Average age

As I get older I find that I pay more attention to the ages of the people who appear each day in the obituary pages of The Times! Now I find that I have started to calculate the average (mean) age at which they died – and I am encouraged if it is greater than my age (it usually is!).

But, I have an interesting mathematical observation, related to the issue of rounding errors, to offer you.

Take this example. Four people are listed in the obituaries, dying at the ages of 69, 73, 76 and 80. What is the average (mean) age at which they died?

Now, 69 + 73 + 76 + 80 = 298. Divide this by 4 and we get 74.5 years.

But, this is NOT the average age of these four individuals.

Remember that when we say that someone died at the age of 69 this means that they could have been as little as one day short of their 70th birthday. Different conventions for rounding up or rounding down are used in various contexts. We always round down to the year below when we give someone's age in years.

So, the best estimate for the average age of the four individuals in this example would be the mean of 69.5, 73.5, 76.5 and 80.5 years. And that gives the mean age as 75 years.

At my age, let me assure you, that extra half a year is quite significant!

Mathematical Problem 4: Solution

This is the solution to the problem set on 9 April.

We observed that if a daughter is born in '66 and her mother in '44 then in the year that the mother turns 66, the daughter turns 44.

Was this an unusual coincidence? Of course not! It happens for any two people! It would be fun to try this as a little investigation with some primary school children.

For example, if A was born in 1957 and B was born in 1971, then in the year that A turns 71 B turns 57. Why? Because the age of A minus the age of B (once they have both had their birthdays) is always equal to the birth year of B minus the birth year of A. If A is born 14 years earlier than B then B is always 14 years younger than A. So when A's age equals B's birth year (ignoring the 1900) then B's age (14 less than this) must equal A's birth year.

Why had I never noticed that before?!

There is a complication if they are born in different centuries, – this requires the older person to pass 100 before the phenomenon occurs. For example, Catherine was born in 1966 and her son, Jack, was born in 2001. So, the year Jack turns 66, Catherine will turn 101!

Mathematical Problem 4: ages and birth years

It's my wife's birthday on Wednesday. Christina will be 66. In October, our daughter, Catherine, will be 44. Now, Christina was born in 1944 and Catherine was born in 1966.

So, daughter born in '66, mother in '44. And in the year that mother turns 66, daughter turns 44!

That's neat, isn't it! "What are the chances of that happening?" Mrs Haylock asked her mathematical husband in the car yesterday.

What do you think? Is it a rare coincidence?