diamond geezer

 Saturday, April 09, 2022

What's the average date of Easter?

I'm going to argue it's today, and also that it was yesterday, and also that it's in ten days time, and also that it might be tomorrow.

First things first.
Easter Day is the first Sunday after the first full moon after the spring equinox.
(or at least that's the headline version, in reality there's an ancient algorithm underneath).

The earliest possible date of Easter is March 22nd and the latest is April 25th.
That's exactly 35 days, the middle one of which is April 8th, i.e. yesterday.
You might therefore expect the average Easter date to be April 8th, but that's not how averages work, it's all about frequency.

If you tally the 100 Easter dates in the 21st century you get this...
22 Mar 0
23 Mar 1
24 Mar 0
25 Mar 2
26 Mar 3
27 Mar 2
28 Mar 3
    29 Mar 3
30 Mar 4
31 Mar 5
1 Apr 3
2 Apr 2
3 Apr 3
4 Apr 4
    5 Apr 4
6 Apr 3
7 Apr 2
8 Apr 3
9 Apr 3
10 Apr 4
11 Apr 4
    12 Apr 4
13 Apr 2
14 Apr 3
15 Apr 5
16 Apr 4
17 Apr 3
18 Apr 3
    19 Apr 3
20 Apr 5
21 Apr 4
22 Apr 2
23 Apr 1
24 Apr 2
25 Apr 1
Which average would you like?

The mode - the most common date(s) - is March 31st, April 15th and April 20th.
Perhaps not coincidentally, these were the dates of Easter in 2002, 2001 and 2003.

The median - the middle date if you list them in order - falls inbetween April 9th and April 10th.
This is suggesting a toss-up between today and tomorrow.

The mean - where you 'add up' the dates and divide by 100 - is April 8.98th.
That's obviously a very silly date, so you could say it's very late in the day on April 8th or you could say it rounds up to April 9th.

So far I'd say April 9th is winning.

But every century is different.

Here's what the 20th century looked like.
22 Mar 0
23 Mar 1
24 Mar 1
25 Mar 1
26 Mar 3
27 Mar 3
28 Mar 2
    29 Mar 3
30 Mar 4
31 Mar 4
1 Apr 4
2 Apr 2
3 Apr 4
4 Apr 4
    5 Apr 3
6 Apr 4
7 Apr 4
8 Apr 2
9 Apr 3
10 Apr 3
11 Apr 4
    12 Apr 5
13 Apr 2
14 Apr 3
15 Apr 3
16 Apr 4
17 Apr 4
18 Apr 3
    19 Apr 4
20 Apr 3
21 Apr 3
22 Apr 3
23 Apr 3
24 Apr 0
25 Apr 1
This time the mode is April 12th
the median is April 9th
and the mean is April 8.74th (i.e. the 8th, or maybe the 9th)

Combine the 20th and 21st centuries and things should get more accurate.

Now the mode is March 31st/April 12th
the median is April 9th
and the mean is April 8.86th (i.e. the 8th, or maybe the 9th)

But to be absolutely accurate we need to calculate the average over a complete Easter cycle.
It turns out a complete Easter cycle is very very long.
The cycle only repeats every 4×25×19×30 centuries, i.e, every 5,700,000 years.

And if you draw a frequency graph of Easter dates over 5,700,000 years you get this.

The graph shows that Easter dates step up from 22nd March and step down to 25th April.
The three weeks in the middle (that's 28th March to 18th April) have very similar frequencies.

I've coloured the bars green and blue because, near enough, this splits the dates in half.
Generally speaking Easter is before 9th April half the time and after 8th April half the time.
But the precise split is 50.04%/49.96%, so the actual median is 8th April, not 9th April.

If you calculate the mean (by adding up all 5,700,000 dates and dividing by 5,700,000) you get April 8.45th.
That is definitely 8th April whether you truncate the decimal or round it down.

But the mode is something else entirely as you can see from the one sticky-up bar.
Over the course of 5,700,000 years the most common date of Easter is absolutely definitely 19th April.

The reason for the mode being 19th April is complicated.
When the Catholic Church switched to the Gregorian calendar in 1583 they needed to amend the rules for Easter to fit with certain century years no longer being leap years. But this might have widened the range of Easter dates to include 26th April and this would have gone against the intentions of the Council of Nicaea in 325AD. Instead they tweaked the algorithm on the rare occasions that the first full moon after the equinox falls on the latest possible date and this is a Sunday, knocking things back a week. This took the tiny bar that would have appeared on 26th April and plonked it on top of the bar for 19th April instead, hence 19th April is the most common Easter date.

To summarise:

Over the entire Easter cycle:
median: 8th April
mean: 8th April
mode: 19th April

But over the period of our lifetimes, i.e. the 20th and 21st centuries:
median: 9th April
mean: 8th/9th April
mode: 31st March/12th April

There's a reason why the average Easter is more 9th than 8th at the moment, which is that the earliest possible date of 22nd March cannot happen in the 20th, 21st and 22nd centuries. This date last cropped up in 1818 and will next occur in 2285. Without 22nd March in the mix our current average Easter nudges one day later to 9th April rather than 8th April.

Easter fell on 8th April in 2007 and in 2012, but hasn't fallen on 9th April since 1950. Such are the twisted wigglings of this festival day. But it will fall on 9th April again in 2023, so we can all look forward to an Average Easter next year.

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