$ ext{sigma}^2 = n imes p imes (1-p)$

Definition

$ ext{sigma}^2$ is the variance of a binomial distribution, which represents the spread or dispersion of the distribution. It is calculated as the product of the number of trials (n), the probability of success (p), and the probability of failure (1-p).

5 Must Know Facts For Your Next Test

1. The variance of a binomial distribution, $ ext{sigma}^2$, is a key parameter that describes the spread of the distribution around the mean.
2. The formula $ ext{sigma}^2 = n imes p imes (1-p)$ shows that the variance increases as the number of trials (n) increases, and as the probability of success (p) approaches 0.5.
3. A higher variance indicates a wider spread of the distribution, meaning the possible outcomes are more dispersed from the mean.
4. The standard deviation, $ ext{sigma}$, is the square root of the variance and represents the average deviation of the distribution from the mean.
5. Knowing the variance of a binomial distribution is important for calculating probabilities, confidence intervals, and making inferences about the underlying population.

Review Questions

• Explain how the formula $ ext{sigma}^2 = n imes p imes (1-p)$ relates to the characteristics of a binomial distribution.
  • The formula $ ext{sigma}^2 = n imes p imes (1-p)$ demonstrates that the variance of a binomial distribution is directly influenced by the number of trials (n) and the probability of success (p). As the number of trials increases, the variance also increases, indicating a wider spread of the distribution. Additionally, the variance is maximized when the probability of success is 0.5, as this represents the greatest amount of uncertainty or dispersion in the possible outcomes.
• Describe how the variance and standard deviation of a binomial distribution are related and how they can be used to make inferences about the distribution.
  • The variance, $ ext{sigma}^2$, and the standard deviation, $ ext{sigma}$, are closely related in a binomial distribution. The standard deviation is the square root of the variance, and it represents the average deviation of the distribution from the mean. Knowing the variance or standard deviation is important for calculating probabilities, constructing confidence intervals, and making inferences about the underlying population. For example, the empirical rule states that approximately 68% of the observations in a normal distribution fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations. This information can be used to assess the spread of a binomial distribution and make probabilistic statements about the likelihood of observing certain outcomes.
• Analyze how changes in the parameters n and p would affect the variance of a binomial distribution, and explain the practical implications of these changes.
  • The formula $ ext{sigma}^2 = n imes p imes (1-p)$ shows that the variance of a binomial distribution is directly influenced by the number of trials (n) and the probability of success (p). As the number of trials (n) increases, the variance also increases, indicating a wider spread of the distribution. This means that with more trials, the possible outcomes become more dispersed from the mean, making it harder to predict the exact number of successes. Conversely, as the probability of success (p) approaches 0 or 1, the variance decreases, resulting in a narrower distribution. This has practical implications, as it allows for more precise predictions and inferences about the likelihood of observing certain outcomes in a binomial experiment. Understanding how changes in n and p affect the variance is crucial for designing and analyzing binomial experiments, as it helps researchers determine the appropriate sample size and interpret the significance of their findings.

Related terms

Binomial Distribution:

A discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure).

Variance:

A measure of the spread or dispersion of a probability distribution, calculated as the average squared deviation from the mean.

Bernoulli Trial: A statistical experiment with only two possible outcomes, typically labeled as 'success' and 'failure', where the probability of success remains constant across all trials.