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Sobel Test Calculator

Calculate the significance of mediation effects instantly. Get Z-statistics, p-values, and confidence intervals using Sobel, Aroian, and Goodman test variants.

Mediation Effect Calculator

a (IV → Mediator coefficient)

SEa (Standard error of a)

b (Mediator → DV coefficient)

SEb (Standard error of b)

Sobel Test Results

Z-Statistic

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(Sobel Test)

One-tailed p

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Two-tailed p

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Indirect Effect (a×b)

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95% Confidence Interval

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Interpretation

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All Test Variants Comparison

Test	Z-Value	p (two-tailed)	Significant?
Sobel	–	–	–
Aroian	–	–	–
Goodman	–	–	–

What Is the Sobel Test?

The Sobel test is a specialized statistical test used in mediation analysis to determine whether the indirect effect of an independent variable (IV) on a dependent variable (DV) through a mediator variable is statistically significant. It calculates a z-statistic by dividing the product of two path coefficients by their pooled standard error.

Named after American statistician Michael E. Sobel who developed it in 1982, this test helps researchers understand whether a mediator truly carries the influence of an IV to a DV, or if the apparent mediation effect occurred by chance.

What Is Mediation Analysis?

Mediation analysis is a statistical method used to examine how an independent variable affects a dependent variable through an intermediate variable called a mediator. Instead of a direct causal relationship (X → Y), mediation analysis tests whether X influences Y indirectly through M (X → M → Y).

Mediation Model Diagram

X
Independent Variable

→

Path a

M
Mediator

→

Path b

Y
Dependent Variable

The indirect effect = a × b | The Sobel test evaluates if a×b is significantly different from zero.

What Is the Sobel Test Formula?

The Sobel test calculates a z-statistic using the unstandardized regression coefficients and their standard errors:

z = (a × b) / √(b² × SEa² + a² × SEb²)

Variable Definitions

a = Unstandardized regression coefficient for IV → Mediator path
b = Unstandardized regression coefficient for Mediator → DV path (controlling for IV)
SEa = Standard error of coefficient a
SEb = Standard error of coefficient b
z = Test statistic (compared to normal distribution)

What Are the Three Sobel Test Versions?

There are three principal versions of the Sobel test, differing in how they calculate the standard error of the indirect effect:

Test Version	Formula (Denominator)	Description
Sobel (Original)	√(b²SEa² + a²SEb²)	Omits the third term. The original version by Sobel (1982).
Aroian	√(b²SEa² + a²SEb² + SEa²SEb²)	Adds the third term. Recommended by Baron & Kenny (1986).
Goodman	√(b²SEa² + a²SEb² − SEa²SEb²)	Subtracts the third term for unbiased variance estimate.

Recommendation: Use the Aroian version as it doesn’t assume the product of standard errors is negligibly small. The Sobel and Aroian tests converge closely with sample sizes greater than 50.

How to Interpret Sobel Test Results?

The Sobel test produces a z-statistic that follows a standard normal distribution. Use these guidelines to interpret your results:

Interpretation Guidelines

|z| ≥ 1.96: Significant at α = 0.05 (95% confidence)
|z| ≥ 2.58: Significant at α = 0.01 (99% confidence)
p < 0.05: The indirect effect is statistically significant
p ≥ 0.05: No evidence of significant mediation

What Does a Significant Result Mean?

A statistically significant Sobel test indicates that the mediator variable significantly carries the influence of the independent variable to the dependent variable. The mediation effect is unlikely to have occurred by chance alone.

What Does a Non-Significant Result Mean?

A non-significant result suggests there is insufficient evidence to conclude that mediation is occurring. However, this does not prove the absence of mediation—the effect may be too small to detect with your sample size.

How to Calculate Sobel Test Step by Step?

Run Regression 1: Regress the mediator (M) on the independent variable (X) to obtain coefficient a and its standard error SEa.
Run Regression 2: Regress the dependent variable (Y) on both X and M to obtain coefficient b and its standard error SEb.
Calculate the indirect effect: Multiply a × b.
Calculate the standard error: SE = √(b²×SEa² + a²×SEb²)
Compute the z-statistic: z = (a × b) / SE
Find the p-value: Compare z to the standard normal distribution.

Example Calculation

Given: a = 0.45, SEa = 0.10, b = 0.60, SEb = 0.12

Indirect effect = 0.45 × 0.60 = 0.27
SE = √(0.36×0.01 + 0.2025×0.0144) = √(0.0036 + 0.00292) = √0.00652 = 0.0807
z = 0.27 / 0.0807 = 3.345
p (two-tailed) = 0.0008 → Significant mediation!

When Should You Use the Sobel Test?

The Sobel test is appropriate when:

You have already obtained regression coefficients and standard errors from your analysis
Your sample size is reasonably large (n > 100, ideally n > 200)
You want a quick assessment of mediation significance
You don’t have access to raw data for bootstrapping

When Should You NOT Use the Sobel Test?

Small samples (n < 100): The test assumes normal sampling distribution, which may not hold
When raw data is available: Bootstrapping methods are more robust and don’t assume normality
Multilevel models: Consult Krull & MacKinnon (1999) for appropriate methods

What Are the Limitations of the Sobel Test?

While widely used, the Sobel test has several important limitations researchers should consider:

Normality assumption: The test assumes the sampling distribution of the indirect effect (a×b) is normally distributed, which is often violated.
Large sample requirement: Works well only with large samples (n > 200). With smaller samples, the test is underpowered.
Conservative: Monte Carlo studies show the Sobel test is generally conservative, meaning it may fail to detect true mediation effects.
Single mediator: The basic Sobel test is designed for simple mediation models with one mediator.

Better Alternative: Bootstrapping

For most applications, bootstrapping methods (available in PROCESS macro for SPSS/SAS, or via R packages) provide more accurate confidence intervals and better statistical power without normality assumptions.

Frequently Asked Questions

Can I use standardized coefficients in the Sobel test?

The Sobel test is designed for unstandardized (raw) regression coefficients. Using standardized coefficients may produce inaccurate results because the standard error estimates are calculated differently for standardized values.

What sample size do I need for the Sobel test?

The Sobel test requires large samples to perform accurately. A minimum of 100 observations is recommended, with 200+ being ideal. For smaller samples, use bootstrapping methods instead.

Is a negative z-value problematic?

No. A negative z-value simply indicates the direction of the indirect effect is negative. The absolute value of z is used to determine significance. Both positive and negative z-values beyond ±1.96 are statistically significant at α = 0.05.

Which version should I report: Sobel, Aroian, or Goodman?

The Aroian version is recommended as it’s the most accurate and was popularized by Baron & Kenny (1986). The original Sobel and Aroian versions converge closely with samples larger than 50. Avoid the Goodman version as it can produce negative variance estimates.

How do I get the coefficients (a, b) and standard errors?

Run two regression analyses: (1) Regress the mediator on the IV to get ‘a’ and SEa, (2) Regress the DV on both the IV and mediator to get ‘b’ and SEb. These values are available in standard regression output from SPSS, R, Stata, or other statistical software.

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