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Silph Study: #040

Published:
02.04.2021

A Guide to Candy XL: Part 1 - Egg Hatching

In November 2020, the long-awaited increase to the trainer level cap was finally unveiled. With it, travelers level 40 and above have access to a new resource for powering up Pokémon: Candy XL. There are currently three practical methods for obtaining Candy XL: hatching eggs, catching Pokémon, and transferring Pokémon. In this three-part series, we’ll explore the workings of Candy XL and propose mathematical models to explain them, as well as suggest a few strategies for making your Pokémon the very best. Without further ado, let’s dive into our first analysis on egg hatching!

Key Findings

  1. Not all egg types give the same amount of Candy XL, and the amount of Candy XL generally increases with egg distance. The drop rate falls into three distinct tiers: 2km, 5km/7km, and 10km/12km.
  2. The distribution of Candy XL from each egg type is well-described by a binomial model (flipping a weighted coin).
  3. There is no correlation between the amount of Candy XL and regular candy or stardust received in a hatch. Furthermore, higher IV Pokémon do not drop more Candy XL, and Adventure Sync eggs are not significantly different from their PokéStop counterparts of the same distance.

Average Candy XL Per Hatch

As many travelers are already aware, the Silph Research Group published a brief article shortly after the Candy XL feature was released globally with a surprising discovery: 10km and 12km eggs dropped less Candy XL than lower distance eggs. A few hours after the article was published, Niantic acknowledged the issue and quickly patched it. On behalf of the community, we’d like to thank those researchers again for their hard work in uncovering the bug.

Since then, researchers have continued hatching eggs at an astounding pace, cracking open 9,562 eggs in a relatively short time. From the data collected by these researchers, clean tiers in the average amount of Candy XL dropped per egg emerged. We found that, on average, 2km eggs give 1.6 Candy XL per hatch, 5km and 7km eggs give 3.2, and 10km and 12km eggs give 4.8. These numbers are shown below in the following table along with their 95% confidence intervals:

Distance Hatches Avg. Candy XL/hatch
2km Egg 3,233 1.60 ± 0.04
5km Egg 2,870 3.24 ± 0.06
7km Egg 1,435 3.18 ± 0.08
10km Egg 589 4.9 ± 0.2
12km Egg 1,435 4.8 ± 0.1

It seems likely that the values are multiples of one another, and 1.6, 3.2, and 4.8 are used for 2km eggs, 5km/7km eggs, and 10km/12km eggs, respectively.

Binomial Model

The analysis above is limited to the average Candy XL per hatch, but how does the amount of Candy XL vary around the average for each egg distance?

Several mathematical models were considered as potentially describing the observed data, including a Poisson distribution, a (rounded and truncated) normal distribution, and a binomial distribution. Both the Poisson and normal distribution were discarded, as no parameter combination could accurately describe the observed data for at least two egg types using these models.¹

A binomial model is also known as a “coin flipping” model. Each sample is taken by the equivalent of flipping a weighted coin several times and counting the number of successes. In this case, getting a Candy XL is defined as a success. The binomial distribution has two parameters to fit: the number of coin flips, and the probability of a successful coin flip.

The average Candy XL from a hatch is equal to the number of coin flips multiplied by the probability of a successful flip. To fit the model to our data, we constrained the probability of a successful flip to the average Candy XL divided by the number of coin flips for each egg distance. We then sequentially varied the number of coin flips and compared the theoretical distribution and observed distribution of Candy XL using a chi-squared goodness-of-fit test. The following numbers of coin flips were sufficiently good fits to the observed data and could not be rejected at a significance level of 0.01:

  • 2km eggs: 6 – 12 coin flips (probabilities of success range from 0.27 to 0.13)
  • 5km eggs: 12 – 30 coin flips (probabilities of success range from 0.27 to 0.11)
  • 7km eggs: 10 – 58 coin flips (probabilities of success range from 0.32 to 0.055)
  • 10km eggs: 14 – any coin flips (probabilities of success range from 0.35 to approaching 0)
  • 12km eggs: 13 – 53 coin flips (probabilities of success range from 0.37 to 0.09)

Little can be concluded from these fittings about the precise parameters being used. A wide range of parameter combinations within a binomial model were sufficient to describe the observed data.

