This document discusses key concepts about the normal distribution and z-scores, including: - Approximately 68%, 95%, and 99% of scores in a normal distribution fall within 1, 2, and 3 standard deviations of the mean, respectively. - A z-score describes how many standard deviations a raw score is above or below the mean, and allows comparison of scores from different distributions. - The normal curve table lists percentages of scores associated with different z-scores and can be used to find percentages of scores above or below a given value.

1. Aron, Coups, & AronStatistics for the Behavioral & Social Sciences- Chapter 2 and 4 z Scores & the Normal Curve

2. Normal distribution

3. The normal distribution and standard deviations Mean -1 +1 In a normal distribution: Approximately 68% of scores will fall within one standard deviation of the mean

4. The normal distribution and standard deviations Mean +1 -1 +2 -2 In a normal distribution: Approximately 95% of scores will fall within two standard deviations of the mean

5. The normal distribution and standard deviations Mean +2 +1 -1 -2 +3 -3 In a normal distribution: Approximately 99% of scores will fall within three standard deviations of the mean

6. The Normal Curve

7. Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and and standard deviation of 10: 100 110 120 90 80 -1 sd 1 sd 2 sd -2 sd What score is one sd below the mean? 90 120 What score is two sd above the mean?

8. Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and and standard deviation of 10: 100 110 120 90 80 -1 sd 1 sd 2 sd -2 sd 1 How many standard deviations below the mean is a score of 90? How many standard deviations above the mean is a score of 120? 2

9. Z scores z scores are sometimes called standard scores Here is the formula for a z score: A z score is a raw score expressed in standard deviation units. What is a z-score?

10. z-score describes the location of the raw score in terms of distance from the mean, measured in standard deviations Gives us information about the location of that score relative to the “average” deviation of all scores A z-score is the number of standard deviations a score is above or below the mean of the scores in a distribution. A raw score is a regular score before it has been converted into a Z score. Raw scores on very different variables can be converted into Z scores and directly compared. What does a z-score tell us?

11. Mean of zero Zero distance from the mean Standard deviation of 1 The z-score has two parts: The number The sign Negative z-scores aren’t bad Z-score distribution always has same shape as raw score Z-score Distribution

12. z = (X –M)/SD Score minus the mean divided by the standard deviation Computational Formula

13. 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, (example from text). To change a raw score to a Z score: Step One: Determine the deviation score. Subtract the mean from the raw score. 8 – 12 = -4 Step Two: Determine the Z score. Divide the deviation score by the standard deviation. -4 / 4 = -1 Steps for Calculating a z-score

14. Using z scores to compare two raw scores from different distributions You score 80/100 on a statistics test and your friend also scores 80/100 on their test in another section. Hey congratulations you friend says—we are both doing equally well in statistics. What do you need to know if the two scores are equivalent? the mean? What if the mean of both tests was 75? You also need to know the standard deviation What would you say about the two test scores if the SDin your class was 5 and the SDin your friend’s class is 10?

15. Calculating z scores What is the z score for your test: raw score = 80; mean = 75, SD= 5? What is the z score of your friend’s test: raw score = 80; mean = 75, S = 10? Who do you think did better on their test? Why do you think this?

16. Transforming scores in order to make comparisons, especially when using different scales Gives information about the relative standing of a score in relation to the characteristics of the sample or population Location relative to mean Relative frequency and percentile Why z-scores?

18. Figure the deviation score. Multiply the Z score by the standard deviation. Figure the raw score. Add the mean to the deviation score. Formula for changing a Z score to a raw score: X= (Z)(SD)+M Computing Raw Score from a z-score

19. Standardizes different scores Example in text: Statistics versus English test performance Can plot different distributions on same graph increased height reflects larger N Comparing Different Variables

20. How Are You Doing? How would you change a raw score to a Z score? If you had a group of scores where M = 15 and SD = 3, what would the raw score be if you had a Z score of 5?

21. Normal Distribution histogram or frequency distribution that is a unimodal, symmetrical, and bell-shaped Researchers compare the distributions of their variables to see if they approximately follow the normal curve.

22. Use to determine the relative frequency of z-scores and raw scores Proportion of the area under the curve is the relative frequency of the z-score Rarely have z-scores greater than 3 (.26% of scores above 3, 99.74% between +/- 3) The Standard Normal Curve

23. Why the Normal Curve Is Commonly Found in Nature A person’s ratings on a variable or performance on a task is influenced by a number of random factors at each point in time. These factors can make a person rate things like stress levels or mood as higher or lower than they actually are, or can make a person perform better or worse than they usually would. Most of these positive and negative influences on performance or ratings cancel each other out. Most scores will fall toward the middle, with few very low scores and few very high scores. This results in an approximately normal distribution (unimodal, symmetrical, and bell-shaped).

24. The Normal Curve Table and Z Scores A normal curve table shows the percentages of scores associated with the normal curve. The first column of this table lists the Z score The second column is labeled “% Mean to Z” and gives the percentage of scores between the mean and that Z score. The third column is labeled “% in Tail.” .

25. Normal Curve Table A-1

26. Using the Normal Curve Table to Figure a Percentage of Scores Above or Below a Raw Score If you are beginning with a raw score, first change it to a Z Score. Z = (X – M) / SD Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage. Make a rough estimate of the shaded area’s percentage based on the 50%–34%–14% percentages. Find the exact percentages using the normal curve table. Look up the Z score in the “Z” column of the table. Find the percentage in the “% Mean to Z” column or the “% in Tail” column. If the Z score is negative and you need to find the percentage of scores above this score, or if the Z score is positive and you need to find the percentage of scores below this score, you will need to add 50% to the percentage from the table. Check that your exact percentage is within the range of your rough estimate.

31. Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 1 Draw a picture of the normal curve and shade in the approximate area of your percentage using the 50%–34%–14% percentages. We want the top 5%. You would start shading slightly to the left of the 2 SD mark. Copyright © 2011 by Pearson Education, Inc. All rights reserved

32. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 2 Make a rough estimate of the Z score where the shaded area stops. The Z Score has to be between +1 and +2.

33. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 3 Find the exact Z score using the normal curve table. We want the top 5% so we can use the “% in Tail” column of the normal curve table. The closest percentage to 5% is 5.05%, which goes with a Z score of 1.64.

34. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 4 Check that your Z score is within the range of your rough estimate. +1.64 is between +1 and +2.

35. Copyright © 2011 by Pearson Education, Inc. All rights reserved Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 5 If you want to find a raw score, change it from the Z score. X = (Z)(SD) + M X = (1.64)(16) + 100 = 126.24