Two-Way ANOVA Calculator (No Replication): Test Two Factors at Once with F and p-Values

Two factors tested together in one table.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the number of rows and columns.

Type the table of measurements.

The calculator splits the variation and gives F and p for the rows and the columns.

Formulas used

  • SS_T = SS_A + SS_B + SS_E
  • F_A = \frac{SS_A/(r - 1)}{SS_E/((r - 1)(c - 1))}
  • F_B = \frac{SS_B/(c - 1)}{SS_E/((r - 1)(c - 1))}

Worked example

With the default numbers in the calculator above:

  1. Grand mean: 30
  2. Sum of squares for the rows (factor A): 152 (df 2)
  3. Sum of squares for the columns (factor B): 20.6667 (df 2)
  4. Sum of squares for the residual error: 13.3333 (df 4)
  5. Total sum of squares: 186
  6. F for the rows: 22.8
  7. p-value for the rows: 0.0065036
  8. F for the columns: 3.1
  9. p-value for the columns: 0.15379

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

When do I use two-way ANOVA?

When you have two factors, such as the machine and the operator, and want to know whether each one affects the result.

What does no replication mean?

There is only one measurement in each cell. The calculation assumes that the two factors do not interact.

What do I do after a significant result?

Compare pairs of levels with an adjusted test to find which ones differ.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any statistics text (two-way analysis of variance, the randomised block design).

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