z-score

   DERIVED SCORES – Z

One important and classic type of standard score is the           Z score. The mean of a z distribution is 0.00 and the value of one standard deviation is 1.00. Z scores are the initial step in converting any raw score into any standard score.

STANDARD SCORE 

                                                Standard score is a type of score whose mean and standard deviation are known or given.

Z SCORE

                        The first standard score whose mean is zero and whose standard deviation is 1.0.

Z DISTRIBUTION

                                         A distribution of scores according to frequency that is perfectly normally distributed based on Z scores.

Most statisticians convert their raw scores into Z scores and then into some other standard scores because Z scores have two unfortunate characteristics.

One is that a Z score of 0 has bad connotations. Eg, if a person takes a major test at school and he/she comes home and reports getting a zero. People might automatically assume that the person did not do very well at all, yet if the person is reporting a Z score, that person scored right at the mean. Z scores, thus, have a strong chance of being misinterpreted by people without statistical training.

A second characteristic with Z scores is that they involve the use of negative numbers .statisticians and non- statisticians have a natural aversion to negative numbers. Negative numbers are more difficult to deal with mathematically than positive numbers.

In developing standard scores, the mean and standard deviation for a distribution are computed.

The deviations above the mean are indicated by a plus sign, the ones below the mean are indicated by a minus sign.

Despite the mathematical and psychological problems presented by negative Z scores, the Z distribution lies at the heart of inferential statistics. This is because statisticians use the Z distribution to test experimental hypotheses, such as a drug is effective or not.

CALCULATING Z SCORES

                                                            In order to change scores into Z scores, the mean and the standard deviation of the scores must be known. If we have the mean and standard deviation for a set of scores, then any individual number can be converted to a Z score by the following formula:

                            

Z = (X – M)/ S

• X =   individual score in the set of original numbers.
• M = the mean of the original set of numbers.
• S = the standard deviation for the original set of numbers.
• Z =        the standard score.

Even other standard scores can be converted into Z scores or vice versa.

            It is important to remember that the Z is a standard score whose distribution is normally distributed. When the raw scores from any distribution are converted to Z scores, the resulting distribution will approximate the normal distribution. This is because the Z distribution is a perfectly normal distribution.

            By converting from raw scores to Z scores, interpretations can then be made as if the scores are normally distributed. The transformations from raw scores to Z scores, in most cases, will be appropriate.

PRACTICE ON CONVERTING RAW DATA INTO Z SCORES

Convert the following raw scores , obtained by 10 students on a psychology exam , into Z scores

67 , 74 , 77 , 81 , 85 , 89 , 92 , 93 , 94 , 99

Z= X – M /S

First obtain the mean and standard deviation

             Mean     =  ∑X / n

                             =851/10

                             = 85.1

          S.D             =

                             =10.17

Thus for the raw score of 67

                             =( 67 – 85.1) /10.17

                             =1.78

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REFERENCE

                             Frederick L, Coolidge,2000. Statistics a Gentle Introduction.New Delhi;The Cromwell Press Ltd.

                             S R Sharma,1994.Statistical Methods in Educational Research . New Delhi;Anmol Publications Pvt Ltd.