Our recent investigation into the preparation time of Dalgona whipped coffee left us with a pile of timing data. Some of that data happened to be compatible with *2ᵏ* factorial experimental design. So, why not complete the analysis and see what we find?

**Question:**

How much does bowl size or beater type affect the time to make Dalgona coffee? Or more precisely, what percent of variation in the preparation time of Dalgona coffee is explained by bowl size, beater type, and interactions between the two.

**Equipment & Materials:
**

- large bowl
- small bowl
- electric mixer (with bare metal beaters)
- electric mixer (with silicone coated beaters)
- Ingredients for Dalgona coffee
- instant coffee
- sugar
- water

**Background:**

To determine the amount of variability attributable to each factor (bowl size, beater type), a *2ᵏ* factorial experimental design can be used [1]. In the *2ᵏ* method, two levels for each of *k* factors are tested. In this case there are two factors denoted A (bowl size) and B (beater type), thus *k*=2. For bowl size there are two experimental levels denoted -1 (small) and 1 (large). For beater type, the two levels are -1 (bare metal) and 1 (silicone coated).

**Procedure:**

- Prepare Dalgona coffee with all four combinations of bowl and mixer, recording time for each combination.

**Data:**

The table below shows the amount of time it took to prepare the Dalgona coffee for each bowl and mixer combination. All times are in seconds.

Beater Type (B) | ||

Bowl Size (A) | Bare Metal | Silicone Coated |

Small | 63 | 90 |

Large | 341 | 367 |

**Analysis:**

Data from a *2ᵏ* experiment is processed using a sign table, as shown below. The *I* column is mostly useless, other than to compute the sum and arithmetic mean of the measurements. The total row is computed by taking the sum of response values (y) multiplied by the sign for each factors. For example, the total row for factor A is: (-1)*63 + 1*41 + (-1)*90 + 1*367 = 555.

I |
A |
B |
AB |
y |

1 | -1 | -1 | 1 | 63 |

1 | 1 | -1 | -1 | 41 |

1 | -1 | 1 | -1 | 90 |

1 | 1 | 1 | 1 | 367 |

861 | 555 | 53 | -1 | Total |

215.25 | 138.75 | 13.25 | -0.25 | Total/2ᵏ |

To get the percent of variation explain by each factor or combination of factors, the *sum of squares total* (SST) needs to be computed. The quantity SST is computed by taking 2*ᵏ *(4 in this case) times the sum of each Total/2*ᵏ* squared, excluding I. Thus, SST = 4*(138.75²+13.25²+(-0.25)²) = 77708.75.

The portion of variation for each factor is computed by dividing the quantity 2*ᵏ* * square of the Total/2*ᵏ* by SST for each factor (e.g., 2*ᵏ* *138.75²/SST = 4*138.75*138.75/77708.75 ≈ 0.991). Those portions as percentages are 99.1%, 0.91%, and <0.01% for A, B and AB (the interaction between A and B) respectively. Thus bowl size accounted for over 99% of the variation in preparation time. Unfortunately, we weren’t willing to repeat each trial sufficient times for a 2*ᵏr* experimental design to determine if the variation due to beater type or its interactions with bowl size were statistically significant.

**Conclusion:**

Using a 2*ᵏ* experimental design, we can conclude that the size of the bowl used when preparing Dalgona coffee has the largest impact on the amount of time required. For the bowls used, the size accounted for over 99% of variation in the preparation time.

**References:**

[1] **R. Jain, “The Art of Computer Systems Performance Analysis: Techniques for Experimental Design, Measurement, Simulation, and Modeling,”** Wiley- Interscience, New York, NY, April 1991, ISBN:0471503361.

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