Final answer:
Variance in the frequency domain is calculated by squaring the deviation of each data value from the mean, multiplying by the frequency, summing these products together, and dividing by the number of observations minus one. For probability distributions, variance considers probabilities as frequencies, using a similar sum of squared deviations weighted by probability.
Step-by-step explanation:
To calculate variance in the frequency domain for a given data set, one must consider the frequency of occurrence of each data value. Variance is a statistical measure that calculates the dispersion of the data points in a data set. To compute the variance, we start by finding the deviation of each data value from the mean. This deviation is squared to ensure all resulting values are positive, as the direction of deviation is not important for variance calculation – only the magnitude of deviation is. The squared deviations are then multiplied by their respective frequencies, and these products are summed together, giving a weighted sum of squared deviations.
The final value is divided by the number of observations minus one, which is the degrees of freedom for a sample variance calculation. This process is encapsulated in the formula:
Variance (s²) = Σ(frequency * deviation²) / (total number of data values - 1)
It is also worth noting that variance of a probability distribution can also be calculated in the frequency domain by considering probabilities as frequencies. The formula for this is given by:
o² = Σ (x − μ)² P(x)
where o² represents the variance, x is the random variable, μ is the mean of the distribution, P(x) represents the probabilities, and Σ denotes the summation over all possible values of x.