Variance and Standard Deviation Practice

Find the mean, population variance, and population standard deviation of the data set: 2, 4, 6, 8, 10.

Mean = 6. Squared deviations: (2-6)^2 = 16, (4-6)^2 = 4, (6-6)^2 = 0, (8-6)^2 = 4, (10-6)^2 = 16. Sum = 40. Variance = 40 / 5 = 8. Standard deviation = sqrt(8) ≈ 2.83.

Two runners have the same average race time of 50 seconds. Runner A has a standard deviation of 1.2 seconds, while Runner B has a standard deviation of 4.5 seconds. Which runner is more consistent?

Runner A is more consistent because a smaller standard deviation indicates less variability and tighter clustering around the mean.

A data set has a mean of 20 and a standard deviation of 4. If you add 5 to every data point, what are the new mean and the new standard deviation?

The new mean is 25 (20 + 5). The new standard deviation remains 4 because adding a constant shifts the data without changing its spread.

A data set has a mean of 15 and a standard deviation of 3. If you multiply every data point by 2, what are the new mean, standard deviation, and variance?

The new mean is 30 (15 * 2). The new standard deviation is 6 (3 * 2). The new variance is 36 (original variance was 9, and 9 * 2^2 = 36).

A frequency table lists values (x) and frequencies (f): x = 1 (f = 2), x = 3 (f = 5), x = 5 (f = 3). Calculate the mean and the population standard deviation.

Total n = 10. Sum of fx = (1*2) + (3*5) + (5*3) = 2 + 15 + 15 = 32. Mean = 3.2. Squared deviations times frequency: 2*(1-3.2)^2 = 9.68; 5*(3-3.2)^2 = 0.20; 3*(5-3.2)^2 = 9.72. Total sum of squares = 19.60. Variance = 19.60 / 10 = 1.96. Standard deviation = sqrt(1.96) = 1.4.