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 (answ: avg=2.8)
b) Find the SD of the 100 transactions. (answ: SD=20.35)
how do i do this? (Answers given but please show work!). Thank you
To find the average of the 100 transactions, we need to calculate the sum of all the transactions and then divide it by 100.
Step 1: Find the sum of the 60 gains:
The average of the gains is 18, which means the total sum of the 60 gains is 18 * 60 = 1080.
Step 2: Find the sum of the 40 losses:
The average of the losses is -20, which means the total sum of the 40 losses is -20 * 40 = -800.
Step 3: Find the sum of all the transactions:
The sum of all the transactions is the sum of the gains and the losses, which is 1080 + (-800) = 280.
Step 4: Find the average of the 100 transactions:
To find the average, we divide the sum of all the transactions by the total number of transactions, which is 280 / 100 = 2.8.
Therefore, the average of the 100 transactions is 2.8.
Now, to find the standard deviation of the 100 transactions, we need to calculate the combined standard deviation of the gains and losses.
Step 1: Find the variance of the gains:
The standard deviation of the gains is 7.5. To find the variance, we square the standard deviation, which is (7.5)^2 = 56.25.
Step 2: Find the variance of the losses:
The standard deviation of the losses is 9.2. To find the variance, we square the standard deviation, which is (9.2)^2 = 84.64.
Step 3: Calculate the combined variance:
To calculate the combined variance, we need to take a weighted average of the variances of the gains and the losses. Since there are 60 gains and 40 losses, the weight for the gains is 60/100 = 0.6, and the weight for the losses is 40/100 = 0.4.
The combined variance is calculated as (0.6 * 56.25) + (0.4 * 84.64) = 33.75 + 33.856 = 67.606.
Step 4: Find the combined standard deviation:
To find the combined standard deviation, we take the square root of the combined variance, which is sqrt(67.606) ≈ 8.227.
Therefore, the standard deviation of the 100 transactions is approximately 8.227.
Note: I made a correction to the standard deviation calculations in the previous response. The correct answer to part b) is SD ≈ 8.227, not 20.35.