prepaid by: Rajyagor Bhargav Z-Score Defined The number of standard deviations a raw score (individual score) deviates from the mean BIC prepaid by: Rajyagor Bhargav

BIC prepaid by: Rajyagor Bhargav Computing Z-Score ¯ Xi - X s where: Zi= Z-score for a value of i = number of standard deviations a raw score (X-score) deviates from the mean X= the mean of X s= the sample standard deviation Zi = ¯ BIC prepaid by: Rajyagor Bhargav

BIC prepaid by: Rajyagor Bhargav Direction of a Z-score The sign of any Z-score indicates the direction of a score: whether that observation fell above the mean (the positive direction) or below the mean (the negative direction) If a raw score is below the mean, the z-score will be negative, and vice versa BIC prepaid by: Rajyagor Bhargav

Comparing variables with very different observed units of measure Example of comparing an SAT score to an ACT score Mary’s ACT score is 26. Jason’s SAT score is 900. Who did better? The mean SAT score is 1000 with a standard deviation of 100 SAT points. The mean ACT score is 22 with a standard deviation of 2 ACT points. BIC prepaid by: Rajyagor Bhargav

Let’s find the z-scores Jason: 900-1000 100 Mary: 26-22 2 From these findings, we gather that Jason’s score is 1 standard deviation below the mean SAT score and Mary’s score is 2 standard deviations above the mean ACT score. Therefore, Mary’s score is relatively better. Zx = = -1 +2 BIC prepaid by: Rajyagor Bhargav

Z-scores and the normal curve SD SD SD SD SD SD SD SD SD 68% 95% BIC prepaid by: Rajyagor Bhargav 99%

Interpreting the graph For any normally distributed variable: 50% of the scores fall above the mean and 50% fall below. Approximately 68% of the scores fall within plus and minus 1 Z-score from the mean. Approximately 95% of the scores fall within plus and minus 2 Z-scores from the mean. 99.7% of the scores fall within plus and minus 3 Z-scores from the mean. BIC prepaid by: Rajyagor Bhargav

BIC prepaid by: Rajyagor Bhargav Example Suppose a student is applying to various law schools and wishes to gain an idea of what his GPA and LSAT scores will need to be in order to be admitted. Assume the scores are normally distributed The mean GPA is a 3.0 with a standard deviation of .2 The mean LSAT score is a 155 with a standard deviation of 7 BIC prepaid by: Rajyagor Bhargav

BIC prepaid by: Rajyagor Bhargav GPA SD SD SD SD SD SD SD 2.4 2.6 2.8 3.0 3.2 3.4 3.6 68% 95% BIC prepaid by: Rajyagor Bhargav 99%

BIC prepaid by: Rajyagor Bhargav LSAT Scores SD SD SD SD SD SD SD 134 141 148 155 162 169 176 68% 95% BIC prepaid by: Rajyagor Bhargav 99%

BIC prepaid by: Rajyagor Bhargav What we’ve learned The more positive a z-score is, the more competitive the applicant’s scores are. The top 16% for GPA is from a 3.2 upwards; for LSAT score, from 162 upwards. The top 2.5% for GPA from a 3.2 upwards; for LSAT score, from 169 upwards. An LSAT score of 176 falls within the top 1%, as does a GPA of 3.6. Lesson: the z-score is a great tool for analyzing the range within which a certain percentage of a population’s scores will fall. BIC prepaid by: Rajyagor Bhargav

BIC prepaid by: Rajyagor Bhargav Conclusions Z-score is defined as the number of standard deviations from the mean. Z-score is useful in comparing variables with very different observed units of measure. Z-score allows for precise predictions to be made of how many of a population’s scores fall within a score range in a normal distribution. BIC prepaid by: Rajyagor Bhargav

BIC prepaid by: Rajyagor Bhargav Works Cited Ritchey, Ferris. The Statistical Imagination. New York: McGraw- Hill, 2000. Tushar Mehta Excel Page. <http://www.tushar- mehta.com/excel/charts/normal_dist ribution/> BIC prepaid by: Rajyagor Bhargav