A list of transactions contains 100 numbers 60 gains and 40

A list of transactions contains 100 numbers: 60 gains and 40 losses. The gains are positive numbers and the the losses are negative numbers. The units are thousands of dollars. For the 60 gains, the average is 18 and the SD is 7.5. For the losses the average is -20 and the SD is 9.2.
a) Find the average of the 100 transactions.
b) Find the SD of the 100 transactions.
(show work)

Solution

Let group 1 = the gains
group 2 = the losses

X1 = group 1 sample mean = 18

X2 = group 2 sample mean = -20

s1 = group 1 sample standard deviation = 7.5

s2 = group 2 sample standard deviation = 9.2

Also, note that

grand mean = (n1 X1 + n2 X2)/(n1 + n2)

= (60(18) + 40(-20))/(60+40)

= 2.8 [ANSWER]

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Also, note that

grand standard deviation = [(n1 - 1)s1^2 + (n2 - 1)s2^2 + (n1 - 1) (u - X1)^2 + (n2 - 1)(u - X2)^2]/[n1 + n2 - 2]

where u = grand mean = 2.8.

Thus, plugging in gthe values we have,

grand standard deviation = 20.335 [ANSWER]