# A simple (?) problem of ratios in a population

#### ElbowPatches

##### New member
Hello everyone,

I'd like to know how to calculate an expected value for the number of a population with a particular property.

Let's suppose that for the whole population of a country 20% of those who choose to study a subject, let's say mathematics at A level, are female. (Let's say the population has a 1:1 ratio of male:female as a whole, though I'd also like to account for there being a 51:49 ratio or such nationally).

If I take a year group that has 50 males and 50 females, I'd expect the 4:1 ratio of male:female choosing to study maths, regardless of the total number that choose it.

What if my group doesn't have a 1:1 ratio, but instead has say 40% males and 60% females. What proportion of males and females would I expect to choose maths?

Would it be 24%, as an additional fifth of female population are there with the same expected chance of choosing maths?

What would be the general formula for working out the expected proportion of m:f in a subject, given an initial proportion of m:f in the yearvgroup?

Thanks for any help everyone.

EP

#### JeffM

##### Elite Member
Hello everyone,

I'd like to know how to calculate an expected value for the number of a population with a particular property.

Let's suppose that for the whole population of a country 20% of those who choose to study a subject, let's say mathematics at A level, are female. (Let's say the population has a 1:1 ratio of male:female as a whole, though I'd also like to account for there being a 51:49 ratio or such nationally).

If I take a year group that has 50 males and 50 females, I'd expect the 4:1 ratio of male:female choosing to study maths, regardless of the total number that choose it.

What if my group doesn't have a 1:1 ratio, but instead has say 40% males and 60% females. What proportion of males and females would I expect to choose maths?

Would it be 24%, as an additional fifth of female population are there with the same expected chance of choosing maths?

What would be the general formula for working out the expected proportion of m:f in a subject, given an initial proportion of m:f in the yearvgroup?

Thanks for any help everyone.

EP
It may be a simple problem, but I do not find it a clear one. So this response may not advance things particularly.

$$\displaystyle m \text { = proportion of males in population, where } 0 \le m \le 1.$$

$$\displaystyle f\text { = proportion of females in population, where } 0 \le f \le 1.$$

$$\displaystyle m + f = 1.$$

$$\displaystyle s \text { = proportion of population who are math students, where } 0 < s \le 1.$$

$$\displaystyle p \text { = proportion of math students who are male, where } 0 \le p \le 1.$$

$$\displaystyle q\text { = proportion of math students who are female, where } 0 \le q \le 1.$$

Are these the variables?

Are you asking what the expected value of male and female students would be in a math class with c students, if the math class was chosen at random out of all math classes? Or are you asking something different?

#### ElbowPatches

##### New member
Thankyou, I think that's right.

Let me put it like this. Nationally, 20% of maths students are female. If I had a sample (any population size) that contained 50% females, I'd expect 20% of the students to be females. The sample in question has 60% females and 40% males. What would be the expected ratio of f:m if the national trend is exemplified?

Let's suppose the ratio of maths student found is 33% female. I want to know if this is in line, below, or above the national trend. By my reckoning (a different answer from my initial post) the expected proportion is:

Fraction of females in sample x 0.2 : Fraction of males in sample x 0.8

So for a sample of 60 girls and 40 boys, I'd expect a ratio of
0.6 x 0.2 : 0.4 x 0.8
= 0.12 : 0.32
So 27.27% of the sample doing Physics would be expected to be female. So this way I can compare the ratio of students to the national ratio, accounting for the fact that there were more females in the sample.

I hope that makes sense.

#### JeffM

##### Elite Member
I am still having problems. Part of it may arise from your word "sample," which may have a technical meaning.

I am also confused by your ratio of female to male students. Does this mean that 20% of math students are female or that 20% of females are math students? If the former, we need to know what percentage of the population studies math.

#### ElbowPatches

##### New member
I am still having problems. Part of it may arise from your word "sample," which may have a technical meaning.

I am also confused by your ratio of female to male students. Does this mean that 20% of math students are female or that 20% of females are math students? If the former, we need to know what percentage of the population studies math.
So, let's say that nationwide 20% of maths students are female. We do not know what the proportion of the total female population study maths. In a particular year group in a school (the sample) 25% of the maths students are female. "Great! We're above average!" They might say. But that year group was 60% female anyway, so if they were in line with the national average you'd expect the proportion to be higher than 20%. How much higher? That's the question.

By my reckoning they'd need to be 27.3% to be in line with the average, so if they were actually only 25%, they'd be below the national average. Oh dear, what appeared to be above average is in fact below average.