# Leap Day Statistics

The following explore different statistical aspects of a leap year. Leap years have one extra day due to an astronomical fact about the earth’s revolution around the sun. Almost every four years it’s a leap year.

It takes roughly 365 and one-quarter days for the earth to revolve around the sun, however, the standard calendar year lasts only 365 days. Were we to ignore the extra quarter of a day, strange things would eventually happen to our seasons - like winter and snow in July in the northern hemisphere. To counteract the accumulation of additional quarters of a day, the Gregorian calendar adds an extra day of February 29 nearly every four years. These years are called leap years, and February 29th is known as leap day.

## Birthday Probabilities

Assuming that birthdays are spread uniformly throughout the year, a leap day birthday on February 29 is the least probable of all birthdays. But what is the probability and how could we calculate it?

We start by counting the number of calendar days in a four-year cycle. Three of these years have 365 days in them. The fourth year, a leap year has 366 days. The sum of all of these is 365+365+365+366 = 1461. Only one of these days is a leap day. Therefore the probability of a leap day birthday is 1/1461.

This means that less than 0.07% of the world’s population was born on a leap day. Given current population data from the U.S. Census Bureau, only about 205,000 people in the U.S. have a February 29th birthday. For the world‘s population approximate 4.8 million have a February 29th birthday.

For comparison, we can just as easily calculate the probability of a birthday on any other day of the year. Here we still have a total of 1461 days for every four years. Any day other than February 29 occurs four times in four years. Thus these other birthdays have a probability of 4/1461.

The decimal representation of the first eight digits of this probability is 0.00273785. We could have also estimated this probability by calculating 1/365, one day out of the 365 days in a common year. The decimal representation of the first eight digits of this probability is 0.00273972. As we can see, these values match each other up to five decimal places.

No matter which probability we use, this means that around 0.27% of the world's population was born on a particular non-leap day.

## Counting Leap Years

Since the institution of the Gregorian calendar in 1582, there have been a total of 104 leap days. Despite the common belief that any year that is divisible by four is a leap year, it’s not really true to say that every four years is a leap year. Century years, referring to years that end in two zeros such as 1800 and 1600 are divisible by four, but may not be leap years. These century years count as leap years only if they are divisible by 400. As a result, only one out of every four years that end in two zeros is a leap year. The year 2000 was a leap year, however, 1800 and 1900 were not. The years 2100, 2200 and 2300 will not be leap years.

## Mean Solar Year

The reason that 1900 was not a leap year has to do with the precise measurement of the average length of earth’s orbit. The solar year, or amount of time that it takes the earth to revolve around the sun, varies slightly over time. it is possible and helpful to find the mean of this variation.

The mean length of revolution is not 365 days and 6 hours, but instead 365 days, 5 hours, 49 minutes and 12 seconds. A leap year every four years for 400 years will result in three too many days being added during this time period. The century year rule was instituted to correct this overcounting.

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