When is Easter this year? I ask this question every year! This year, Easter occurs on April 16, 2017. Next year, Easter falls on April 1, 2018. I have a hard time remembering when it occurs in any given year. I think that March Easters are both early and unusual. Is that true?

Being a statistician, my first thought is to study the distribution of Easter dates. By analyzing the distribution, we can determine which dates are rare and which are common. How unusual are Easter dates in March? Are there patterns in the dates?

Let’s first look at the official determination for when Easter happens. The holiday falls on the Sunday after the Paschal Full Moon date of that year. What? To clarify, the Paschal Full Moon is the Ecclesiastical Full Moon that falls after March 20. Huh? This definition isn’t helping me! Let’s graph the data. That’ll clarify things!

In this post, I study the dates of when the Western Easter occurs to answer all of these questions. I’ve obtained all the Easter dates that occur within the Gregorian calendar, which runs from 1583 to 4099. You can download the CSV data file for Easter_dates.

## Graphing Counts of Easter Dates: 1583 to 4099

Graphing any dataset is a great way to get a quick sense of the data. Below, I graph the distribution of all Easter dates from 1583 to 4099. At a glance, this chart tells us which dates are common and rare. To fit on the graph, I coded the dates. For example, M27 is March 27 and A16 is April 16.

The X-axis shows us that Easter dates can range from March 22 to April 25. There are 35 possible dates. The distribution has an extended center where the count of Easters by date stays around the average of 72. The date with the most Easters is April 10, which has 102 (4.05%). The date with the fewest Easters is March 23, which has only 14 (0.56%).

It’s only in the extreme tails of the distribution where you find Easter dates that are very rare. The earliest three Easter dates (March 22 – 24) and the latest three Easter dates (April 23 – 25) are pretty unusual. The graph shows us that the earliest three dates are rarer than the latest three dates.

To demonstrate this, Easter happens on the first possible date of March 22 only 0.60% of the time. The holiday last occurred on March 22 in 1818, and it won’t fall on this date again until 2285. That’s a span of 467 years!

Conversely, Easter occurs on the last possible date of April 25 1.03% of the time. It last fell on March 22 in 1943 and won’t fall on this date again until 2038. That’s a period of 95 years. I might be alive to celebrate that one!

## How Rare are March Easters?

I suspect that when Easter falls in March people say it’s early. The previous graph showed us that most Easters fall within April. Let’s use a probability distribution plot to see exactly how often it falls in March.

The red area shows us that about 23% of all Easters happen in March. That’s about 1 out of 4 years. That’s not incredibly rare, but it is still a bit unusual. The last time it was in March was 2016, and we have to wait until March 31, 2024 for the next time. That eight-year gap between March Easters is unusually long!

Now we know which Easter dates are common and rare, let’s move on to patterns!

## Are There Patterns in the Easter Dates?

Do patterns exist in Easter dates? If Easter falls on a specific date this year, can we predict when it will fall on that date again? To answer this question, I’ll use another graph. The plot below displays the rate of first recurrences by the number of years.

The large spike at 11 years really pops out! This peak tells us that for 1,141 Easters (45%) the first time Easter repeats its date is in 11 years.

Let’s see how this 11-year pattern fits the next 10 Easters. The dates below indicate that 6 of the next ten Easters first reoccur on the same date in 11 years. That’s 60% of the Easters, which is a bit higher than the overall percentage (45%). In statistics, it’s crucial to understand the larger context before drawing conclusions.

Another thing that leaps out at me is the small number of possibilities for when Easter can first repeat a date. If a value does not appear on the first recurrence graph, Easter never repeats the date in that number of years. For example, an Easter date cannot first reappear in 1, 2, 3, 4, 7, 8, 9 years, and so on. In the 2517 Easters in the Gregorian calendar, there are only 24 possible values for when it can first repeat a date.

The official determination for when Easter happens only served to confuse me. However, standard data analysis procedures provided crystal clear answers about Easter dates.

