# Standard deviation of discrete variable

A start-up looking to get into the sleeveless shirt market is looking for \$10,000 from investors to get their company started. If you choose to invest this \$10,000, at the end of 5 years the company will either go broke (i.e., you lose \$10,000), they are profitable and they will give you back your money plus another \$10,000 (i.e., you gain \$10,000), or the company is sold to another company and you get your original money back plus \$100,000 (i.e., you gain \\$100,000). Based on your market research into sleeveless shirts, you believe there is a 0.2 chance the company goes broke, a 0.6 chance that they become profitable and a 0.2 chance that they are sold to another company. What is the standard deviation on the amount of money you would earn from this investment?

The answer I came up with is 24066.57 but the correct answer is 38781.44. I do not understand how! I have spent almost 3 hours trying to figure this question out. Help please!

The mean return is $$\mu = \sum XP(X) = (-10,000 \times 0.2 + 10,000 \times 0.6 + 100,000 \times 0.2) = 24000$$ The variance is $$\sigma^2 =\sum (X-\mu)^2P(X) =$$ $$(-34,000^2 \times 0.2 + -14,000^2 \times 0.6 + 76,000^2 \times 0.2) = 1.504 \times 10^{6}$$
which, taking the square root, gives the standard deviation of $$38781.44$$