Soft Pretzels Lab
What moves together in 173 soft pretzel recipes
When a recipe uses more of one ingredient, does it tend to use more or less of another? For every pair of the 5 ingredient ratios (grams per 100 g of flour), this page ranks all 173 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).
10 pairs in all: 1 rises together at rho 0.20 or more, 2 move in opposite directions at −0.20 or less, and 4 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 173 recipes (p of 0.05 or more).
All 10 pairs as a table
| Pair | Rho | Strength | p |
|---|---|---|---|
| Water and yeast | 0.35 | weak | under 0.001 |
| Water and butter | −0.23 | weak | 0.003 |
| Water and salt | −0.21 | weak | 0.005 |
| Salt and yeast | −0.19 | very weak | 0.013 |
| Butter and salt | 0.19 | very weak | 0.014 |
| Butter and yeast | −0.10 | very weak | 0.19 |
| Butter and sugar | 0.10 | negligible | 0.21 |
| Water and sugar | 0.08 | negligible | 0.29 |
| Sugar and yeast | 0.07 | negligible | 0.39 |
| Sugar and salt | −0.06 | negligible | 0.41 |
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 water tend to use more yeast
rho 0.35 across 173 recipes · weak · p under 0.001
The third of recipes with the most water use a median of 1.4% yeast; the third with the least water, 1.2%.
Recipes with more water tend to use less butter
rho −0.23 across 173 recipes · weak · p 0.003
The third of recipes with the most water use a median of 5.4% butter; the third with the least water, 8%.
Recipes with more water tend to use less salt
rho −0.21 across 173 recipes · weak · p 0.005
The third of recipes with the most water use a median of 1.6% salt; the third with the least water, 1.9%.
What moves on its own
- Sugar. Nearly independent of the rest: its strongest correlation with any other ingredient is 0.10, with butter. A recipe's amount of sugar says almost nothing about its other amounts.
The pairs closest to unrelated: sugar and salt (−0.06), sugar and yeast (0.07), water and sugar (0.08) and butter and sugar (0.10).
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 10 pairs tested, about 0.5 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.