We’re skipping section 4.1 on percentile ranks – conceptually, it’s very similar to the formula used for calculating the median when there is replication of the middle value.

 

Standard Scores (4.2)

 

-     Standard Scores provide a method for interpreting single scores within the set or distribution

-     Chapters 2 and 3 focused on statistical indices for describing an entire set of scores

 

-     Standard Scores can tell us how much different one person’s score is from another’s.

 

-     A standard score is computed by taking into account how far a given score deviates from the mean.

-     The deviation from the mean is expressed in terms of standard deviation units

-     This is done by computing: standard score =

-     This is the deviation score (for that specific case) divided by the standard deviation for the distribution

-     Standard score tells you the number of standard deviation units above or below the mean a particular score is

 

EXAMPLE:  Back to the set of scores for how old you think I am.

 

Score (X) – Raw Scores

23                               We previously computed

24

24                               M = 25.67

24                               SD = 2.32

25

25

25

25

26

27

28

32

 

-     What is the standard score for the raw score of 27?

Standard score =   (.57 SD’s above the mean)

 

-     What is the standard score for the raw score of 23?

Stand. score =  (1.15 SD’s below the mean)

 

-     What is the standard score for the raw score of 28?

Stand. Score =     (1 SD above the mean)

 

Properties of Standard Scores

-     Sign

-     Negative – score is below the mean

-     Zero – score is equal to the mean

-     Positive – score is above the mean

 

-     The mean of any set of standard scores will always be zero ()

 

-     The variance and SD of any set of standard scores will always be 1.00

**If you turned all your raw scores into standard scores, then calculated , , and , then , and

 

Uses of Standard Scores

-     Standard scores allow us to make comparisons between scores from distributions that have different means and standard deviations

**  Remember, when we convert to standard scores, M = 0 and SD = 1.0, so it is possible to compare two scores that were originally from two different distributions.

 

EXAMPLE: You give two tests.

 

Verbal Ability                                                   Quantitative Ability

100 questions                                                   100 questions

X = 85                                                             X= 95

M = 65                                                                        M = 80

SD = 8.00                                                        SD = 13.00

std. score =                  std. score =

 

-     Looking at the raw scores, it looks like this person did better on the Quantitative Ability test.

-     But when take into account how others did on these tests, this person actually did better on the verbal ability test

 

Standard Scores and the Normal Distribution (4.3)

 

-     Normal Distribution (Chapter 2: 2.9)

-     Normal distributions have the following properties:

-     Are bell shaped

-     Are symmetrical around the mean

-     Have Mean = Median = Mode

 

-     There is a different normal distribution for every unique combination of M and SD, but all have the 3 properties above

 

-     When convert all scores to standard scores, there are some known properties of standard scores and the normal distribution.

-     In any normal distribution, the proportion of scores falling above or below a given standard score is always the same.

-     Example: 50% of the scores always occur above or below a standard score of zero (i.e., the raw score equals the mean) for any normal distribution

-     The proportion of scores that occur between two specific standard scores is the same for all normal distributions

-     Example: The same proportion of scores will occur between a standard score of 1 and a standard score of 2 for all normal distributions.

 

-     Discussion of Figure 4.1 – p. 110

-     .3413, or about 34% of the scores fall between standard scores of 0 and 1(or 0 and -1)

-     .6826, or about 68% of scores fall between standard scores of –1 and +1

-     .9544, or about 95% of scores fall between –2 and +2

-     .9974, or about 99% of scores fall between –3 and +3

 

-     When we are talking about standard scores in a normal distribution, we call the standard scores z-scores

-     Is computed the same way as before:

 

-     Appendix B (p. 562-571) gives proportions falling under more specific proportions of the normal curve than are presented in Figure 4.1 (for many more z scores than the whole numbers listed in Figure 4.1)

 

-     This table gives us the following information:

-     Column 1 – absolute value of z scores, starting w/.00 through 4.00

-     Column 2 – the proportion of scores falling between any positive z score and its negative value

-     Example: Recall from Fig. 4.1, the proportion of scores between z’s of –1 and +1 is .6826 or 68.26%

-     Example: look at z=1.96 – the proportion of scores between z’s of –1.96 and +1.96 is .9500 or 95%

