what is a z-score
- How far away from the mean the raw score lies measured in units of standard deviation.
- When you see these, think standard deviation, because that is what this is telling you, how many standard deviations away a number is from the mean
-Z-Scores do not go past 3
what do the different z scores mean (0, 1, -1)
- z = 0 corresponds to a score exactly at the mean
- z = +1.00 corresponds to a score that is one standard deviation above the mean
-z = -1.00 corresponds to a score that is one standard deviation below the mean
how do you calculate z-scores
- Subtract the Mean (M) from the raw score
- Divide this difference by the standard deviation (s)
the sign is very important
what is a T-score
Transforms the z-scores so that the M corresponds to T = 50 and the standard deviation corresponds to 10 T-score units
T = 10z + 50
Simplified
- The T-score is simply a linear transformation of the z-score
- It is used in many psychological tests, including the MMPI
How do we obtain the t-score
To obtain the T-score we simply multiply the z-score by 10 and then add 50
- To find out how many standard deviation units away from the mean a given T-score falls, reverse the formula and calculate the z-score from the T-score
- z = T – 50 / 10
what does a t-score of 50 correspond to a raw score that is exactly at the mean
- 60 = one standard deviation above the mean
- 40 = one standard deviation below the mean
How many standard deviations away from the mean is a T-score of 73?
73-50 = 23 / 10 = 2.3
How to calculate deviation IQ scores (standard scores)
M = 100
SD = 15
SS = 15z + 100
(always rounded)
z = -.75, SS = (15)(-.75)+100 = -11.25+100 = 88.75 rounded to 89.
how to you convert standard scores to z-scores
reverse the formula
z = SS – 100 / 15
general forumlas/similarities
T = 10z + 50
SS = 15z + 100
Y = Az + B
Where A will be the standard deviation of the new distribution and B will be the mean of the new distribution
converting a standard score to a t-score
STEP 1. Convert the SS to a z-score
z = (SS-100)/15 = (120-100)/15 = 20/15 = 1.33
STEP 2. Convert this z-score to a T-score
T = 10z+50 = (10)(1.33)+50 = 13.3 + 50 = 63.3 rounded to 63.
what is normalizing scores?
- All of the transformations we’ve done so far (T, SS) are LINEAR transformations that preserve the shape of the original distribution
- It is possible to take a non-normal distribution and normalize the scores, i.e., transform the original distribution into one that more closely resembles a normal distribution.
steps to normalize scores
1) Step One. Obtain the percentile rank that corresponds to each score
- Use the frequency distribution of scores
2) Step Two. Consult a Table of Normal Frequencies (see Appendix)
- Find the z-score that corresponds to the raw score’s percentile rank
- This z-score is the normalized z-score
- We can use this z-score to compute a normalized T-score using the standard formula
are normalized T-scores and linear T-scores different?
yes
- Since M=80, sd=12, a raw score of 86 corresponds to a linear z-score of
- (86-80)/12 = +.50, which corresponds to a linear T-score of 10(.50) + 50 = 55
advantages and disadvantages of normalizing scores
- Advantage: Use properties of normal curve to interpret standard scores
- Disadvantage: Losing information about the shape of the original distribution
how to use a table when the percentile rank is below 50
- Step One. Subtract the PR from 100 (=60)
- Step Two. Look this value up in the table. The z-score that corresponds to this is .25
- Step Three. Put a negative sign in front of this z-score = -.25
