Question

Average Household Size for Owner-Occupied Units Table P.11 in Exercise P.82 gives the probability function for the random variable giving the household size for an owner-occupied housing unit in the US. (a) Find the mean household size. (b) Find the standard deviation for household size.

Short answer

The mean of the household size is calculated by summing up the products of each household size and its corresponding probability. To calculate the standard deviation, first calculate the variance, which is equal to the difference between the mean of the squares of the sizes and the square of the mean size. The standard deviation is the square root of this calculated variance.

Step-by-step solution

Calculate Mean

The mean or expected value of a random variable \(X\) for a discrete probability distribution is calculated using the formula: \[ E(X) = \sum [x \cdot P(X = x)] \] where \(x\) represents each outcome and \(P(X = x)\) is the probability associated with that outcome. Sum up all the products of each household size multiplied by its corresponding probability.

Calculation of X^2

The next step involves calculating the square of the values of \(x\). Multiply each outcome (the household size) by itself.

Calculation of E(X^2)

Now, calculate the expected value of \(X^2\), similar to how the mean was calculated. Take each squared outcome, multiply it by its corresponding probability, and sum up these products. This gives you \(E(X^2)\).

Calculation of Variance

The Variance of a random variable \(X\) is calculated by the formula: \[ Var(X) = E(X^2) - [E(X)]^2 \] Substitute the values calculated in steps 1 and 3 into this formula to find the variance.

Calculation of Standard Deviation

Finally, to find the standard deviation, take the square root of the calculated variance. This is because the standard deviation is defined as the square root of variance. This gives the required standard deviation for household size.

Key concepts

Mean (Expected Value)

Understanding the mean, or expected value, of a probability distribution is akin to identifying the central point around which the values of a random variable are clustered. In layman terms, it represents the average outcome if an experiment is repeated multiple times. For a random variable, such as household size in owner-occupied units in the US, the mean is calculated by multiplying each possible size by the probability of that size occurring, then summing all these products together.

This process, detailed in Step 1 of the exercise, highlights how the mean conveys a picture of a typical outcome within the distribution, making it a foundational concept in the study of probability. As a way to enhance understanding, it's useful to think of it as a balancing point; if the distribution was a physical object, the mean is where it would balance perfectly.

Variance

Variance is a measure of how spread out the numbers in a distribution are around the mean. In simpler terms, it tells us how much, on average, the outcomes of a random variable differ from the mean or expected value. You can envision each data point tugging the mean in different directions; variance measures the intensity of this tug-of-war.

To compute the variance, as described in Steps 2 to 4 of the exercise, you start with finding the square of each outcome to magnify the influence of larger deviations. You then calculate the mean of these squared differences, known as the expected value of the square of the random variable, and subtract the square of the mean of the random variable. A key take-home point is that high variance indicates a wide spread of data points, while low variance indicates they are closely bunched together around the mean.

Standard Deviation

Standard deviation is a highly relevant concept as it paints a picture of the amount of variability or dispersion in a set of data from a probability distribution. If variance is the average squared deviation from the mean, then standard deviation is the average deviation itself. It is the square root of variance, a step crystallized in Step 5 of the exercise.

The beauty of standard deviation lies in its ability to be interpreted in the same units as the original data, making it a more intuitive measure of spread compared to variance. For instance, when considering household sizes, standard deviation would tell us, in number of people, how much the sizes tend to vary from the average household size. Frequent use of standard deviation in reporting has made it a key stalwart in summarizing data distributions, and it aids in understanding the probability of deviations from the expected value.