster-randomized designs. Journal of Educational & Behavioral
Statistics32 (2007). 341–370.
Hill, C. J., Bloom, H. S., Black, A. R., & Lipsey, M. W. (2007, July). Empirical benchmarks for
interpreting effect sizes in research. New York: MDRC.
Ray, J. W., and Shadish, W. R.How interchangeable are different estimators of effect size.
Journal of Consulting & Clinical
Wilkinson, L.APA Task Force on Statistical Inference. Statistical methods in psychology
journals: Guidelines and explanations. American Psychologist54 (1999). 594–604.
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1. Calculate the sample size needed given these factors:
2. Compromise function

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