Mean & Standard Deviation of a Discrete Random Variable
The Mean of a discrete random variable , also known as the Expected Value , is the weighted average of all possible outcomes where the weights are the probabilities of those outcomes.
It is denoted by the Greek letter and represents the value one would expect to see on average if the random experiment were repeated many times.
The Standard Deviation measures the variability or 'spread' of the distribution, indicating how much the values of typically deviate from the mean .
The Variance is the average of the squared deviations from the mean and serves as the mathematical precursor to finding the standard deviation.
Formula:
Formula:
Show Your Setup: Even if using a calculator, always write out the first few terms of the summation (e.g., ) to secure partial credit.
Calculator Efficiency: Use the '1-Var Stats' function on a graphing calculator. Input the outcomes in one list (L1) and the probabilities in the frequency list (L2).
Sanity Check: The mean must always fall between the minimum and maximum possible values of the random variable. If your calculated mean is outside this range, check your arithmetic.
Population vs. Sample: For probability distributions, always use the population standard deviation () provided by the calculator, never the sample standard deviation ().
Forgetting to Square: A common error is forgetting to square the deviations before multiplying by the probabilities, which would result in a sum of zero for any distribution.
Probability Sum: Always verify that the sum of probabilities equals 1 before starting calculations; if it does not, the distribution is invalid or missing data.
Rounding Too Early: Rounding the mean before using it to calculate the variance can lead to significant 'rounding drift' in the final standard deviation. Keep as many decimals as possible until the final step.