However, one parameter combination has a particularly simple scheme:

Distance Coin Flips Probability
2km Egg 8 0.2 or 1/5
5km Egg7km Egg 16 0.2 or 1/5
10km Egg12km Egg 24 0.2 or 1/5

The observed data compared to this scheme is shown in the following plots:

Figure showing the distribution of Candy XL From 2KM Eggs

Figure showing the distribution of Candy XL From 5 and 7KM Eggs

Figure showing the distribution of Candy XL From 10 and 12KM Eggs

This scheme is alluring because it requires varying a single parameter between egg distances instead of two. There are several other models that also fit this criteria, including 6, 12, and 18 flips (probability of success of 0.27); 10, 20, and 30 flips (probability of success of 0.16); and 12, 24, and 36 flips (probability of success of 0.13).

Notably, this model has the (possibly intended) quirk that the average regular candy per hatch is approximately the same as the number of total coin flips for each egg distance. Might this trend continue for other methods of obtaining Candy XL?

Factors that do not influence Candy XL

The following parameters were examined for an effect on Candy XL, and no significant effects were found:

  • 5km and 10km eggs obtained from Adventure Sync rewards awarded the same amount of Candy XL as their PokéStop counterparts (Student’s t-test, both p-values>0.05).
  • The comparison at the end of the previous section between the average regular candy and average Candy XL does not apply to individual hatches; there was no correlation between Candy XL and regular candy from egg to egg. Stardust (which is directly related to regular candy) was also uncorrelated with Candy XL (tested using Pearson’s product moment correlation coefficient, all p-values>0.05).
  • While researchers did not record exact IVs of each hatch, we used the CP of each hatch relative to its possible CP range as an estimate for whether each Pokémon had high or low IVs. This “relative CP” was unrelated to the amount of Candy XL received (tested using Pearson’s product moment correlation coefficient, all p-values>0.05).

Parting Words

Whew, that was a lot of math! So what does this mean for optimal egg hatching strategies? 2km eggs (and PokéStop eggs in general) reward the most Candy XL per kilometer walked, but the fact that Candy XL is species-specific will always mean that the species in each egg pool will be more important than the exact number of Candy XL received.

One potential question about the proposed binomial model is that if there were truly 24 coins being flipped every time a 10km or 12km egg hatches, then why have we never seen a hatch with 24 Candy XL? Simply put, this is very unlikely to occur. Flipping 24 heads in a row on a fair coin (50% chance of success) is already extremely unlikely (~1 in 17 million trials), but the probability of that outcome with our weighted coin with a 20% chance of success is astronomically low (~1 in 60 quadrillion). In other words, 1 out of every 60,000,000,000,000,000 hatched 10/12km eggs would be expected to reward 24 Candy XL. If you were only hatching 10km eggs with an infinite incubator, you’d have to walk the distance from Earth to Pluto and back approximately five million times! Perhaps one day a traveler will experience this once in a lifetime event.

We hope that you’ve enjoyed this deep dive into the mechanics of Candy XL from egg hatching. Look out for the second article in the series on Catching Pokémon which will be coming soon!


Acknowledgements

Article author: Scientist Titleist
Analysis: Scientists CaroKann and Titleist
Project Leaders: Scientists Cham1nade and DarkMighty
Graphics and figure: Scientists WoodWoseWulf and Titleist
Editing: Scientists Cham1nade and skyeofthetyger
Data collection: 120 tenacious researchers collected data for this article. These 11 have the blisters to show for it:

  • AlertedFancy
  • archer
  • butters
  • CmonINeedAName
  • CrispenWah
  • nWo wolf (nordypr1)
  • Pancake
  • Paradox.of.fate
  • racheliptak
  • Rockaholic2
  • Zambia

Footnotes

¹ The Poisson distribution is a single parameter model, with λ equal to both the mean and variance. We first fit λ to each egg distance using the maximum likelihood estimate (the sum of all Candy XL divided by the total number of hatches for that egg distance). The expected counts (0, 1, 2, etc.) of Candy XL given the sample size and Poisson probability mass function were then compared to the observed counts using a chi-squared goodness of fit test. The distributions of 2km, 5km, 7km, and 12km eggs were rejected as their p-values fell below the chosen significance level of 0.01. The distribution of 10km eggs was not rejected (p-value = 0.40), likely owing to the smaller sample size.

Similarly, for the truncated, rounded normal distribution, a mean and standard deviation were fit to each egg distance before comparing the observed and expected counts (0, 1, 2, etc.) of Candy XL with a chi-squared goodness of fit test. The distributions of 2km and 5km eggs were rejected as their p-values fell below the chosen significance level of 0.01. The distributions of 7km, 10km, and 12km eggs were not significantly different from the normal distribution, with p-values of 0.012, 0.11, and 0.81, respectively. We expect that with more data, the observed distributions would eventually diverge from the normal distribution for the remaining egg distances.