Effect Size Formula: Definition & Solved Example

Effect size is a crucial concept in mathematical research and statistical analysis. It provides a standardized way to measure the strength and magnitude of the difference between groups or the relationship between variables, regardless of the sample size. 

• By quantifying the practical significance of a finding, effect size helps researchers to interpret the meaning of their results and to compare them across studies. 
• Effect size can be computed in various ways and used in different fields of mathematics, from experimental design to meta-analysis. 
• Understanding effect size is crucial for researchers and practitioners who want to draw valid conclusions from their data and make evidence-based decisions. 
• The formula for effect size depends on the statistical test being used.

Moreover, effect size is becoming increasingly important in education, psychology, social sciences, and other fields where quantitative research is common.

Key Terms: Statistics, Data, Effect Size, Average, Mean, Standard Deviation, Variance, Correlation.

---

Effect Size in Statistics

[Click Here for Sample Questions]

In statistics, effect size refers to the magnitude or strength of a relationship or difference between two or more variables. It provides a standardized way to measure the practical significance of a finding, which may be statistically significant but not necessarily practically significant.

• Effect size is important because it helps researchers and practitioners to interpret the results of statistical analyses, and to compare findings across different studies. 
• Effect size can be calculated in different ways depending on the research question and the type of data, and there are various effect size measures, such as Cohen's d, eta-squared, and partial eta-squared. 
• Understanding effect size is crucial for anyone conducting quantitative research, as it provides a powerful tool for interpreting and communicating the practical significance of their findings.

---

Effect Size Formula

[Click Here for Sample Questions]

There are different formulas for calculating effect size depending on the type of analysis and the research question. Here are a few commonly used effect size formulas in statistics:

• Cohen's d: Cohen's d is a measure of the standardized mean difference between two groups. The formula is:

d = (M1 - M2) / s

where M1 and M2 are the means of the two groups, and s is the pooled standard deviation of the two groups.

• Pearson's correlation coefficient: Pearson's correlation coefficient is a measure of the strength and direction of the linear relationship between two continuous variables. The formula is:

r = (Σ(x - x̄)(y - ȳ)) / √(Σ(x - x̄)²Σ(y - ȳ)²)

where x and y are the two variables, x̄ and ȳ are their respective means, and Σ denotes the sum of the values.

• Eta-squared: Eta-squared is a measure of the proportion of variance in the dependent variable accounted for by the independent variable in an ANOVA analysis. The formula is:

η² = SSbetween / SStotal

where SSbetween is the sum of squares between groups, and SStotal is the total sum of squares.

Note: Cohen's d is a commonly used effect size measure in statistics, especially for comparing means between two groups. It is versatile and easy to interpret, working with continuous, binary, and ordinal data. It also provides a standardized metric for comparing effect sizes across studies. However, the choice of effect size measure should always depend on the specific research question, type of data, and statistical analysis being conducted.

Also Read:

---

Cohen’s d Effect Size Formula

[Click Here for Sample Questions]

Cohen's d is a commonly used effect size measure that quantifies the standardized mean difference between two groups. It is computed by taking the difference between the means of two groups and dividing it by the pooled standard deviation of the groups. The formula for Cohen's d is:

d = (M1 - M2) / s

where M1 and M2 are the means of the two groups being compared, and s is the pooled standard deviation.

The pooled standard deviation is calculated using the following formula:

s = √((s1² + s2²) / 2)

where s1 and s2 are the standard deviations of the two groups.

Cohen's d can take on positive or negative values, with positive values indicating that the mean of group 1 is higher than the mean of group 2, and negative values indicating the opposite. The magnitude of Cohen's d indicates the practical significance of the difference between the groups, with larger values indicating a greater difference. Generally, a Cohen's d value of 0.2 is considered a small effect size, 0.5 a moderate effect size, and 0.8 or above a large effect size.

Step-by-Step Calculation Procedure:

Step 1: Calculate the mean of each group.

Step 2: Calculate the standard deviation of each group.

Step 3: Calculate the difference between the means of the two groups.

Step 4: Calculate the pooled standard deviation.

Step 5: Calculate Cohen's d.

Step 6: Interpret the effect size.

Solved Example

Example: Calculate Cohen's d to determine the standardized mean difference between the scores of two groups A and B. Group A has the following scores: 12, 16, 18, 14, and 20. Group B has the following scores: 15, 19, 21, 17, and 23.

Solution: Step 1: Calculate the mean of each group.

Mean of Group A: (12 + 16 + 18 + 14 + 20) / 5 = 16

Mean of Group B: (15 + 19 + 21 + 17 + 23) / 5 = 19

Step 2: Calculate the standard deviation of each group.

For Group A:

Calculate the deviations of each score from the mean: (12 – 16) = – 4, (16 – 16) = 0, (18 – 16) = 2, (14 – 16) = – 2, and (20 – 16) = 4.

