The mean is the arithmetic average: add the observed values and divide by their count. It describes the central level of a dataset. When outcomes have known or estimated probabilities, the expected value is a probability-weighted mean: multiply each outcome by its probability, then add the products. section 14.1.2 section 10.1.4
Choose the form of the mean that matches the decision:
- For historical stock returns, calculate the mean log return to represent the observed daily trend. Pair it with standard deviation, which captures fluctuations around that trend; both feed a Monte Carlo simulation. section 9.5.4
- For quality control, use the process mean as a control chart’s center line. Compare plotted observations with that reference to spot patterns or shifts. section 14.1.2
- For choices under uncertainty, calculate each option’s expected payoff from its possible payoffs and their probabilities, then compare the options. section 10.1.4
Compare investment alternatives
Suppose a stock-market investment pays $100,000 in a strong economy and loses $120,000 in a weak one. With probabilities of 0.75 and 0.25, its expected value is 0.75 × $100,000 + 0.25 × (−$120,000) = $45,000. Check that the scenario probabilities sum to one before comparing this result with other investments. The highest expected payoff is a decision criterion, not a guarantee of that payoff. section 10.1.4
>>> strong_econ = 0.75 #1
>>> weak_econ = 0.25 #2
>>> if strong_econ + weak_econ != 1:
>>> raise ValueError(‘The probabilities ‘ \
>>> must sum to 1.’) #3
>>> payoff_table['EV'] = \
>>> (payoff_table['Strong Economy'] * strong_econ) + \
>>> (payoff_table['Weak Economy'] * weak_econ) #4
>>> print(payoff_table) #5
Investment Alternatives Strong Economy Weak Economy EV
0 Stock Market 100000 -120000 45000.0
1 Professional Soccer Club 60000 -20000 40000.0
2 Stock Market / Soccer Club (50/50) 80000 -70000 42500.0
3 Savings Account 35000 35000 35000.0
>>> max_ev = payoff_table['EV'].max() #6
>>> print("Maximum Expected Value:", max_ev) #7
Maximum Expected Value: 45000.0
The Python payoff table checks the probabilities, calculates an expected value for each investment, and finds the largest. section 10.1.4
A mean does not describe the full range of outcomes. In staffing simulations, an expected need near two employees can coexist with more variable days; a ten-run simulation may also be skewed by anomalies. Use enough simulations and consider variability and the cost of a shortfall before acting on an average alone. section 9.3.6
