Measures of Relative Standing Percentiles Percentiles z-scores z-scores T-scores T-scores

Percentiles/Quantiles Percentiles and Quantiles Percentiles and Quantiles The sample k th percentile (P k ) is value such that k% of the observations in the sample are less than P k and (100 – k)% are greater E.g. 90 th percentile (P 90 ) is a a value such that 90% of the observations have smaller value and 10% of the observations are greater in value. Quantile is just another term for percentile, e.g. JMP refers to percentiles as quantiles.

Quartiles Quartiles are specific percentiles Quartiles are specific percentiles Q 1 = 1 st quartile = 25 th percentile Q 1 = 1 st quartile = 25 th percentile Q 2 = 2 nd quartile = 50 th percentile Q 2 = 2 nd quartile = 50 th percentile Q 3 = 3 rd quartile = 75 th percentile Q 3 = 3 rd quartile = 75 th percentile = Median

Boxplot Minimum = x (1) Q1Q1 Q3Q3 Median IQR = Interquartile Range which is the range of the middle 50% of the data Outliers Maximum = x (n)

Comparative Boxplots

Definition of z-score Population z-scoreSample z-score In either case, the z-score tells us how many standard deviations above (if z > 0) or below (if z < 0) the mean an observation is.

Interpretation of z-Scores If z = 0 an observation is at the mean. If z = 0 an observation is at the mean. If z > 0 the observation is above the mean in value, e.g. if z = 2.00 the observation is 2 SDs above the mean. If z > 0 the observation is above the mean in value, e.g. if z = 2.00 the observation is 2 SDs above the mean. If z < 0 the observation is below the mean in value, e.g. if z = the observation is 1 SD below the mean. If z < 0 the observation is below the mean in value, e.g. if z = the observation is 1 SD below the mean.

% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (z-scores) 0.1% 2.4% 13.5% z-score

The Empirical Rule (z-scores) Therefore for normally distributed data: 68% of observations have z-scores between 68% of observations have z-scores between and % of observations have z-scores between 95% of observations have z-scores between and of observations have z-scores between 99.7 of observations have z-scores between and 3.00

Outliers based on z-scores When we consider the empirical rule an observation with a z-score 2.00 might be characterized as a mild outlier. When we consider the empirical rule an observation with a z-score 2.00 might be characterized as a mild outlier. Any observation with a z-score 3.00 Any observation with a z-score 3.00 might be characterized as an extreme outlier. might be characterized as an extreme outlier.

Example: z-scores Q: Which is more extreme an infant with a gestational age of 31 weeks or one with a birth weight of 1950 grams? Calculate z-scores for each case. Gestational Age = 31 weeks Birthweight = 1950 grams Because the z-score associated with a gestational age of 31 weeks is smaller (more extreme) we conclude that it corresponds to more extreme infant.

Standardized Variables We can convert each observed value of a numeric variable to its associated z-score. This process is called standardization and the resulting variable is called the standardized variable. Note: When standardized the mean is 0 and standard deviation is 1!

T-Scores ~ Another “Standardization” Facts About T-scores Have a mean of 50. Have a mean of 50. Have a standard deviation of 10. Have a standard deviation of 10. May extend from 0 to 100. May extend from 0 to 100. It is unlikely that any T-score will be beyond 20 or 80  (i.e. 3 SD’s above and below the mean) It is unlikely that any T-score will be beyond 20 or 80  (i.e. 3 SD’s above and below the mean) Definition of T-Score

Empirical Rule: z- and T-scores 68% 95% 99.7%

T-Scores T-scores may be used in same way as z-scores, but may be preferred because: Only positive whole numbers are reported. Range from 0 to 100. However, they are sometimes confusing because 60 or above is good score, BUT not if we are taking a 100 point exam!