Every Pokémon fan “knows” Legendary Pokémon are stronger, and that Dragon and Fighting types hit hardest.

But is that real, or just a few famous examples?

Hypothesis testing lets us answer with data instead of anecdotes:

1. Do Legendary/Mythical Pokémon have a higher average base stat total than regular Pokémon? (two-sample t-test)
2. Does a Pokémon’s primary type affect its average Attack? (one-way ANOVA)
3. Do types differ in how they attack (physical vs. special) or in how much? (paired t-tests, effect sizes)

The Legendary box sits far to the right, but a few Legendaries (like Cosmog, BST 200) are weaker than most regular Pokémon.

Let \(\mu_L\) be the mean BST of Legendary/Mythical Pokémon and \(\mu_R\) that of regular Pokémon.

\[H_0: \mu_L = \mu_R \qquad \text{vs.} \qquad H_a: \mu_L \neq \mu_R\]

The groups have different sizes and spreads, so we use Welch’s t-test, which does not assume equal variances:

\[t = \frac{\bar{x}_L - \bar{x}_R}{\sqrt{\dfrac{s_L^2}{n_L} + \dfrac{s_R^2}{n_R}}} \qquad \nu \approx \frac{\left(\dfrac{s_L^2}{n_L} + \dfrac{s_R^2}{n_R}\right)^2} {\dfrac{(s_L^2/n_L)^2}{n_L - 1} + \dfrac{(s_R^2/n_R)^2}{n_R - 1}}\]

We reject \(H_0\) if the p-value \(= P(|T_\nu| \geq |t|)\) is below \(\alpha = 0.05\).

With \(k = 18\) types, testing every pair with t-tests would inflate false positives. ANOVA tests all the means at once:

\[H_0: \mu_1 = \mu_2 = \cdots = \mu_{18} \qquad \text{vs.} \qquad H_a: \text{at least one } \mu_j \text{ differs}\]

\[F = \frac{\text{MS}_{\text{between}}}{\text{MS}_{\text{within}}} = \frac{\sum_{j=1}^{k} n_j(\bar{y}_j - \bar{y})^2 \,/\, (k-1)} {\sum_{j=1}^{k}\sum_{i=1}^{n_j} (y_{ij} - \bar{y}_j)^2 \,/\, (N-k)}\]

Effect size: the share of variation in Attack explained by type,

\[\eta^2 = \frac{SS_{\text{between}}}{SS_{\text{total}}}\]

##               Df Sum Sq Mean Sq F value Pr(>F)    
## type_1        17 102239    6014   7.513 <2e-16 ***
## Residuals   1007 806048     800                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• \(F(17, 1007) = 7.51\), p-value \(< 0.0001\), so we reject \(H_0\): type does matter.
• But \(\eta^2 = 0.113\): type explains only about 11% of the variation in Attack.
• Fighting and Dragon do lead, but the spread within each type is much larger than the gaps between types.

Each Pokémon has both stats, so the two values are paired. For Pokémon \(i\) of a given type, let

\[D_i = \text{Atk}_i - \text{SpA}_i, \qquad H_0: \mu_D = 0 \quad \text{vs.} \quad H_a: \mu_D \neq 0\]

\[t = \frac{\bar{D}}{s_D / \sqrt{n}}, \qquad \text{df} = n - 1\]

We run this test once for each of the 18 types. To keep the overall false-positive rate at 5%, we use the Bonferroni correction:

\[\alpha^{*} = \frac{0.05}{18} \approx 0.0028 \qquad \Longleftrightarrow \qquad p_{\text{adj}} = \min(18p,\ 1) < 0.05\]

Adding the stats together, type explains only 7% of total offense: the types converge. Taking the gap between them, type explains 23%: the types diverge. Types differ in how they attack far more than in how much. (Atk and Sp. Atk correlate only weakly, \(r = 0.28\).)

• Legendaries really are stronger. The difference is huge and statistically clear (about 180 BST points).
• Type matters for Attack, but only a little. The effect is real but small (\(\eta^2 \approx 0.11\)).
• Types differ in style, not power. Fighting and Ground are physical and Psychic is special, but total offense is nearly the same across types.
• Statistical significance ≠ practical significance. With 1,000+ observations, even small effects get tiny p-values. Always check the effect size.
• Caveats:
  • Base stats are designer choices, not random samples, so “population” here means the full roster.
  • Group variances differ, which is why we used Welch’s test. We ran 18 paired tests, so we used a Bonferroni correction.

• Pokémon data: darkmatternet, Ultimate Pokémon Dataset 2025, Kaggle. https://www.kaggle.com/datasets/darkmatternet/ultimate-pokmon-dataset-2025 (file pokemon_complete_2025.csv; underlying data from PokéAPI, https://pokeapi.co)