Z-Scores and Standard Normal Distribution

Question 1

What is the formula for transforming a raw score into a z-score?

Answer 1

Subtract the mean of the distribution from the score you want to transform and divide the result by the standard deviation of the distribution.

Raw-score - M / SD

Question 2

Define what a z-score is

Answer 2

A score that is expressed in standard deviation units

Question 3

What is a characteristic of the standard normal distribution?

Answer 3

Mean = 0

Standard deviation = 1

Question 4

When transforming raw scores into z-scores as a linear transformation, what happens to the distribution?

Answer 4

Distribution of z-scores has the same shape as the distribution of raw scores.

The same even though has been transformed.

Question 5

What is the bell shaped distribution of z-scores called?

Answer 5

Standard normal distribution

Question 6

In a normal distribution how many cases fall between the values of –1 and 1 standard deviation?

Answer 6

68%

Question 7

Linear transformation

Answer 7

Transforming a raw score into a z-score.

Every score is changed by multiplying or dividing by a constant, or adding or subtracting a constant, or both.

Relationships do not change.

Question 8

What do you do in order to find out the percentage of scores falling above a z-score of 0.84?

Answer 8

• Look at table
• Read first column and find 0.8
• Read along second column and find 0.4
• Gives a number of 0.200
• This is equal to 20%
• 20% of scores fall above

Question 9

What are the uses of z-scores?

Answer 9

• Knowing the percentile of a score on a variable
• Comparing a score on a variable with a score on another variable
• Knowing which score on a variable is at a specific percentile

Question 10

Using z-scores, how do you work out the percentile of a score on a variable?

Answer 10

Raw score - mean score/ SD

Then look at table

Question 11

Using z-scores, how do you compare a score on a variable with a score on another variable?

Answer 11

• Useful when have different units and want to compare
• Transform both scores into z-scores and compare.
• Formula Z = x - X / S

Question 12

Using z-scores, how do you know which score on a variable is at a specific percentile?

Answer 12

E.g. want to select the top 25% of students. How to obtain a raw-score when you know the z-score.

Formula = X = (z) (Sx) + X

• Use table
• Search for value which then gives you the z-score
• Apply formula which applies to the z-score