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The variance and standard deviation are two measures of dispersion (spread) of a random variable in a probability distribution. They help us understand how the data points are distributed around the mean. The variance of a discrete random variable X, denoted by \(Var(X) \), is calculated using the following formula: \[Var(X) = \sum{(x_i -\mu)^2 * P(x_i)}\] Where: - \(x_i\) are the possible values of the random variable X - \(\mu\) is the expected value (mean) of the random variable X - \(P(x_i)\) is the probability of each value \(x_i\) The standard deviation of the random variable X, denoted by \(SD(X)\), is the square root of the variance: \[SD(X) = \sqrt{Var(X)}\]

As seen from the formulas above, both the variance and standard deviation take into account the possible values of the random variable and their respective probabilities. By doing so, they measure the average distance or deviation of each data point from the mean. The main difference between these two measures of dispersion is that variance is calculated in squared units while the standard deviation is in the same unit as the mean. As a result, the interpretation of the standard deviation may be more intuitive, especially for practical applications, since it is in the same scale as the data. However, both measures capture the same underlying property of the distribution - the spread of the random variable. In this sense, we can say that they indeed measure the spread of a probability distribution.

The statement is true. Both the variance and the standard deviation of a random variable measure the spread of a probability distribution. The variance measures the average of the squared differences from the mean, while the standard deviation is the square root of the variance, making it more interpretable for practical applications. Nonetheless, both capture the degree of dispersion in the distribution, thus measuring the spread.