III. Understanding Norms and Test Scores

     To understand norms and statistical assessment one first needs to understand standardization. Standardization is the process of testing a group of people to see the scores that are typically attained. With a standardized test, the participant can compare where that score fell compared to the standardization group's performance. With standardization the normative group must reflect the population for which the test was designed. The group's performance is the basis for the tests norms.


    Many major psychological measures are norm-based, meaning that the score for an individual is interpreted by comparing his/her score with the scores of a group of people who define the norms for the test. Sir Francis Galton developed the logic for norm-based testing in the mid 1800s.

    To organize and summarize data for normative purposes, begin by grouping the data into a frequency distribution. The information provided by frequency distributions can be presented graphically in the form of a bell-shaped normal distribution curve, as long as it approximates that symmetrical form.

    A group of scores can be summarized by a measure of central tendency. The most familiar of these measures is the arithmetic average, more technically known as the mean (M), and is found by adding all the scores (X) and dividing the sum by the total number of items (N), (M = X/N).

    Another measure of central tendency is the mode, or the most frequent score. In a frequency distribution, the mode is the midpoint of the class interval with the highest frequency. A third measure of central tendency is the median, or middle score of the distribution. For a normal distribution the median is equal to the mean; this is not true for every distribution.

    Further description of a set of test scores is given by measures of variability, or the extent of individual differences around the central tendency. A familiar way for reporting variability is in terms of the range between the highest and lowest score. However, a much more
serviceable measure of variability is the standard deviation
(symbolized by SD or s). The
standard deviation (SD) is an index of the width of a frequency

 distribution. The smaller the standard deviation, the closer the scores cluster around the mean score. The greater the standard deviation, the greater the differences between the scores and the mean. The standard deviation is computed by calculating the average squared distance of the scores from the mean and taking the square root of this value.


Method for Determining a Standard Deviation 
Step 1:
Place scores in order 
Participant #    Score for test #1
1.                              2
2.                              2
3.                              3
4.                              3
5.                              3
6.                              4
7.                              4
8.                              4
9.                              6
10.                            9


Step 2:
Find the mean, median, mode, and range 

Mean = M =  X/N = 40/10 = 4.00 
                   
Median= 3.5 (score in the middle position) 
Mode = 3 & 4 (the most frequently occurring score/s; in this case there are two modes)
Range =  9-2 = 7 (the largest score minus the smallest score) 


Step 3: a.
Subtract the mean from the individual scores (X - M)
Participant #      Test #1 mean difference (X - M)
    1.                     2 - 4 = -2
    2.                     2 - 4 = -2
    3.                     3 - 4 = -1
    4.                     3 - 4 = -1
    5.                     3 - 4 = -1
    6.                     4 - 4 = 0
    7.                     4 - 4 = 0
    8.                     4 - 4 = 0
    9.                     6 - 4 = 2
   10.                    9 - 4 = 5


b.
Square individual differences

-2 x -2 = 4
-2 x -2 = 4 
-1 x -1 = 1
-1 x -1 = 1
-1 x -1 = 1
0 x 0 = 0 
0 x 0 = 0 
0 x 0 = 0
2 x 2 = 4
5 x 5 = 25

Use the sum of squares (SS) defining formula to find variability 

Σ(X - M)2 
{in this example our answer is 40}
*remember: always work formulas inside to outside, and right to left!*


Step 4:
Find the variance by dividing the answer for the sum of squares by the number of scores. SS/N
SS=40 N=10
SS/N = 40/10 = 4


Step 5:
Find the standard deviation (SD).  ( SS/N)
SD = (SS/N) = 4 = 2


Within-Group Norms
   
Nearly all standardized tests provide some form of within-group norms. Within-group norms--which evaluate an individual's performance in terms of the most nearly comparable standardization group--have a uniform and clearly defined quantitative meaning, and can be used in most statistical analyses. The main types of within-group norms are percentiles (click on converting to percentiles and back) and standard scores. Percentile scores express an individual's relative position within the standardization group in terms of the percentage of persons whose scores fall below that of the individual.

Standard scores
 
    Standard scores express the distance between an individual's score and the group mean. If we equate a standard deviation to one, we can express a raw score as being x number of standard deviations above or below the mean. To do so we change our raw scores into a type standard score, i.e.,  z scores.

The equation for finding the z score is: 

X - M / SD (See step 3 under Correlation)

Correlation
    The term correlation simply refers to the degree to which two or more sets of data show a tendency to vary together. The strength of the positive (increase, increase) correlation coefficient can vary from .00 to 1.00.

Determining the Correlation Coefficient 
   
(example in the form of test-retest reliability)

Step 1
   
List scores. Find the mean (M) and Standard Deviation (SD) for each set of scores.

Participant #        1st Scores     2nd Scores

   1                            4                     6
   2                            6                     7
   3                            2                     3
   4                            2                     4
   5                            4                     2
   6                            4                     7
   7                            9                     9
   8                            3                     5
   9                            3                     6
   10                          3                     5

                        M  = 4               M 2  = 5.4

                        SD 1  = 2            SD 2 = 1.96

                        n 1 = 10                n 2= 10

 

Step 2: Find the z score for each test score using the formula:

(X - M)/ SD

About Z Scores

    Z scores are a type of standard score. The z score is useful when attempting to compare items from distributions with different means and standard deviations. The z score for a test score indicates how far and in what direction that test score is from its distribution's mean, expressed in units of its distribution's standard deviation. The z scores will have a mean of zero and a standard deviation of one.

Participant #    lst Test Z Score    2nd Test Z Score

    1                        0                             .31
    2                        1                             .82
    3                       -1                          -1.22
    4                       -1                            -.71
    5                        0                          -1.73
    6                        0                              .82
    7                        2.5                        1.84
    8                       -0.5                         -.20
    9                       -0.5                          .31
   10                      -0.5                        -.20

Step 3
   
Sum the scores from each test. Apply scores to the defining formula for the Pearson r:   

r = Σ ZxZy / N

 


Participant #     ZxZy
    1.                     0
    2.                       .82
    3.                     1.22
    4.                        .71
    5.                     0
    6.                     0 
    7.                      4.60
    8.                        .10
    9.                       -.16
    10.                      .10
ZxZy  =             7.39

 (ZxZy) / 10 = 0.74*

*This answer is the correlation coefficient

About Pearson r
    The Pearson r is the most commonly used measure of correlation, sometimes called the Pearson Product Moment correlation. It is simply the average of the sum of the Z score products and it measures the strength of linear relationship between two characteristics. The positive (increase, increase) correlation coefficient can range from 0.00 to 1.00. The closer to 1 the stronger the relationship. 

 

 

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