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Z-scores Z-score or standard score A statistical techniques that uses the mean and the standard deviation to transform each score (X) into a zscore Why z-scores are useful? Z-scores and location in a distribution The sign of the z-score (+ or –) The numerical value corresponds to the number of standard deviations between X and the mean The relationship between z-scores and locations in a distribution Transforming back and forth between X and z The basic z-score definition is usually sufficient to complete most z-score transformations. However, the definition can be written in mathematical notation to create a formula for computing the z-score for any value of X. X– μ Deviation score z = ──── Standard deviation σ Example If = 95 and = 16 then a score of 124 has a z-score of Example For a population with =100 and = 8, find the z-score for each of the following 1. 2. X= 84 X=104 What if we want to find out what someone’s raw score was, when we know their z-score? Example: Distribution of exam scores has a mean of 70, and a standard deviation of 12. If an individual has a z-score of +1.00, then would score did they get on the exam? Another example … If an individual’s z-score is -1.75, then what score did they get on the exam? Again using a mean of 70, and a standard deviation of 12. Transforming back and forth between X and z (cont.) So, the terms in the formula can be regrouped to create an equation for computing the value of X corresponding to any specific z-score. X = μ + zσ X=+z So if =80 and = 12 what X value corresponds to a z-score of -0.75? X=+z =80 = 12 Z-scores +0.80 -2.34 +1.76 -0.03 X Distribution of z-scores shape will be the same as the original distribution z-score mean will always equal 0 standard deviation will always be 1 Using z-scores to make comparisons Example: You got a grade of 70 in Geography and 64 in Chemistry. In which class did you do better? Z-scores and Locations The fact that z-scores identify exact locations within a distribution means that z-scores can be used as descriptive statistics and as inferential statistics. As descriptive statistics, z-scores describe exactly where each individual is located. As inferential statistics, z-scores determine whether a specific sample is representative of its population, or is extreme and unrepresentative. z-Scores and Samples It is also possible to calculate z-scores for samples. z-Scores and Samples Thus, for a score from a sample, X–M z = ───── s Using z-scores to standardize a sample also has the same effect as standardizing a population.