Final answer:
The square of the standard deviation is called the variance. Variance (σ²) for a set of data is calculated using the mean (μ) of the data, values of the data (x), and their probabilities (P(x)), while the standard deviation is the square root of variance, which measures the spread of data around the mean.
Step-by-step explanation:
The square of the standard deviation is known as the variance. The formula to compute the variance (σ²) of a discrete random variable X involves squaring each deviation from its expected value, multiplying it by its probability, and then summing up all these products. Mathematically, it can be represented as σ² = Σ (x − μ)² P(x), where 'x' are the values of the random variable X, 'μ' is the mean of X, and P(x) symbolizes the probability of x occurring.
The standard deviation is the square root of the variance, which provides a measure of how spread out the values in a set of data are around the mean. For calculations involving a complete population, we use the symbol σ for standard deviation and σ² for variance, whereas for a sample, we use 's' for standard deviation and 's²' for variance. Variance is an important measure as it indicates the variability in a set of data, representing the mean of the squared deviations from the mean.
Variance has the unique characteristic of being a squared measure, which means that it does not share the same units as the data itself. To reconcile this issue and to understand the spread of the data in the same units as the data, the standard deviation is used, representing the square root of the variance.