Blueberry Muffins Lab
What moves together in 272 blueberry muffin recipes
When a recipe uses more of one ingredient, does it tend to use more or less of another? For every pair of the 7 ingredient ratios (grams per 100 g of flour), this page ranks all 272 recipes on each and measures how closely the two rankings agree: Spearman's rho, from −1 (the more of one, always the less of the other) through 0 (unrelated) to 1 (always rise together).
21 pairs in all: 6 rise together at rho 0.20 or more, none move in opposite directions at −0.20 or less, and 9 are close to unrelated (between −0.10 and 0.10). This describes recipes as they are published; it does not say what changing one amount would do to a bake.
Every pair at once
Read a cell by its row and column. Caramel cells rise together, slate cells move in opposite directions, and the deeper the color the stronger the relationship. Pale cells with grey numbers are not distinguishable from no relationship with 272 recipes (p of 0.05 or more).
All 21 pairs as a table
| Pair | Rho | Strength | p |
|---|---|---|---|
| Sugar and fat | 0.38 | weak | under 0.001 |
| Blueberries and sugar | 0.37 | weak | under 0.001 |
| Sugar and salt | 0.37 | weak | under 0.001 |
| Fat and egg | 0.33 | weak | under 0.001 |
| Blueberries and salt | 0.26 | weak | under 0.001 |
| Blueberries and egg | 0.24 | weak | under 0.001 |
| Dairy and leavening | 0.17 | very weak | 0.004 |
| Blueberries and fat | 0.16 | very weak | 0.009 |
| Fat and salt | 0.14 | very weak | 0.024 |
| Egg and leavening | 0.12 | very weak | 0.043 |
| Blueberries and leavening | 0.12 | very weak | 0.047 |
| Fat and dairy | −0.10 | very weak | 0.098 |
| Sugar and egg | 0.10 | negligible | 0.12 |
| Sugar and dairy | −0.08 | negligible | 0.19 |
| Dairy and salt | −0.07 | negligible | 0.26 |
| Egg and dairy | −0.06 | negligible | 0.30 |
| Egg and salt | 0.06 | negligible | 0.31 |
| Leavening and salt | −0.03 | negligible | 0.62 |
| Fat and leavening | 0.03 | negligible | 0.65 |
| Blueberries and dairy | −0.01 | negligible | 0.80 |
| Sugar and leavening | −0.01 | negligible | 0.80 |
The three clearest relationships
The strongest pairs that reach rho 0.20 either way with a p-value under 0.01. Each dot is a recipe; the line joins the median of the second ingredient within each fifth of recipes ranked by the first.
Recipes with more sugar tend to use more fat
rho 0.38 across 272 recipes · weak · p under 0.001
The third of recipes with the most sugar use a median of 45% fat; the third with the least sugar, 32%.
Recipes with more blueberries tend to use more sugar
rho 0.37 across 272 recipes · weak · p under 0.001
The third of recipes with the most blueberries use a median of 76% sugar; the third with the fewest blueberries, 57%.
Recipes with more sugar tend to use more salt
rho 0.37 across 272 recipes · weak · p under 0.001
The third of recipes with the most sugar use a median of 1.1% salt; the third with the least sugar, 0.3%.
What moves on its own
Every ingredient moves with at least one other: none has its strongest correlation under 0.15.
The pairs closest to unrelated: blueberries and dairy (−0.01), sugar and leavening (−0.01), fat and leavening (0.03) and leavening and salt (−0.03).
How the numbers are made
Spearman's rho is the ordinary correlation of the two rankings, so it measures whether one amount tends to rise with the other, in a straight line or not, and a single extreme recipe moves it little. A recipe with none of an ingredient ranks lowest on it. An ingredient fewer than 12 recipes use is left out of the matrix.
The p-value is the chance of a rho at least this far from zero if the two amounts were unrelated. With 21 pairs tested, about 1.1 would pass 0.05 by chance alone even if no ingredient were related to another, which is why the relationships explained above must pass 0.01 and reach 0.20.
Every number is divided by the same flour weight, so a recipe light on flour reads high on every ingredient at once. That nudges every correlation toward positive: part of a positive rho can be flour, not the two ingredients. A negative one holds despite it.
A correlation describes the recipes as published. “Recipes with more blueberries tend to use…” is a statement about what bakers write down together, not a rule that adding one forces the other. Plot any two ingredients to see the recipes behind a number.