Combined individual error This is from my college project management course. Reading through an example question here it says:

When estimating in parts, the total error will be less than the sum
of the part errors.

This makes sense to me.

For a 1000-hour job with estimating accuracy of ± 50%, the estimate
range is from 500 to 1500 hours.

This also makes sense

If the estimate is independently made in 25 parts, each with 50%
error, the total would be 1000 hours, as before and the estimate range
would be from 900 to 1100 hours

Okay, How did we get that?

To combine independently-made estimates

*

*Add the estimated values.

*Combine the variances (squares) of the errors.


Cool, following you...

With 25 estimates for a 1000-hour job

*

*Each estimate averages 40 hours

*The standard deviation is 50%, or 20 hours


Here's where it stops making sense. It goes on to take the square root of sum of the variances which makes sense, but where did he get the standard deviation being 50%?
I could be missing something here?
 A: You got it right: since the standard deviations are $20$ hours, the variances are $20^2=400$ (squared hours).  The sum of the variances is $25\times 400=10000$ squared hours and its square root is $\sqrt{10000}=100$ hours.  Apparently from this the authors would construct a range by subtracting and adding this to the mean, giving $1000-100=900$ to $1000+100=1100$ hours.

In general, when $n$ independent and equal estimates are summed to equal $1000$ and each has $\alpha\%$ error, then the corresponding means obviously equal $1000/n$ and their variances all equal
$$(\alpha 1000/n)^2.$$
When $n$ of these are summed and you take the square root you get
$$\alpha 1000/\sqrt{n}.$$
Relative to the total of $1000$ this equals
$$\alpha/\sqrt{n}.$$
In words, this says the relative error in the sum is only $1/\sqrt{n}$ times as large the relative errors in the components.
The authors probably proceeded by wondering how many components with $\alpha=50\%$ error would have to be involved in order to cut the absolute error down from $500$ to $100$.  Because that's a factor of five reduction, the number of components would have to be $n=5^2=25$ (because $1/\sqrt{n}=1/5$).  That's where the numbers in the example came from.

Incidentally, the quoted analysis is ambiguous.  If the parts were not all exactly $1/25$ of the whole, the answer would be different.  For instance, if one part contributed $999$ (with a $50\%$ error of $0.50\times 999=499.5\approx 500$ hours) and the other $24$ parts contributed $1/24$ each, all of them with $50\%$ error, then it should be clear that only the first part really makes any difference and that the overall error in the sum of the $25$ parts is actually nearly $500$ hours, not $100$ hours.
This of course is just common sense rendered mathematically.
A: Well, as I see this is just given to you. You suppose it is known from somewhere: "If the estimate is independently made in 25 parts, each with 50% error" - it starts from here and just used as a condition in the following sentences.
A: Perhaps the problem is trying to point out that with the same relative error (50%) there is a big difference in whether the estimate is made as a whole or in parts. You've already seen the whole example. 
For the in parts case, you have each estimate is on average 40 hours, with variance of 20*20 = 400. For 25 parts, this would be a total variance of 25(400) = 10000, which gives a standard deviation of 100 hours. Thus the range from 900 to 1100 hours, which is plus or minus one standard deviation. 
In my opinion this is a poor example presented poorly (the text, not your question), but at least we can understand what they are doing. 
