Summation of uncertainty intervals I have the following data based on the 6 independent regions that make up the world. I want to compile them to find a global estimate. I can add best estimates from each region to give the best estimate for the the world. I am not sure how to deal with the uncertainty intervals as I feel adding the low and high estimates does not seem correct




Region
Best Estimate
Low
High




African Region
379
321
441


Region of the Americas
19
18
20


Eastern Mediterranean Region
80
68
92


European Region
21
20
22


South-East Asia Region
698
650
747


Western Pacific Region
87
80
93




I agree with Allan that having the data that makes up the estimates would be ideal. However, if you don't have the data I'm not sure you'd want to make the assumption of these all being Uniform distributions since this gives equal weight of all values between the lower/upper limits given. Since the column name is Best Estimate then I would think that you may not agree with that assumption.
Making a different assumption that these could each be Normally distributed and are independent of each other would allow you to take advantage of the sum of normally distributed random variables.
$$
\begin{aligned}
X \sim \mathcal N(\mu_X, \sigma^2_X) \\
Y \sim \mathcal N(\mu_Y, \sigma^2_Y) \\
Z = X + Y
\end{aligned}
$$
Then
$$Z \sim \mathcal N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$$
What you don't have is the variance for each but you could make another rough estimate using the range rule. For this you simply take the range for each and divide it by $4$ to have a rough estimate of your standard deviation.
Putting this all together in Julia you would have:
using Distributions

means = [379, 19, 80, 21, 698, 87]
lows = [321, 18, 68, 20, 650, 80]
highs = [441, 20, 92, 22, 747, 93]

# using the range rule
stds = (highs - lows) / 4
vars = stds.^2

# using the assumption of independent Normal R.V.s
world = Normal(sum(means), sqrt(sum(vars)))

# what's our 95% interval for the grand total?
quantile(world, [0.025, 0.5, 0.975])

Results in:
 1207.2072954499056
 1284.0
 1360.7927045500944

