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PSYC 5: Chapter 5

PSYC 5: Chapter 5. z-scores & Standardized Distributions Learning objectives: Define z-scores Describe the benefit of using z-scores Calculate z-scores. New Statistical Notation.

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PSYC 5: Chapter 5

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  1. PSYC 5: Chapter 5 z-scores & Standardized Distributions • Learning objectives: • Define z-scores • Describe the benefit of using z-scores • Calculate z-scores

  2. New Statistical Notation • The absolute value of a number is the size of that number, regardless of its sign. That is, the absolute value of +2 is 2 and the absolute value of -2 is 2. • The symbol means “plus or minus.” Therefore, 1 means +1 and/or -1.

  3. Understanding z-Scores

  4. Frequency Distribution of Attractiveness Scores

  5. z-Scores • Like any raw score, a z-score is a location on the distribution. A z-score also automatically communicates the raw score’s distance from the mean • A z-score describes a raw score’s location in terms of how far above or below the mean it is when measured in standard deviations

  6. z-Score Formula • The formula for computing a z-score for a raw score in a sample is

  7. Computing a Raw Score • When a z-score and the associated and are known, this information can be used to calculate the original raw score. The formula for this is

  8. Interpreting z-ScoresUsing the z-Distribution

  9. A z-Distribution A z-distribution is the distribution produced by transforming all raw scores in the data into z-scores.

  10. z-Distribution of Attractiveness Scores

  11. Characteristics of the z-Distribution • A z-distribution always has the same shape as the raw score distribution • The mean of any z-distribution always equals 0 • The standard deviation of any z-distribution always equals 1

  12. Comparison of Two z-Distributions, Plotted on the Same Set of Axes

  13. Relative Frequency • Relative frequency can be computed using the proportion of the total area under the curve. • The relative frequency of a particular z-score will be the same on all normal z-distributions.

  14. The Standard Normal Curve The standard normal curve is a perfect normal z-distribution that serves as our model of the z-distribution that would result from any approximately normal raw score distribution

  15. Proportions of Total Area Under the Standard Normal Curve

  16. Percentile The standard normal curve also can be used to determine a score’s percentile.

  17. Proportions of the Standard Normal Curve at Approximately the 2nd Percentile

  18. Using z-Scores to Describe Sample Means

  19. Sampling Distribution of Means A distribution which shows all possible sample means that occur when an infinite number of samples of the same size N are randomly selected from one raw score population is called the sampling distribution of means.

  20. Central Limit Theorem The central limit theorem tells us the sampling distribution of means • forms an approximately normal distribution, • has a m equal to the m of the underlying raw score population, and • has a standard deviation that is mathematically related to the standard deviation of the raw score population.

  21. 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

  22. z-Score Formula for a Sample Mean The formula for computing a z-score for a sample mean is

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