Introduction to Effect Size in Inferential Statistics

Effect size is a quantitative measure of the magnitude of a phenomenon. In inferential statistics, it helps researchers understand the strength of an effect observed in a study, beyond merely determining whether an effect exists. This measure is crucial in fields such as psychology, education, and health sciences, guiding decision-making and policy formulation. Below are three diverse examples that illustrate the concept of effect size in inferential statistics.

Example 1: Comparing Exam Scores Between Two Teaching Methods

Context

A school district wants to evaluate the effectiveness of two different teaching methods on student performance in mathematics. They conduct a study involving two classes, each taught using one of the methods.

Example

• Method 1 (Traditional Teaching): 20 students, Mean score = 75, Standard deviation = 10
• Method 2 (Interactive Teaching): 20 students, Mean score = 85, Standard deviation = 12

To calculate the effect size, we can use Cohen’s d:

1. Calculate the pooled standard deviation:

   • Pooled SD = √[((n1 - 1) * SD1² + (n2 - 1) * SD2²) / (n1 + n2 - 2)]
   • Pooled SD = √[((20 - 1) * 10² + (20 - 1) * 12²) / (20 + 20 - 2)]
   • Pooled SD = √[(19 * 100 + 19 * 144) / 38] = √[4616 / 38] ≈ 11.47

2. Calculate Cohen’s d:

   • Cohen’s d = (M1 - M2) / Pooled SD
   • Cohen’s d = (75 - 85) / 11.47 ≈ -0.87

Notes

An effect size of -0.87 indicates a large effect, suggesting that the interactive teaching method significantly improves student performance compared to traditional teaching.

Example 2: Evaluating a New Medication’s Impact on Blood Pressure

Context

A clinical trial tests a new medication designed to lower blood pressure. The trial includes two groups: one receiving the medication and the other receiving a placebo.

Example

• Medication Group: 30 participants, Mean systolic BP = 120 mmHg, Standard deviation = 8 mmHg
• Placebo Group: 30 participants, Mean systolic BP = 130 mmHg, Standard deviation = 10 mmHg

Applying Cohen’s d:

1. Calculate the pooled standard deviation:

• Pooled SD = √[((30 - 1) * 8² + (30 - 1) * 10²) / (30 + 30 - 2)]
• Pooled SD = √[(29 * 64 + 29 * 100) / 58] = √[(1856 + 2900) / 58] = √[4756 / 58] ≈ 10.05

2. Calculate Cohen’s d:

   • Cohen’s d = (M1 - M2) / Pooled SD
   • Cohen’s d = (120 - 130) / 10.05 ≈ -0.99

Notes

An effect size of -0.99 suggests a large and clinically significant effect of the medication on lowering systolic blood pressure compared to the placebo.

Example 3: Analyzing Customer Satisfaction Between Two Service Providers

Context

A market research firm assesses customer satisfaction levels between two competing service providers in the telecommunications industry.

Example

• Provider A: 50 customers, Mean satisfaction score = 4.2 (on a scale of 1 to 5), Standard deviation = 0.5
• Provider B: 50 customers, Mean satisfaction score = 3.8, Standard deviation = 0.6

Calculating Cohen’s d:

1. Calculate the pooled standard deviation:

• Pooled SD = √[((50 - 1) * 0.5² + (50 - 1) * 0.6²) / (50 + 50 - 2)]
• Pooled SD = √[(49 * 0.25 + 49 * 0.36) / 98] = √[(12.25 + 17.64) / 98] = √[29.89 / 98] ≈ 0.55

2. Calculate Cohen’s d:

   • Cohen’s d = (M1 - M2) / Pooled SD
   • Cohen’s d = (4.2 - 3.8) / 0.55 ≈ 0.73

Notes

An effect size of 0.73 indicates a medium to large effect, signifying that Provider A has a significantly higher customer satisfaction rating than Provider B, which could impact marketing strategies.

Conclusion

These examples demonstrate how effect size provides valuable insights in inferential statistics, allowing researchers and practitioners to gauge the strength of relationships and the impact of interventions in various fields.