Chapter 5 Notes- Z-Scores

Z- Scores Raw Scores and Standing • raw scores do not give an accurate picture of the piace of a score in a sample of scores - eg., on exams 1-3 youscore 85, 70, and 566 → is your performance decreasing? Example Test Score •are your scores really getting worse? - meaning of zero → exam I was 85 above zero, exam 2 was 70 above zero - problem is that this assumes worst Score on test was zero • whatif tests differ in difficuty? - Zero would not mean the same thing → i.e., raw score on the tests are not comparable Purpose of 2- Scores • identify and describe location of every score in the distribution • Standardize an entire distribution • indexes te location of an observation in a distribution relative to otter distributions • it we know the M and SD of a distribution, then we can transform nem to a 2-score Z - Scores and Location in a Distribution • exact location is described by 2-score - Sign tells whether score is located above or below the mean - number tells distance between Score and mean in Standard deviation units 2- Score Equation for a Papulation • numerator is a deviation score - u • denominator expresses deviation in Standard z = deviation units, how far in Standard deviations • example - Jacob spoke to other children 8 times in an hour, the mean number of times Children Speak is 12, and the standard deviation is 4, Find Jacob's 2 Score. • determining a rau Score (x) from a 2-Score - x= M +20 • raw score example - a child had a 2 Score of 1.5 on the number of times spoken with anoter Child during an hour. How many times did that child Speak in an hour (M=12, 50 =4) : Computing 2- scores for Samples - • populations are most common Context for computing 2-scores • it is possible to compute 2-scores for sampres - indicates relative position of score in sample - indicates distance from sample mean • Sample distribution can be transformed into 2-scores - Same shape as original distribution - Same mean M aud Standard deviation s Learning Check a 2-score of 2= t 1.00 indicates a position in a distribution. a) above te mean by l point b) above the mean by a distance equal to l 50 c) below tie mean by l point d) belous ne mean by a distance equal to 150 decide if each of the following Statements is true/faise a negative 2-score always indicates a location below tre mean a score crose to tre mean nas a T/F 2- Score close to 1.00 Other Relationships Between z, X, the mean, and the Standard Deviation • a z-score establisnes a relationsnip between tre raw score, the mean, and te Standard deviation * for example: given mean, raw score, and 2-Score, you can compute standard deviation • another example: given standard deviation, raw score, and 2-Score, you can compute tre mean • 2= (x-m)/s • x= m+z (s) • m= x - 2(5) • S = (x-m)/z Using 2-Score to Standardize a Distribution • every X value can be transformed to a 2-score • Characteristics of z-score transformation: - Same Shape as original distribution - mean of 2-score dismibution is always o - Standard deviation is auways 1.00 • a z-Score distribution is caled a standardized distriloution 2- Scores for Making Comparisons • all z-scores are comparable to each other • Scores from different distributions can lol converted to 2-scores • 2-scores (Standardized Scores) allow the direct comparison of scores from two different distributions because they have been converted to the same scale Other Standardized Distributions Based on 2-Scores • proces of standardization is widely used Messies Golitie - SAT has M= 500 and 0 - 100 - 1a has M= 100 and o= 15 points 45-30 $2=1 •M - 100 • Standardizing a distribution has two steps - original raw scores transformed to 2-scores - the 2-scores are transformed to new X values so that he specific predetermined M and o are attained