Interpreting Statistical Questions: Ice Hockey Player Diff.

[Pouyan]

Homework Statement

In ice hockey, you have five outfielders on the field. How many percent more outfielders does the opposing team have if you get a player sent off?

Relevant Equations

I interpret it this way: we have two groups, team A and team B. Both have 5 players, team A loses a 1. How many percent more do B players have than A?

The solution in my book:
5/4 = 1.25. That is 25 % more.

What I came up with:
I thought that now we have totally 9 players. So A: 4/9 and B: 5/9. The difference is 1/9 which is about 11%!

A friend told me :

The difference between B & A is 5-4=1
The changing rate is (5-4)/5 = 0.2 !
So B has 20 % more than A.

Now it's confusing. How should I interpret such a statistical question ?! I get three different answers

 

[PeroK]

Homework Statement:: In ice hockey, you have five outfielders on the field. How many percent more outfielders does the opposing team have if you get a player sent off?
Relevant Equations:: I interpret it this way: we have two groups, team A and team B. Both have 5 players, team A loses a 1. How many percent more do B players have than A?

The solution in my book:
5/4 = 1.25. That is 25 % more.

That's what 25% more means. $5$ is 25% more than $4$.

What I came up with:
I thought that now we have totally 9 players. So A: 4/9 and B: 5/9. The difference is 1/9 which is about 11%!

I've never seen anything like this. 11% more equates to a factor of 1.11.

A friend told me :

The difference between B & A is 5-4=1
The changing rate is (5-4)/5 = 0.2 !
So B has 20 % more than A.

Now it's confusing. How should I interpret such a statistical question ?! I get three different answers

A has 20% less than B. With percentages you need to know what your reference value is.

When you say B has x% more than A, then A is your reference value and B = A + x% of A.

When you say A has y% less than B, then B is your reference value and A = B - y% of B.

That's the convention for pay rises, interest rates etc.

 

[PeroK]

PS note the following interesting point:

If you earn $20 per hour and get a 50% pay rise, then you earn $30 per hour. If you then get a 33.3% pay cut, then you are back to $20 per hour. I.e. a 50% pay rise, followed by a 33.3% pay cut cancel each other out. If you got a 50% pay rise followed by a 50% cut, you'd be down to $15 per hour.

Likewise, a 50% pay cut takes you to $10 per hour and you would need a 100% pay rise to get back to $20 per hour.

 

[Pouyan]

That's what 25% more means. $5$ is 25% more than $4$.

I've never seen anything like this. 11% more equates to a factor of 1.11.

A has 20% less than B. With percentages you need to know what your reference value is.

When you say B has x% more than A, then A is your reference value and B = A + x% of A.

When you say A has y% less than B, then B is your reference value and A = B - y% of B.

That's the convention for pay rises, interest rates etc.

Thank you for the help! Now I understand that we have to take care of details.

But I don't still get it...

Why can't we say:

We have 9 players, 4 for A and 5 for B, in percent:

B = 5/9 = 56 %
A = 4/9 = 44%

Why can't we say B has about 11% more than A?

 

[PeroK]

Thank you for the help! Now I understand that we have to take care of details.

But I don't still get it...

Why can't we say:

We have 9 players, 4 for A and 5 for B, in percent:

B = 5/9 = 56 %
A = 4/9 = 44%

Why can't we say B has about 11% more than A?

That's a different calculation and it's not what is meant when we say 11% more. Everything in mathematics needs a clear definition so there is no argument. In that case B has 25% more than A, by definition of what 25% more means.

You could say that B has 11% more of the total than A. Then the total number of players is the reference. That's a different calculation.

To take an example of where your calculation would be inappropriate. Suppose you work for a company whose total salary bill for the year is $10 million. You earn $50,000 dollars a year and your friend earns $100,000 dollars per year. Everybody would say that your friend earns 100% more than you (i.e. double what you earn). But:

$100,000 is only 1% of the total salary bill, and $50,000 is only 0.5% of the total salary bill. It's not correct to say that your friend earns only 0.5% more than you.

Instead you can say that your friend's extra earnings represent only 0.5% of the total salary bill. That would be the accurate description of such a calculation.

 

[Mondayman]

Homework Statement:: In ice hockey, you have five outfielders on the field.

You have five skaters on the ice.