Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.
Dogs The number of dogs per household in a neighborhood

Verified step by step guidance

1. Step 1: Understand the problem. You are given a probability distribution table for the number of dogs per household in a neighborhood. The goal is to calculate the mean, variance, and standard deviation of this distribution and interpret the results.
2. Step 2: Calculate the mean (expected value). Use the formula for the mean of a probability distribution: $E(X)\; =\; \Sigma (x\; \times \; P(x))$, where x represents the number of dogs and P(x) represents the probability of x dogs. Multiply each value of x by its corresponding probability and sum the results.
3. Step 3: Calculate the variance. Use the formula for variance: $Var(X)\; =\; \Sigma ((x\; -\; E(X))²\; \times \; P(x))$. First, subtract the mean (E(X)) from each value of x, square the result, multiply by the corresponding probability, and sum these values.
4. Step 4: Calculate the standard deviation. The standard deviation is the square root of the variance: $SD(X)\; =\; \surd Var(X)$. Take the square root of the variance calculated in the previous step.
5. Step 5: Interpret the results. The mean represents the average number of dogs per household in the neighborhood. The variance and standard deviation provide measures of how spread out the number of dogs is around the mean. A smaller standard deviation indicates that the number of dogs per household is more consistent, while a larger standard deviation indicates greater variability.

Key Concepts

Mean

The mean, or expected value, of a probability distribution is calculated by multiplying each outcome by its corresponding probability and summing these products. It represents the average outcome one would expect if the experiment were repeated many times. In this context, it provides insight into the typical number of dogs per household in the neighborhood.

Variance

Variance measures the spread of a probability distribution around its mean. It is calculated by taking the average of the squared differences between each outcome and the mean, weighted by their probabilities. A higher variance indicates greater variability in the number of dogs per household, while a lower variance suggests that the number of dogs is more consistently close to the mean.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the data. It indicates how much individual data points typically deviate from the mean. In this scenario, the standard deviation will help interpret the variability in the number of dogs per household, giving a clearer picture of how concentrated or spread out the data is.