Suppose you simulate 25 games and record the values of the random variable $x$ for each game. Which of the following is the correct formula to approximate the mean of $x$ based on your simulation results?

• A. Add all 25 observed values of $x$ and divide by 24
• B. Add all 25 observed values of $x$ and divide by 25: $\frac{\sum x_i}{25}$
• C. Select the largest observed value of $x$ as the mean
• D. Multiply all 25 observed values of $x$ together and take the 25th root

Answer: (B)

Verified step by step guidance

1. Identify that the mean (or average) of a set of values is calculated by summing all the observed values and then dividing by the number of observations.
2. Let the observed values be \( x_1, x_2, \ldots, x_{25} \). The sum of these values is expressed as \( \sum_{i=1}^{25} x_i \).
3. Since there are 25 observations, the mean is found by dividing the total sum by 25, not 24, because the divisor must equal the number of data points.
4. Write the formula for the mean as: \[ \bar{x} = \frac{\sum_{i=1}^{25} x_i}{25} \] where \( \bar{x} \) represents the sample mean.
5. Understand that other options like dividing by 24, selecting the largest value, or taking the 25th root of the product correspond to different statistics (like biased divisor, maximum value, or geometric mean) and are not correct for calculating the arithmetic mean in this context.