How is the Standard Deviation Affected by Gender Differences in Life Span?

[I like Serena]

Suppose the men in a country have a mean life span of 79 with a standard deviation of 8.
And the women in the country have a mean life span of 83 and also a standard deviation of 8.
Furthermore, suppose the life spans are normally distributed.

What is the mean life span of the entire population?
And what is their standard deviation?
Hint: it is not √32.

 

[Caracrist]

what is the men/women birth ratio?

 

[I like Serena]

what is the men/women birth ratio?

50-50.

 

[davee123]

I get a mean of 81 (trivial) and a standard deviation of SQRT(68)?

[edit]Huh.. and if men had a mean of 80, and women 82 (still std dev 8), the new standard deviation is SQRT(65). And if they were means of 78 and 84, std deviation goes to SQRT(73). Seems to make sense, I guess, going up by the square of the distance from the mean (64+1, 64+4, 64+9, etc).
[/edit]

DaveE

 

[I like Serena]

I get a mean of 81 (trivial) and a standard deviation of SQRT(68)?

[edit]Huh.. and if men had a mean of 80, and women 82 (still std dev 8), the new standard deviation is SQRT(65). And if they were means of 78 and 84, std deviation goes to SQRT(73). Seems to make sense, I guess, going up by the square of the distance from the mean (64+1, 64+4, 64+9, etc).
[/edit]

DaveE

Yep. Right on. :)
So how did you get there?

And what if men have a standard deviation of 7, women an std dev of 9, and if the male-female ratio is 0.7 (that ratio is typical for age 65+)?

 

[davee123]

So how did you get there?

Excel. I effectively made a normal distribution curve formula between 0-160 for both male/female-- effectively doing a plot of the formula for those points. From those, I created the desired dataset (which is a pretty close approximation), and found the standard deviation for it. Sorta the cheaty method, but admittedly the math was looking a bit too abstract for me when I tried to solve it more generically.

And what if men have a standard deviation of 7, women an std dev of 9, and if the male-female ratio is 0.7 (that ratio is typical for age 65+)?

I get a wacky value of approx 8.4675, although I'd point out that the ratio should be pretty close to 50/50. Even if the ratio at age 65+ were 99% female, so long as the ratio at BIRTH were 50/50, it wouldn't invalidate all the tons of men who died young-- they're still valid data points.

DaveE

 

[I like Serena]

Excel. I effectively made a normal distribution curve formula between 0-160 for both male/female-- effectively doing a plot of the formula for those points. From those, I created the desired dataset (which is a pretty close approximation), and found the standard deviation for it. Sorta the cheaty method, but admittedly the math was looking a bit too abstract for me when I tried to solve it more generically.



I get a wacky value of approx 8.4675, although I'd point out that the ratio should be pretty close to 50/50. Even if the ratio at age 65+ were 99% female, so long as the ratio at BIRTH were 50/50, it wouldn't invalidate all the tons of men who died young-- they're still valid data points.

DaveE

Pretty accurate. My exact calculation gives 8.46752395554428;).

That leaves the question what a nice formula is for the standard deviation...?

I'd like to point out though that I intentionally used the phrase "life span" instead of "life expectancy". The difference is that "life span" is the natural age a human being can get to, which excludes accidents at birth, accidents in traffic, consequences of doing heavy work, obesity, etcetera.
If you make a graph of the probability density, you'll see that the first 60 years of life are rather unpredictable, but after 60 it looks pretty much like a normal distribution.

The interesting thing is that it is more or less the same the world over. So even in some African countries where life expectancy is as low as 40 years, their life span is still above 70, and they have 100+ people ("centenarians") as well.

Furthermore, the ratio at BIRTH is NOT 50/50. The male-female ratio at birth is about 1.05 and is correlated to race.
The current hypothesis is that it is caused by the Y-chromosome being smaller than the X-chromosome, which makes male swimmers faster than female swimmers :O.

 

[davee123]

Pretty accurate. My exact calculation gives 8.46752395554428;).

Yeah, I left off the last few digits that I got, but that matches.

I'd like to point out though that I intentionally used the phrase "life span" instead of "life expectancy". The difference is that "life span" is the natural age a human being can get to, which excludes accidents at birth, accidents in traffic, consequences of doing heavy work, obesity, etcetera.

So... it's really more like "life span of people 65+" rather than "life span of people"?

Furthermore, the ratio at BIRTH is NOT 50/50. The male-female ratio at birth is about 1.05 and is correlated to race.
The current hypothesis is that it is caused by the Y-chromosome being smaller than the X-chromosome, which makes male swimmers faster than female swimmers :O.

I recall being told a long time ago that there were statistically more of one type of sperm, but that the other type were slightly more prone to surviving to fertilization. I honestly don't recall which was which, but it worked out to ~roughly~ 50/50, but not exactly.

DaveE

 

[I like Serena]

So... it's really more like "life span of people 65+" rather than "life span of people"?

Very sharp! I guess we're indeed talking more or less about 65+ people.
However, if we only look at 65+ we have the problem that it is not a normal distribution anymore, since a significant part is missing, which btw would make the male mean life span shift upward more than the female mean.

I recall being told a long time ago that there were statistically more of one type of sperm, but that the other type were slightly more prone to surviving to fertilization. I honestly don't recall which was which, but it worked out to ~roughly~ 50/50, but not exactly.

DaveE

I was not aware of that explanation. But what could be the explanation that there might be more of one type of sperm?

Sperm is made out of cells with exactly 1 X- and 1 Y-chromosome, which are split into 1 X-sperm and 1 Y-sperm. This means that at least just before sperm is being created the expected ratio is exactly 50/50.

Btw, I did hear from a colleague, that apparently female sperm is more persistent than male sperm, that is, female sperm lives longer, although male sperm is faster.
So you can influence your chances one way or the other! :).
Explanation for the persistence would be that male sperm spends his limited energy source faster.
I found a couple of references on the internet to support this, but as yet no scientific ones.