If the score to be standardized is larger then the mean deviation will be a

# If the score to be standardized is larger then the

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value. If the score to be ”standardized” is larger then the mean, the deviation will be a positive one and the value of the z-score will be positive; the reverse is also true. (2) the magnitude of the z -score communicates an observation’s relative distance to the mean, as compared to other data values. (3) we can use z -scores to standardize entire distributions. By converting each score in a distribution to a z -score we obtain a standardized distribution.Such a standardized distribution, regardless of its original values, will have a mean of 0.0 and a variance and standard deviation of 1.0. (4) they can be mathematically related to probabilities. If the standardized distribution is a normal distribution we can state the probability of occurrence of an observation with a given z -score. Lecture Number 3 Data Description October 19, 2016 74 / 99
Measures of Position - Standard Scores Example 12: A student scored 65 in a Biometry test that had a mean of 50 and a standard deviation of 10; she scored 30 in a Soil Science test with a mean of 25 and a standard deviation of 5. Compare her relative positions in the two tests. Solution for Example 12: First, find the z -scores. For Biometry the z -score is: z = x - ¯ x s = 65 - 50 10 = 1 . 5 For Soil Science, the z -score is: z = x - ¯ x s = 30 - 25 5 = 1 . 0 Since the z -score for Biometry is larger, her relative position in the class is higher than in the Soil Science class. Lecture Number 3 Data Description October 19, 2016 75 / 99
Measures of Position - Percentiles Percentiles divide a data set into 100 equal parts. Definition of Percentile The p th percentile of a set of n measurements arranged in order of magnitude is that value that has at most p % of the measurements below it and at most (100 - p )% above it. Percentiles are frequently used to describe the results of achievement, test scores and the ranking of a person in comparison to the rest of the people taking an examination. Note that percentiles are not the same as percentages! e.g.If a student gets 73% out of 100% in a Biometry test, there is no indication of his/her position with respect to the rest of the class. The computation of percentiles is illustrated on the next slide. Lecture Number 3 Data Description October 19, 2016 76 / 99
Measures of Position - Percentiles Each data value corresponds to a percentile for the percentage of the data values that are less than or equal to it. Let x (1) , x (2) , x (3) , ..., x ( n ) denote the ordered observations for a data set; that is, x (1) x (2) x (3) ... x ( n ) . The i th ordered observation, x ( i ) , corresponds to the 100( i - 0 . 5) / n percentile. Percentile = 100( i - 0 . 5) n (11) Alternative formula: Percentile = ( No . of values below x i )+0 . 5 Total No . of values * 100% These formulas are used in place of assigning the percentile 100 i / n so that we avoid assigning the 100th percentile to x ( n ) , which would imply that the largest possible data value in the population was observed in the data set, an unlikely happening.

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