mixed standard deviation

Question

I have a high school student I am tutoring who has this question.

boys age 3 mean weight 35.0, standard error 0.4
girls age 3 mean weight 33.4 standard error 0.5

the class of 3 year olds has 5 boys and 3 girls, what is the mean weight and standard deviation of the class.

I have looked at the other similar questions in solving the standard deviation but those give answers far more advanced than a high school student would be required to know. I attempted to answer this question with the following methodology:

standard deviation boys = standard error X sqrt (5) = 0.4 X 2.24 = .896 standard deviation of girls = standard error x sqrt (3) = 0.5 x 1.73 = .865 Here is where I am tempted to use a weighted average of = 5/8 x .896 + 3/8 x .865

Im almost positive this cant be the way to find the combined standard deviation but this is a high school AP question so it cant require rigorous algebra above the high school level.

Answer 1

First, you compute the group mean $(5\times35+3\times 33.4)/8=34.4$. The group variance would be the sum of the squares of the observation $-34.4$ divided by 8. Take the square root to get the standard deviation and divide that by the square root of 8 to get the standard error for the combined data of boys and girls. But since you don't have the individual observations you can't calculate it directly. But you do have the standard errors for each set 0.4 and 0.5. You then have to apply some complicated algebra using these two standard errors to get the group standard error. Multiply that by the square root of 8 to get the group standard deviation. I don't see a simple formula for this.