Survey

Survey

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Chapter 5 DESCRIBING DATA WITH Z-SCORES AND THE NORMAL CURVE Going Forward Your goals in this chapter are to learn: • What a z-score is • What a z-distribution is and how it indicates a score’s relative standing • How the standard normal curve is used with z-scores to determine relative frequency, simple frequency, and percentile • What the sampling distribution of means is and what the standard error of the mean is • How to compute z-scores for sample means and then determine their relative frequency Understanding z-Scores A Frequency Distribution z-Scores • A z-score is a location on the distribution • A z-score automatically communicates its distance from the mean • A z-score indicates how far a raw score is above or below the mean when measured in standard deviations z-Score Formula The formula for transforming a raw score into a z-score is XX z SX Computing a Raw Score When a z-score and the associated S X and X are known, this information can be used to calculate the original raw score. The formula for this is X ( z )( S X ) X Using the z-Distribution to Interpret Scores A z-Distribution A z-distribution is the distribution produced by transforming all raw scores in the data into zscores. z-Distribution of Attractiveness Scores Characteristics of the z-Distribution 1. A z-distribution always has the same shape as the raw score distribution 2. The mean of any z-distribution is 0 3. The standard deviation of any z-distribution is 1 Using the z-Distribution to Compare Different Variables Comparison of Distributions Using the z-Distribution to Compute Relative Frequency Relative Frequency • The relative frequency of a particular z-score will be the same on all normal z-distributions • Relative frequency can be computed using the proportion of the total area under the curve The Standard Normal Curve The standard normal curve is a perfect normal z-distribution that serves as our model of any approximately normal z-distribution. Area Under the Standard Normal Curve Using the z-Table Using z-Scores to Describe Sample Means Sampling Distribution of Means A frequency distribution of all possible sample means occurring when an infinite number of samples of the same size N are selected from one raw score distribution is called the sampling distribution of means. Central Limit Theorem The central limit theorem tells us 1. A sampling distribution is always an approximately normal distribution 2. The mean of the sampling distribution equals the mean of the underlying raw score population used to create the sampling distribution 3. The standard deviation of the sampling distribution is mathematically related to the standard deviation of the raw score population Standard Error of the Mean The standard deviation of the sampling distribution of means is called the standard error of the mean. The formula for the true standard error of the mean is X X N z-Score Formula for a Sample Mean The formula for computing a z-score for a sample mean is z X X Example Using the following data set, what is the z-score for a raw score of 13? What is the raw score for a z-score of -2? 14 14 13 15 11 15 13 10 12 13 14 13 14 15 17 14 14 15 Example z-Score XX z SX We know (from prior chapters) and S X 1.56 for these data. X 13.67 13 13.67 z 0.43 1.56 Example Raw Score from a z-Score X ( z )( S X ) X We know (from prior chapters) and S X 1.56 for these data. X 13.67 X (2)(1.56) 13.67 10.55 Example z-Score for a Sample Mean We know X 13.67 . . If m = 12 and X = 2.5, what is the z-score for this sample mean? X X N 2.5 X 0.589 18 z X X 13.67 12 z 2.84 0.589