The Normal Curve & Z-Scores

what is a normal curve?

a theoretical, symmetrical, bell-shaped distribution with one peak

in a perfectly normal distribution:

• mean = median = mode

• more scores cluster around the mean

• fewer scores occur far from the mean

the normal curve provides a model for estimating how common or unusual a score is

what are characteristics of a normal curve?

• bell-shaped

• unimodal

• symmetrical

• unskewed

• most of the values are clustered around the mean

what is a z-score?

a standardized score for showing how many standard deviations a value lies above or below the mean

• positive z = above the mean

• negative z = below the mean

• z = 0 = exactly at the mean

how do you convert a z-score back to a raw score?

x = x̄ +(z)(s)

what does the area under a normal curve represent?

the proportion of observations within a particular region of the distribution

• every possible outcome belongs somewhere under the curve, so all the regions together account for the entire distribution

how do you find the probability of a score being below under a given value?

1. convert the raw score to a z-score

2. use the normal curve table to find the area below the z-score

3. express the area as a proportion or percentage

what can we find for every z-score?

• the proportion of area between the z-score and the mean → column b

• the percentage of area between 2 scores → must calculate

• the percentage of the area beyond a z-score

• beyond and above typically mean the same area for positive scores

• beyond and below typically refer to the same area for negative

how do we find the area below a positive score?

add 0.50 to the column b area

how do we find the area above a positive score?

use column c only

how do we find the area below a negative score?

look in column c and

for above a negative score add 0.50 to column c

how do we find the area between 2 scores?

add the area between each score (column b) and the mean

if it’s on the same side of the mean → use column b and subtract the smaller area from the larger

if it’s on the opposite sides of the mean → add the area between each score and the mean using column b

how do we find the area with a proportion?

• use column c for both above and below

• use the same rules for opposite and same sides of the mean using column b

• if it’s asking for between use column b

what happens when a variable is normally distributed?

• areas under the curve encompass the same percentage of scores

• 68% between ± 1 standard deviations

• 95% between ± 2 standard deviations

• 99% between ± 3 standard deviations

what does using the normal curve as a standard reference do?

• determine the area above, below, and between any scores in the distribution

• determine the probability of occurrence for any score above, below, and between

what should the interpretations look like?

• approximately x% of cases fall (between/below/above) a score of x

what are probabilities represented as?

• proportions and range from 0.00 to 1.00

• the higher the value, the greater the probability

  • we can find probabilities that are above, below, and between any scores

how do we find probabilities?

1) calculate z-score

2) find the z-score in column c

what do we use the normal curve to convert a distribution for?

its used to change scores to a different mean and different standard deviation

how do we use the normal curve to convert a distribution?

1. compute the z-score for each score in the distribution using the formula

2. for each z-score computed, multiply it by the new standard deviation and add the new mean

   1. Z-score x s₁ + mean₁ → s₁ and mean ₁ refer to the standard deviation and mean in the new distribution

3. every score will be adjusted so that the mean in the new distribution will be the new one

• extreme scores have larger deviations

• scores that are closer to the mean will have smaller deviations

• extreme scores move closer to the mean in the new distribution

what are the effects of moving one distribution to another?

• all scores will be adjusted upward or downward relative to the new mean and new standard deviation

• all scores will be adjusted relative to their deviation from the mean

• when the standard deviation in the new distribution is increased, scores are pushed away from the mean

• when the standard deviation in the new distribution is decreased, scores move closer to the mean