When is Easter this year? I ask this question every year! The next Easter occurs on April 17, 2022. And then, in the next year, Easter falls on April 9, 2023. 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 April 25 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 5 of the next ten Easters first reoccur on the same date in 11 years. That’s 50% of the Easters, which is slightly 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 Bluxome says

Strictly speaking, the median date would be the 18th date of the 35 possible dates from 3/22-4/25 inclusive, so that would be 4/8 as the median date by my count.

Chris says

Hello, great bit of data/statistics Jim. Quick question, do you have the median date of when Easter falls? It looks like it’s around Apr 8th but it could be either side. People always seem to say one of either ‘Easter is late this year’ or ‘Easter is early this year’ whatever the date happens to be. So I would love to know where the middle is from this data! ðŸ™‚

Thanks so much

Jim Frost says

Hi Chris, like David, I get a median date that is closest to April 8. 1232 Easters occur on March 22 – April 8. And, 1285 occur from April 9 – April 25. It’s not quite 50/50 but as close as we can get given that the each day is the smallest chunk in our data.

Deepak Kannan says

Can we get the data set or the excel file containing the dates?

Jim Frost says

Hi Deepak, thereâ€™s a link in the article to download the dates.

Jeremy says

o the cycles of the moon. Also, from something I read years agre are periodic corrections in the Gregorian calendar. One of them is Leap Year every four years, but I think there’s an extra day added every 400 years too, and maybe some others. This might have a minor effect on those dates. It would be interesting to see how the calendar dates of the solste Easters and equinoxes vary throughout the years. They don’t occur on the exact same dates every year (e.g. vernal equinox, or first day of spring, can be March 20 or March 21, and it’s exact time may even slip into March 19 or March 22 in some years).

And Kevin, WHY aren’t you working in cryptology for the NSA? (or ARE you . . . . )

Jim Frost says

Hi Jeremy,

Yes, it’s all interesting how the two calendar systems interact! The list of Easter dates incorporates Leap Years and other factors that you mention. That’s partially why the distribution is so unusual.

George Anogeianakis says

Dear Jim,

Many years ago, when I was working with the EEG I learned that before one can apply the classical statistical methodology to cyclical phenomena, one must take out the effect of cycling out of the data. I have been away from that type of analysis for a long time to be able to recall how exactly that is done but I remember that such methods are used by astronomers in their observation of celestial motions. Maybe you can look into that and comment on it?

Apropos, I really enjoy your mails and although I have not been a customer yet, I am sure I will soon be as I have many young friends that are getting into research now and some of your books are a perfect gift for them

Chuck says

The history around establishing the date for celebrating Easter is an interesting one, and shows how practices can complicate things.

The Jewish calendar is a lunar calendar–their month began with the new moon. In addition, the beginning of a new day is at sundown, not midnight. Passover is prepared during the day on the 14th of Nisan (from the new moon in March to the new moon in April) and then observed that evening, after sundown, on the 15th of Nisan, during the full moon. Differences between the lunar calendar and the solar calendar means that there is a range of days in the calendar in which there is a full moon, which occurs 75% in April.

Early on (at the Council of Nicea in 325, I believe) it was decided that Easter would always be celebrated on a Sunday, while still being connected to the date of the Passover. That poses a problem in that Passover, being based on the lunar calendar, is not tied to any particular day of the week. There are years where Passover and Easter are a week apart (it would be interesting to see how the time between Passover and Easter varies across this time span).

Finally, you have two different calendars–the Gregorian Calendar (which most of the world follows today) established in 1582, which was a reform to the Julian Calendar, which has 3 more days every 400 years than the Gregorian Calendar. All this means that currently the Gregorian Calendar is 13 days ahead of the Julian Calendar, which is used by the Eastern Orthodox churches.

So to sum up: Easter is always celebrated on the first Sunday after the Passover, which is based on the lunar calendar and therefore is not the same calendar date every year, meaning that Easter cannot be on the same calendar date every year. And because Eastern and Western churches also use different calendars (Julian and Gregorian respectively) the Sunday that Easter is celebrated on is different between those churches every year.

As I said, complicated.

Jim Frost says

Hi Chuck,

Thanks so much for sharing that information. It

iscomplicated but very interesting! This is one of those cases where I’m realizing how little I knew about the subject area!Kevin says

Iâ€™d be interested in knowing how a lot of the pieces of the formula came about, and how they relate. The only thing Iâ€™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 itâ€™s still neat how everything works out!

In case you were curious, Iâ€™m the same Kevin who posted the Welch ANOVA formula. Iâ€™m not a statistician by trade, but itâ€™s been a passion of mine ever since high school, so for at least 25 years. Iâ€™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. Letâ€™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? Itâ€™s somewhat complex in that there are a lot of steps, but there is no complex arithmetic involved. Itâ€™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!