The Variance Var(X) - Geometric Distribution

Definition

The variance in a geometric distribution quantifies how the number of trials needed to get the first success varies around its expectation.
It measures the degree of spread around the expected value or mean of the distribution.
The higher the variance, the more the values are spread out, and the more unpredictable the results become.

For a geometric distribution with parameter p (probability of success on each trial), the variance Var(X) is typically denoted as Var(X) = (1-p) / p².
Here, p represents the probability of success on each trial, and p² signifies the square of this probability.

If the probability of success (p) increases, the variance decreases, meaning the results will be more concentrated around the mean. Conversely, if p decreases, Var(X) increases.
The variance provides the measure of the volatility of the trials. It’s used to estimate how standard the trials are from an average trend.
A small variance indicates that the data points tend to be very close to the mean, while a high variance indicates that the data points are spread out over a wider range.

Useful in situations where we are conducting a series of binary experiments (success/failure, yes/no), and we want to know how much the results can vary.
Commonly seen in fields like quality control, risk management, and statistical research.

Variance is highly sensitive to extreme values or outliers which can inflate the measure even if the rest of the data is closely clustered.
It doesn’t give us an immediate sense of how spread out our data is because it’s in squared units. To get a more intuitive measure of spread, we use the square root of variance, i.e., the standard deviation.