Final answer:
Variance and standard deviation are measures of the spread of data in a probability distribution, both being zero or positive. Variance is the average of squared deviations from the mean, while standard deviation is the square root of variance, reflecting how data values deviate from the mean.
Step-by-step explanation:
The variance and standard deviation of a discrete random variable probability distribution both measure the spread or dispersion of the data. The variance, often symbolized as ², is the mean of the squared deviations from the mean. The standard deviation, denoted as s for sample standard deviation or σ for population standard deviation, is equal to the square root of the variance and indicates how far data values are from their mean. It is important to note that both standard deviation and variance are always either zero or positive. When the standard deviation is zero, all data values are the same; when it is greater than zero, it indicates more spread in the data.
To calculate the variance (σ²) of a discrete probability distribution, we compute the sum of each outcome's squared deviation from the expected value, multiplied by its probability. The formula is σ² = Σ (x − μ)² P(x), where x represents values of the random variable X, μ is the mean (expected value) of X, P(x) represents the corresponding probability, and Σ denotes the summation of the products. The standard deviation (σ) is then found by taking the square root of this variance.