Square each deviation: ( – 4)² = 16, 0² = 0, 2² = 4, (-2)² = 4, and 4² = 16.

Calculate the sum of the squared deviations: 16 + 0 + 4 + 4 + 16 = 40.

Divide the sum of the squared deviations by n-1 (where n is the number of scores in the group minus 1): 40 / 4 = 10.

Take the square root of the result: √10 = 3.162.

So, the standard deviation of Group A is 3.162.

For Group B:

Calculate the deviations of each score from the mean: (15 – 19) = – 4, (19 – 19) = 0, (21 – 19) = 2, (17 – 19) = – 2, and (23 – 19) = 4.

Square each deviation: (-4)² = 16, 0² = 0, 2² = 4, (-2)² = 4, and 4² = 16.

Calculate the sum of the squared deviations: 16 + 0 + 4 + 4 + 16 = 40.

Divide the sum of the squared deviations by n-1 (where n is the number of scores in the group minus 1): 40 / 4 = 10.

Take the square root of the result: √10 = 3.162.

So, the standard deviation of Group B is 3.162.

Step 3: Calculate the difference between the means of the two groups.

Difference = Mean of Group A - Mean of Group B = 16 - 19 = -3

Step 4: Calculate the pooled standard deviation.

Pooled Standard Deviation = √((s1² + s2²) / 2) = √((3.162² + 3.162²) / 2) = 3.162

Step 5: Calculate Cohen's d.

Cohen's d = Difference / Pooled Standard Deviation = – 3 / 3.162 = -0.9487

Step 6: Interpret the effect size.

Since Cohen's d is negative, it indicates that the mean of Group A is lower than the mean of Group B. The magnitude of d |-0.9487| which is 0.9487 is less than 1, which means that the effect size is small to medium in size, according to Cohen's conventions. This suggests that there is a noticeable difference between the means of the two groups, but it may not be practically significant.

Example 2: Compare the heights of two groups of individuals (Group A and Group B) in centimeters. Group A has the following heights: 165.3, 170.2, 172.1, 168.4, 171.5, and 169.9. Group B has the following heights: 175.8, 178.4, 176.5, 173.9, 179.1, and 177.2. Calculate Cohen's d to determine the standardized mean difference between the two groups.

Solution: Step 1: Calculate the mean of each group.

Mean of Group A: (165.3 + 170.2 + 172.1 + 168.4 + 171.5 + 169.9) / 6 = 169.2 cm

Mean of Group B: (175.8 + 178.4 + 176.5 + 173.9 + 179.1 + 177.2) / 6 = 176.5 cm

Step 2: Calculate the pooled standard deviation (sp).

sp = sqrt(((n1 – 1)*s1² + (n2 – 1)*s2²) / (n1 + n2 – 2))

sp = sqrt(((6-1)*3.6² + (6-1)*2.9²) / (6+6-2))

sp = sqrt(77.92)

sp = 8.83 cm

Step 3: Calculate Cohen's d.

d = (M1 - M2) / sp

d = (169.2 - 176.5) / 8.83

d = – 0.83

The negative sign indicates that the mean height of Group A is lower than the mean height of Group B. The absolute value of Cohen's d is approximately 0.94, which means that the effect size is large according to Cohen's conventions. This suggests that there is a significant difference between the heights of the two groups.

---

Some Common Effect Size Formulas

[Click Here for Sample Questions]

Some common effect size formulas are as follows:

• Cohen's d: This formula is used to calculate the effect size for the difference between two means.

Cohen's d = (mean difference) / (pooled standard deviation)

where the pooled standard deviation is calculated as the square root of [(sample 1 variance + sample 2 variance) / 2]

• Pearson's r: This formula is used to calculate the effect size for the correlation between two variables.

Pearson's r = (covariance between the two variables) / (product of their standard deviations)

• Odds Ratio: This formula is used to calculate the effect size for the association between two categorical variables.

Odds ratio = (odds of the event in the exposed group) / (odds of the event in the unexposed group)

• Eta-squared: This formula is used to calculate the effect size for the difference between two groups in an ANOVA test.

Eta-squared = (sum of squares between groups) / (total sum of squares)

• R-squared: This formula is used to calculate the effect size for the proportion of variance in the dependent variable explained by the independent variable in a regression analysis.

R-squared = (explained variance) / (total variance)

Also Read:

---

Things to Remember

1. Effect size measures the magnitude of difference between two groups, rather than simply indicating whether a difference exists.
2. There are various effect size measures available, and the choice of measure depends on the research question and the type of data being analyzed.
3. Standardization of effect size measures can facilitate the comparison of effects across different studies and different types of data.
4. Cohen's d-effect size measure is widely used in statistics, particularly in studies involving the comparison of means between two groups.
5. The interpretation of effect size measures depends on the context and the field of study. It is important to understand what constitutes a small, medium, or large effect size in the specific research area.
6. Effect size measures can provide valuable information beyond traditional hypothesis testing and p-values, particularly when determining practical significance.