Student Study Guide to Accompany Statistics Alive!. Wendy J. Steinberg

       Understand the properties of the normal curve

       Summarize the history of the normal curve and how it relates to current statistical practice

       Describe the common uses of the normal curve

      Module Summary

       The normal curve is a symmetric, bell-shaped curve that has an inflection point (meaning the curve bends) at 1 standard deviation (SD) above and below the mean. The shape of the normal curve is how a distribution with an infinite number of scores would appear. This means that the normal curve is theoretical, as opposed to an actual distribution of scores. However, we can expect that scores will distribute themselves in a manner similar to that of the normal distribution, especially as the size of the sample grows beyond n > 30.

       The data sets in this book will rarely fall in a perfect normal curve. However, the properties of the normal curve are robust to distributions that may violate its shape. This indicates that although our data may not look perfectly normal, you can still use the special features of the normal curve to understand our sample.

       The benefit of using the normal curve is that you are able to determine the proportion (percentage) of scores that will fall within 1, 2, and 3 SD of the mean. In a normal distribution, you can always expect that about 68% of the scores will fall within ±1 SD of the mean, about 95% of the scores will fall within ±2 SD of the mean, and about 99% of the scores will fall within ±3 SD of the mean. This means that a score that falls 1 SD above the mean will fall in the same place on the normal curve regardless of what is being measured. Aside from knowing the proportion of scores at a specific standard deviation, you can use the normal curve to determine the exact proportion of the curve above and below any specific score.

      Computational Exercises

      1 The mean grade on a French test was 83, with SD = 8. If you scored 1.5 SD above the mean, what was your grade on the test?

      2 What percentage of the normal distribution is greater than 2 SD away from the mean? Greater than 1 SD away from the mean?

      3 What percentage of the normal distribution is greater than the mean? What percentage of the normal distribution is less than 2 SD below the mean? What percentage of the normal distribution is within 1 SD above the mean?

      4 Jessica has a score on a science test that is 2 SD above the mean. What proportion of the distribution falls above this score?

      5 In a normal curve, 13.59% of scores fall between +1 and +2 SD above the mean. What percentage of scores fall between −1 and −2 SD below the mean?

      Computational Answers

      1 12

      2 Approximately 5% of the distribution is greater than 2 SD away from the mean. Approximately 32% of the normal distribution is greater than 1 SD away from the mean.

      3 Fifty percent of the normal distribution is greater than the mean. Approximately 2.5% of the normal distribution is less than 2 SD below the mean. Approximately 34% of the normal distribution is within 1 SD above the mean.

      4 2.28%

      5 13.59%

      True/False Questions

      1 The majority of the scores of the normal curve occur within 1 SD of the mean.

      2 The end points of the normal curve represent scores with a frequency of 0.

      3 You can still use the assumptions associated with the normal distribution for distributions whose shapes deviate slightly from normality.

      4 Approximately 99% of scores occur within 3 SD of the mean.

      5 The mean of IQ scores is 100, with SD = 15. The mean of the Minnesota Multiphasic Personality Inventory (MMPI) scores is 50, with SD = 10. An IQ score of 85 and an MMPI score of 40 fall in the same place on the normal curve.

      True/False Answers

      1 True

      2 False

      3 True

      4 True

      5 True

      Short-Answer Questions

      1 What is an inflection point? Why is it important in the normal curve?

      2 What aspect of the normal curve makes it very useful in statistics?

      3 With regard to statistics, what does it mean for something to be robust?

      4 What does it mean for a score to fall at the 76th percentile of the normal curve?

      5 If you were to find that you have a score that is worse than 2.5% of the population, where would you fall in the normal distribution?

      6 What types of variables may not follow a normal curve?

      Answers

      1 An inflection point is the place on the normal curve where the frequency curve drastically changes direction. On the normal curve, the first major inflection point represents 1 SD above and below the mean.

      2 The most useful aspect of the normal curve is that the exact proportion (percentage) of the curve above and below any given point is known. This tells us how many scores we can expect between pairs of standard deviations away from the mean. Also, it helps us determine the number of people above and below any particular score when the data are normally distributed.

      3 It means that you can still use the statistical procedure even though some of the required assumptions of the procedure have been violated. For example, you can still use the rules of the normal curve with data that do not perfectly conform to the shape of the normal curve.

      4 It means that 76% of the scores are either equal to or less than this particular score. Also, 24% of the scores are greater than this score.

      5 You would fall 2 SD above the mean, because this represents a z score of 2.

      6 Variables that include rare or unusual events may not follow a normal distribution.

      Multiple-Choice Questions

      1 Almost all of the scores in a distribution (≈0.99) fall within how many standard deviations of the mean?0123

      2 As you move farther away from the mean in a normal distribution, the frequency of scoresincreases.decreases.stays the same.The answer depends on the data set.

      3 The majority of scores in a normal distribution fall where in relation to the inflection point?Outside the inflection pointsIn between the inflection pointsPrecisely at the inflection pointThe answer depends on the data.

      4 The population for the SAT has mean = 500 and s = 100. A score of 300 has as many people below it as what score above it?300500600700

      Multiple-Choice Answers

      1 D

      2 B

      3 B

      4 D

      Module Quiz

      1 What proportion of the population falls within 1 SD above and below the mean? 2 SD above the mean?

      2 The population mean for SAT scores has M = 500 and SD = 100. Between which two scores would ≈99% of the population fall?

      3 Harry’s score on a math test is 1 SD above Ron’s. If Ron scored 1 SD above the mean and the descriptive statistics for the class were M = 30 and SD = 10, what were Ron’s and Harry’s raw scores?

      4 Although Harry did better