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The mean value, often referred to as the expected value in probability distributions, acts as a central point around which all the data from the distribution are balanced. To calculate the mean value for the given scenario, we use the formula:

• Mean, denoted as \( E(X) = \sum_{i=1}^{n} x_i \cdot p(x_i) \)

This formula requires us to multiply each possible number of boxes \( x_i \) by its corresponding probability \( p(x_i) \), then sum all these products. In the case of the hardwood flooring company, this involves calculating:

• \( 1 \cdot 0.2 + 2 \cdot 0.4 + 3 \cdot 0.3 + 4 \cdot 0.1 = 2.3 \)

The mean value of 2.3 indicates that on average, a customer orders 2.3 boxes. Although one can't actually order a fraction of a box, the mean provides a useful measure of central tendency. It offers a single value representing the average or typical number of boxes customers tend to order from this company.

Variance tells us how much the values in a probability distribution spread out or deviate from the mean value. It is calculated as the expected value of the squared deviation of each value from the mean. In simpler terms, it indicates how far apart the values are from the average. The formula for variance is:

• \( Var(X) = E(X^2) - (E(X))^2 \)

First, we have to determine \( E(X^2) \), which involves squaring each possible value of \( x \) and multiplying by its probability:

• \( E(X^2) = 1^2 \cdot 0.2 + 2^2 \cdot 0.4 + 3^2 \cdot 0.3 + 4^2 \cdot 0.1 = 6.1 \)
• Then, subtract the square of the mean \((E(X))^2 \)
• \( Var(X) = 6.1 - 2.3^2 = 0.81 \)

A variance of 0.81 implies that the number of boxes ordered by customers typically deviates from the mean by an average squared amount of 0.81. A smaller variance indicates that the data points tend to be closer to the mean.

Standard deviation is a widely used measure that quantifies the amount of variation or spread in a set of values. In a probability distribution, it tells us, on average, how much the values differ from the mean. The standard deviation is found by taking the square root of the variance:

• \( SD(X) = \sqrt{Var(X)} \)
• For our example, \( SD(X) = \sqrt{0.81} = 0.9 \)

The result, 0.9, indicates that the number of boxes ordered by customers typically varies by about 0.9 boxes from the mean number of boxes. Unlike variance, the standard deviation shares the same unit as the data, which makes it easier to interpret in terms of the data itself. In summary, it's a practical measure of how spread out the orders are around the mean.