Question

Assume the following 5 scores represent a sample: 2,3,5, 5, 6. Transform these scores into z-scores.

a) Calculate the mean
b) Calculate the standard deviation
c) Calculate z-scores

Answer 1

To calculate the z-scores, first calculate the mean and standard deviation. Then, use the formula z = (x - mean) / standard deviation to calculate the z-score for each score. The z-scores for the given scores (2, 3, 5, 5, 6) are -0.77, -0.42, 0.28, 0.28, and 0.63 respectively.

Step-by-step explanation:

To calculate the z-scores for the given scores, we need to first calculate the mean and standard deviation.

a) Calculate the mean:

The mean is calculated by adding up all the scores and dividing by the total number of scores. For the given scores (2, 3, 5, 5, 6), the mean is:

Mean = (2 + 3 + 5 + 5 + 6) / 5 = 21 / 5 = 4.2

b) Calculate the standard deviation:

The standard deviation is a measure of the spread of the scores. We can calculate it using the formula:

Standard deviation = sqrt(1/n * sum((x - mean)^2))

For the given scores, the standard deviation is:

Standard deviation = sqrt(1/5 * ((2-4.2)^2 + (3-4.2)^2 + (5-4.2)^2 + (5-4.2)^2 + (6-4.2)^2)) = sqrt(8.16) ≈ 2.86

c) Calculate the z-scores:

The z-score for each score can be calculated using the formula:

z = (x - mean) / standard deviation

For the given scores, the z-scores are:

Z(2) = (2 - 4.2) / 2.86 ≈ -0.77

Z(3) = (3 - 4.2) / 2.86 ≈ -0.42

Z(5) = (5 - 4.2) / 2.86 ≈ 0.28

Z(5) = (5 - 4.2) / 2.86 ≈ 0.28

Z(6) = (6 - 4.2) / 2.86 ≈ 0.63