-     Column 3 – the proportion of scores falling above a positive z or below a negative z

-     Example: look at z = 1.96 --- .0250 or 2.5% of scores fall above +1.96 and 2.5% fall below –1.96

-     Column 4 – the proportion of scores above its positive value and below its negative value (in other words, Column 3 multiplied by 2)

-     Example: look again at z = 1.96 --- .0500 or 5% of scores fall above +1.96 and below –1.96 combined

-     Column 5 – the proportion of scores between any z score and the mean

-     Example: look again at z = 1.96 --- .4750 or 47.5% of the scores fall between zero and +1.96 (or between 0 and –1.96)

-     Notice that column 2 shows these two proportions combined -- .9500

 

-     Many Normal Distribution tables only give you three columns (1, 3 & 5), but because of the symmetry of the normal distribution, these are the only columns that you NEED.

 

-     Table is used to find proportions of scores in a given range under the normal curve.  This is used when we have converted our raw scores to z scores (standard scores)

-     If we can assume a set of scores is normally distributed, we can use the table to find the percentage of scores in a given portion of the normal distribution.

 

-     The z score of +/- 1.96 turns out to be very important in psychological research

-     We will see this importance when we get into inferential statistics

-     Know that 95% of scores fall between z’s of –1.96 and +1.96

 

-     How do we use Appendix B for an actual problem?

 

-     For example, assume that verbal GRE scores approximate a normal curve with a mean of 430 and a standard deviation of 100.

 

-     What proportion of scores are greater than 500?

a.   Convert to z score: (500 – 430)/100 = .70

b.   Go to Appendix B – Column 3

c.   .2420 (or 24.2%) of the scores are greater than 500

 

-     What proportion of scores are lower than 445?

a.   Convert to z score: (445 – 430)/100 = .15

b.   Go to Appendix B – No one column will answer this!

c.   Column 5: .0596 (proportion between .15 and mean)

d.   We know that .5000 of scores below the mean

e.   .0509 + .5000 = .5509

f.    .5509 (or 55.09%) of scores are lower than 445

 

-     What proportion of scores are between 450 and 470?

a.   Convert 450 to z score: (450-430)/100 = .20

b.   Convert 470 to z score: (470-430)/100 = .40

c.   Go to Appendix B – Great, Now what?

d.   Column 3 for .20: .4207; Column 3 for .40: .3446

e.   Subtract smaller proportion from larger proportion: .0761

f.    .0761 (or 7.61%) of scores between 450 and 470

-     What proportion of scores are between 400 and 450?

a.   Convert 400 to z score: (400-430)/100 = -.30

b.   Convert 450 to z score: .20

c.   Go to Appendix B – But this isn’t the same as above!

d.   Column 5 for -.30: .1179 (remember, Column A is the absolute value)

e.   Column 5 for .20: .0793

f.    Add proportions together (because on either side of mean)

g.   .1972 (or 19.72%) of scores between 400 and 450

 

-     What proportion of scores are less than 400 and more than 500?

a.   Convert 400 to z score: -.30

b.   Convert 500 to z score: .70

c.   Go to Appendix B – This is getting easier, but…

d.   Column 3 for -.30: .3821; Column 3 for .70: .2420

e.   Add proportions together

f.    .6241 (or 62.41%) of scores are less than 400 and more than 500

 

-     Now you see why many tables only have columns 1, 3 & 5 – rarely do you actually use columns 2 & 4 and you can easily get that information by doubling the values of column 3 (equals column 4) and doubling the values of column 5 (equals column 2)

 

-     Remember: It is often best to sketch a normal distribution and then shade in the section you are trying to find.  This will help you determine the computations you need to do in order to answer the question.

 

-     Remember:  To convert proportions into percentages, simply multiply by 100.  Often easier to talk in terms of percentages than proportions.

 

Standard Scores and the Shape of the Distribution (4.4)

-     The process of standardizing scores does not change the fundamental shape of the distribution

-     a positively skewed set of scores will remain positively skewed when standardized

-     a platykurtic set of scores will remain platykurtic

 

-     Standardization affects the central tendency and variability of the scores but not the skewness or kurtosis of scores