How to understanding Z-scores

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As a trainee or novice ESL teacher, you will eventually have to develop a basic understanding of statistics for the purpose of analysing your students’ examination/test results. One very important aspect of statistics that you should become familiar with is Z-scores.

In this short non-technical article you will learn how to calculate and interpret Z-scores. By definition, a Z-score (or standard score as it is sometimes called) may be considered as a measure of a value’s position vis-à-vis the average value in a group of values.

A Z-score can be calculated using the following formula: Assuming that all the ‘values’ relate to examination/test scores: ‘x’ represents the individual student’s score (raw score); ‘avrg’ is the class’s average (mean) score; and ‘stdev’ is the standard deviation of the class’s scores.

The stdev is a measure of the spread of the Z-scores from the avrg: each time you calculate a Z-score, you are actually calculating the number of stdevs the score lies away from the average. It should be noted that it will be necessary to calculate an individual Z-score for every student who has sat the examination/test: the Z-score calculations can be done easily on most pocket calculators and on some online sites.

Visit http://easycalculation.com/statistics/z-score-standard.php The following example illustrates the usefulness of Z-scores for comparative purposes: A student scored 84 in grammar test-A in which the average was 76 and the stdev was 10. In grammar test-B, in which the average was 82 and the stdev was 16, the student scored 90. In which test was the student’s relative standing higher? The Z-score for test-A was 0.8, and for test-B, it was 0.5; therefore, the student’s relative standing was better in test-A because the Z-score was 0.8 standard deviations above the average – whereas in test-B it was only 0.5. Without the use of Z-scores, the novice ESL teacher might incorrectly rush to assume that the student performed relatively better in test-B. Z-scores will also tell you exactly how relatively well a particular student has performed in a test/examination, e.g. a Z-score of –1.5 stdevs means that the student has scored well below the average.

In general, Z-scores that are stdevs (or more) beyond average are called outliers: very few students should get such Z-scores. Statistically speaking, approximately 68% of the Z-scores should lie within stdevs of the average. Finally, the following important point is worth noting: if x = avrg then Z = 0, so the average is equal to a Z-score of zero; however, a Z-score = 0 does not indicate that the student’s raw score is zero, e.g. if the average = 62% and the student scored 62%, then only the Z-score will equal zero.

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