Bagels Lab
What moves together in 150 bagel recipes
When a recipe uses more of one ingredient, does it tend to use more or less of another? For every pair of the 4 ingredient ratios (grams per 100 g of flour), this page ranks all 150 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).
6 pairs in all: 1 rises together at rho 0.20 or more, none move in opposite directions at −0.20 or less, and 3 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 150 recipes (p of 0.05 or more).
All 6 pairs as a table
| Pair | Rho | Strength | p |
|---|---|---|---|
| Water and salt | 0.21 | weak | 0.011 |
| Water and yeast | 0.13 | very weak | 0.10 |
| Water and malt or sugar | −0.10 | very weak | 0.21 |
| Malt or sugar and yeast | 0.09 | negligible | 0.30 |
| Malt or sugar and salt | −0.07 | negligible | 0.37 |
| Salt and yeast | 0.01 | negligible | 0.89 |
No strong relationships
No pair reaches rho 0.20 with a p-value under 0.01, the bar for calling a relationship here. The nearest is water and salt (rho 0.21, p 0.011), short of it with 150 recipes.
What moves on its own
- Malt or sugar. Nearly independent of the rest: its strongest correlation with any other ingredient is −0.10, with water. A recipe's amount of malt or sugar says almost nothing about its other amounts.
- Yeast. Nearly independent of the rest: its strongest correlation with any other ingredient is 0.13, with water. A recipe's amount of yeast says almost nothing about its other amounts.
The pairs closest to unrelated: salt and yeast (0.01), malt or sugar and salt (−0.07) and malt or sugar and yeast (0.09).
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 6 pairs tested, about 0.3 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 water 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.