Whole Wheat Bread Lab116 recipes

Whole Wheat Bread Lab

What moves together in 116 whole wheat bread recipes

When a recipe uses more of one ingredient, does it tend to use more or less of another? For every pair of the 6 ingredient ratios (grams per 100 g of flour), this page ranks all 116 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).

15 pairs in all: 4 rise together at rho 0.20 or more, 1 moves in opposite directions at −0.20 or less, and 5 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 116 recipes (p of 0.05 or more).

Whole wheat share and liquid: rho −0.14, very weak, not distinguishable from zero−0.14Whole wheat share and sweetener: rho −0.03, negligible, not distinguishable from zero−0.03Liquid and sweetener: rho −0.04, negligible, not distinguishable from zero−0.04Whole wheat share and fat: rho 0.15, very weak, not distinguishable from zero0.15Liquid and fat: rho −0.14, very weak, not distinguishable from zero−0.14Sweetener and fat: rho 0.56, moderate0.56Whole wheat share and salt: rho −0.04, negligible, not distinguishable from zero−0.04Liquid and salt: rho 0.24, weak0.24Sweetener and salt: rho −0.11, very weak, not distinguishable from zero−0.11Fat and salt: rho −0.14, very weak, not distinguishable from zero−0.14Whole wheat share and yeast: rho 0.06, negligible, not distinguishable from zero0.06Liquid and yeast: rho −0.10, negligible, not distinguishable from zero−0.10Sweetener and yeast: rho 0.49, moderate0.49Fat and yeast: rho 0.36, weak0.36Salt and yeast: rho −0.27, weak−0.27LiquidSweetenerFatSaltYeastWhole wheat shareLiquidSweetenerFatSalt
All 15 pairs as a table
Spearman correlation for every pair of ratios, strongest first
PairRhoStrengthp
Sweetener and fat0.56moderateunder 0.001
Sweetener and yeast0.49moderateunder 0.001
Fat and yeast0.36weakunder 0.001
Salt and yeast−0.27weak0.003
Liquid and salt0.24weak0.010
Whole wheat share and fat0.15very weak0.12
Liquid and fat−0.14very weak0.13
Whole wheat share and liquid−0.14very weak0.14
Fat and salt−0.14very weak0.14
Sweetener and salt−0.11very weak0.23
Liquid and yeast−0.10negligible0.31
Whole wheat share and yeast0.06negligible0.53
Whole wheat share and salt−0.04negligible0.64
Liquid and sweetener−0.04negligible0.69
Whole wheat share and sweetener−0.03negligible0.76

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.

  1. Recipes with more sweetener tend to use more fat

    rho 0.56 across 116 recipes · moderate · p under 0.001

    The third of recipes with the most sweetener use a median of 7.6% fat; the third with the least sweetener, 0%.

    0%10%20%30%0%5%10%15%20%Sweetener, % of flour →↑ Fat, % of flour

  2. Recipes with more sweetener tend to use more yeast

    rho 0.49 across 116 recipes · moderate · p under 0.001

    The third of recipes with the most sweetener use a median of 1.6% yeast; the third with the least sweetener, 0.8%.

    0%10%20%30%0%1%2%3%Sweetener, % of flour →↑ Yeast, % of flour

  3. Recipes with more salt tend to use less yeast

    rho −0.27 across 116 recipes · weak · p 0.003

    The third of recipes with the most salt use a median of 1.1% yeast; the third with the least salt, 1.5%.

    0%1%2%3%4%0%1%2%3%Salt, % of flour →↑ Yeast, % of flour

Also strong, and left out above because both of its ingredients already appear: fat and yeast (rho 0.36).

What moves on its own

  • Whole wheat share. Nearly independent of the rest: its strongest correlation with any other ingredient is 0.15, with fat. A recipe's amount of whole wheat share says almost nothing about its other amounts.

The pairs closest to unrelated: whole wheat share and sweetener (−0.03), liquid and sweetener (−0.04), whole wheat share and salt (−0.04) and whole wheat share and yeast (0.06).

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 15 pairs tested, about 0.8 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 whole wheat share 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.