Quiz-Variance of a Discrete Random Variable

What does the variance of a random variable measure?

• The spread or dispersion of the random variable's values around the mean

• Probability of the random variable
• Standard deviation of the random variable
• Average value of the random variable

Answer explanation

Variance quantifies how much the values of a random variable deviate from the mean.

How is the variance of a random variable calculated?

• By multiplying the mean by the standard deviation
• By taking the square root of the sum of squared differences
• By summing the squares of deviations from the mean and then dividing by the number of observations

• By dividing the sum of all values by the number of values

Answer explanation

By summing the squares of deviations from the mean and then dividing by the number of observations

A fair coin is tossed 10 times. What is the variance of the number of heads obtained?

• 4.0
• 3.0
• 1.0
• 2.5

Answer explanation

For each toss, the random variable can take values 0 or 1 (representing tails and heads, respectively), with probabilities 0.5 each. The mean is 10 0.5 = 5. The variance is calculated as (0^2 0.5 + 1^2 0.5) / 10 = 0.5 / 10 = 0.05 10 = 2.5.

In a class of 30 students, the scores on a test have a mean of 75 and a variance of 100. What is the standard deviation of the scores?

• 15
• 10
• 5
• 20

Answer explanation

The standard deviation is the square root of the variance. So, sqrt(100) = 10.

Which of the following statements about variance is true?

• Variance can be negative.
• Variance is always an integer.
• Variance is not affected by outliers.
• Variance is always non-negative.

Answer explanation

Variance represents the average of the squared differences from the mean, so it cannot be negative. It measures the spread of values around the mean and is not necessarily equal to the mean.

What does the standard deviation of a random variable measure?

• Number of observations in the dataset
• Dispersion or spread of values around the mean
• Average value of the random variable
• Probability of the random variable

Answer explanation

Standard deviation quantifies the amount of variation or dispersion of a set of values. It measures how much the values deviate from the mean.

How is the standard deviation of a random variable related to its variance?

• The standard deviation is the square root of the variance.
• The standard deviation is the sum of the variance and the mean.
• The standard deviation is the reciprocal of the variance.
• The standard deviation is the product of the variance and the mean.

Answer explanation

Standard deviation is calculated as the square root of the variance. It provides a measure of the average distance of the values from the mean.

A fair six-sided die is rolled 12 times. What is the standard deviation of the number obtained?

• 2.45

• 0.9876
• 1.2345
• 1.7078

Answer explanation

The mean of rolling a fair six-sided die is (1+2+3+4+5+6)/6 = 3.5. The variance is calculated as [(1-3.5)^2 + (2-3.5)^2 + ... + (6-3.5)^2] / 6 = 35/12. Therefore, the standard deviation is sqrt(35/12) ≈ 2.45.

In a class of 40 students, the scores on a test have a mean of 85 and a standard deviation of 10. What is the variance of the scores?

• 75
• 110
• 95
• 100

Answer explanation

Since standard deviation is the square root of variance, to find the variance, we square the standard deviation. So, 10^2 = 100.

Which of the following statements about standard deviation is true?

• Standard deviation is always non-negative.

• Standard deviation measures the central tendency of a dataset
• Standard deviation is always equal to the mean of a dataset
• Standard deviation is not affected by outliers in a dataset

Answer explanation

Standard deviation is a measure of dispersion and cannot be negative. It is always non-negative, representing the spread of values around the mean. It is not necessarily equal to the mean and is affected by the spread of values around the mean.