"Weighted average of the volatility" is an odd baseline. It presumes that the assets are perfectly correlated. But assuming 0 correlation isn't defensible either. If you didn't have information about the pairwise correlations, what default value would you use for investment volatility?
If I didn't have the value, I would not use that approach. I'd use the sum of the historical returns multiplied by their respective proportions and take the standard deviation of those daily returns. That would be my volatility for the portfolio.