Kevin says

I’d be interested in knowing how a lot of the pieces of the formula came about, and how they relate. The only thing I’ve really been able to figure out is that the 19 in the formula involves the Metonic cycle of the moon phases, which repeats in a 19 year period. But it’s still neat how everything works out!

In case you were curious, I’m the same Kevin who posted the Welch ANOVA formula. I’m not a statistician by trade, but it’s been a passion of mine ever since high school, so for at least 25 years. I’m still amazed at what the field can show you when applied correctly. There are some smart people who laid all this groundwork!

Jim Frost says

Hey Kevin, I was wondering if you had posted before. The combination of name and email looked familiar! Thanks for posting the Welch ANOVA formula. I know it was very helpful for the other reader!

The Easter formula is kind of mysterious in a cosmic way! Great discovery about 19 and the Metonic Cycle (which I didn’t know about)! It really doesn’t seem like it should work out but apparently it does.

I’m a bit biased, but I do think statistics is an amazing field. Unfortunately, I think statistics classes all too often take the fun out of it. But, there’s the excitement of learning from data that I think is pretty amazing. For me, I’ll never forget the first real research project I worked on where I was the first person to know the results! Kind of thrilling considering it was a multimillion dollar grant with a decent number of people on it! That’s the type of excitement I wish was conveyed about statistics a bit more.

Kevin says

Okay, here goes. Let’s take 1953 as an example.

1) Let A be the remainder when the year is divided by 19; that is, 1953 mod 19. So here A = 15.

2) Divide the 4 digit year by 100 and round down. Call the answer B. So B = 19.

3) Divide the year by 100 and take the remainder, calling it C. So here C = 53.

4) Divide B by 4 and round down. Call the answer D. So here D = (19/4) rounded down, or 4.

5) Divide B by 4 again, but now take the remainder and call it E. So here E = 3.

6) Divide (B+8) by 25 and round down. Call the answer F. So here F = (27/25) rounded down, or 1.

7) Divide (B-F+1) by 3 and round down. Call the answer G. So here G =

(19-1+1)/3 rounded down, or 6.

8) Calculate 19A+B-D-G+15. Divide this by 30 and take the remainder. Call it H. So here (19*15 + 19 – 4 – 6 + 15)/30 = 10 remainder 9, and H = 9.

9) Divide C by 4, round down, and call the result I. So here I = 13.

10) Divide C by 4, take the remainder and call it K. So here K = 1.

11) Calculate (32 + 2E + 2I – H – K) and divide this by 7, calling the remainder L. So here (32 + 6 + 26 – 9 – 1)/7 = 54/7 = 7r5, so L = 5.

12) Calculate (A + 11H + 22L), divide this by 451, and round down. Call the result M. So here (15 + 99 + 110)/451 is rounded down to 0 and M = 0.

13) The month of Easter in the given year is the whole number part of the quotient when (H + L – 7M + 114) is divided by 31. It will be either 3 (March) or 4 (April). The date of the month is the remainder of the same division plus 1. So here, (9 + 5 – 7(0) + 114)/31 = 128/31 = 4 remainder 4, so Easter 1953 is April 5th.

Not a terribly compact formula, agreed, but it works!

Jim Frost says

Hi Kevin,

Thanks for sharing that! It kind of hurts your brain, but it does what it is supposed to do. At some point, I’ll have to look into how it works.

Kevin says

Hi Jim,

Are you familiar with the Meuss-Jones-Butcher algorithm for calculating the date of Easter in the Gregorian calendar? It’s somewhat complex in that there are a lot of steps, but there is no complex arithmetic involved. It’s pretty interesting, really – it involves a lot of modular arithmetic, which is to be expected given that Easter dates are based on the lunar cycle. Let me know if you want an example!

Jim Frost says

Hi Kevin,

I’m actually not familiar with that algorithm. It sounds interesting. An example would be great! Thanks!