# The Normal Distribution and z Scores

This applet will allow you to explore the normal distribution by changing values of the mean, the standard deviation, the observation, or z itself, and examine the areas under the curve. When you change any value you must press the Enter (or Return) key to have that value take effect.

 Cumulative from minus infinity to the z-score One-Tailed from the z-score to positive infinity Two-Tailed scores more extreme (i.e., further from the middle) than the z-score Middle scores less extreme (i.e., closer to the middle) than the z-score
• On the left of the display are the definitions of the way the different tails of the distribution can be displayed. Make selections from the box that is currently labeled "Two-Tailed" to illustrate these various choices.
• Next change the entry in the box labeled “prob:” to 0.01. Notice that the entry in the box for "z" changes accordingly, and is the two-tailed critical value for z to cut off the extreme 1% of the distribution.
• In Exercise 6.14 I give an example of a year in which the mean Graduate Record Exam score was 489 and the standard deviation was 126. Use this display to calculate the percentage of students who would be expected to have a score of 500 or higher. (You simply enter the appropriate numbers in the boxes and press the Enter key after each entry.)
• What about the percentage expected to score over 700? (Be sure that you select the proper tail of the distribution in computing your percentages.
• Suppose that your instructor just handed back the last statistics exam and you had a 73. That probably won't cheer you up. But suppose that the class mean was 68 (it was a really hard exam), and the standard deviation was 3.6. Now what do you think of your grade. What percentage of the class scored at least as low as you did?