IntroStatsCalc

One-Way ANOVA

Compare three or more group means with an F-test and ANOVA table.

Inputs

Type or paste each group's values on its own line, separated by spaces or commas.

Significance level α

Results update as you type.

Results

Reject H₀: F = 9.9199, p-value = 0.0042

The p-value (0.0042) is less than or equal to α = 0.05, so the data give enough evidence to conclude that at least one group mean is different from the others.

F-distribution with 2 and 10 degrees of freedom curve. The shaded p-value region is the area to the right of F = 9.92, with area 0.0042. Dashed line marks the critical value 4.103.0246810F = 9.92critical 4.103

Results

F9.9199
df (between, within)2, 10
p-value0.0042
Critical value4.103
Decision at α = 0.05Reject H₀
ANOVA table
SourceSSdfMSFp-value
Between groups (factor)492.62312246.31159.91990.0042
Within groups (error)248.31024.83
Total740.923112
Group summaries
GroupnMeanSD
Group 1483.756.7515
Group 2572.23.4928
Group 3485.754.5735

On the TI-84

Put each group in its own list (L1, L2, …). STAT → TESTS → H:ANOVA( then type the lists separated by commas.

ANOVA(L1, L2, L3)

Show steps
  1. Hypotheses. H₀: μ1 = μ2 = μ3   Ha: at least one mean is different
  2. Grand mean. All 13 values together average 79.9231.
  3. Between groups. SSbetween = Σ ni(x̄i − grand mean)² = 492.6231, df = k − 1 = 2, MS = 246.3115.
  4. Within groups. SSwithin = Σ (ni − 1)si² = 248.3, df = N − k = 10, MS = 24.83.
  5. F. F = MSbetween ÷ MSwithin = 246.3115 ÷ 24.83 = 9.9199
  6. p-value. Right-tailed test: area to the right of the test statistic: p-value = 0.0042. Since 0.0042 ≤ 0.05, reject H₀.
  7. Critical-value method. At α = 0.05, the rejection region is F ≥ 4.103. The test statistic F = 9.9199 is in the rejection region, so we reject H₀. (Same decision as the p-value method.)
  8. Conclusion. The p-value (0.0042) is less than or equal to α = 0.05, so the data give enough evidence to conclude that at least one group mean is different from the others.

About this calculator

When to use ANOVA

One-way ANOVA compares the means of three or more independent groups in a single test. Running many two-sample t-tests instead would raise the chance of a false alarm.

The idea

F compares the variation between the group means to the variation within the groups. If the group means are far apart compared with how spread out each group is, F is large and the p-value is small. ANOVA tells you that at least one mean differs, not which one.

Conditions: independent random samples, each population approximately normal, and similar standard deviations (a common check: the largest SD is no more than twice the smallest).

More chi-square & anova calculators: χ² Goodness of Fit · χ² Independence