Area under the Normal Curve using Printed Tables Example problem curve the left of. Area = 0.9147 Probability = 0.9147 or 91.47% curve the right of. Area = 0.0853 Probability = 0.0853 or 8.53% curve between and. Area = 0.2590 Probability = 0.2590 or 25.90% Look in row 1.30 and column 0.07 where it says 0.4147 0.4147 is the area between z=0 and z=1.37 0.5000 more is in the left half (where z is negative) 0.9147 tal area Look in row 1.30 and column 0.07 where it says 0.4147. 0.4147 is the area between z=0 and z=1.37 The area the right of z=1.37 is: 0.5000 tal area in the right half - 0.4147 area between z=0 and z=1.37 = 0.0853 area the right of z=1.37 First, find area the left of z=1.85: Row 1.80 Column 0.05 says 0.4678 Add 0.5000 for the left half get 0.9678 tal Then find area the left of z=0.55: Row 0.50 Column 0.05 says 0.2088 Add 0.5000 for the left half get 0.7088 tal Then subtract find the area in between those two z values: 0.9678 minus 0.7088 equals 0.2590 net area between.
Find the tal area under the normal curve. You should know this very basic special fact, that the tal area under a probability distribution is always exactly precisely = 1. What is the area under the right half of the normal curve? and What is the area under the left half of the normal curve? You should know these very special facts about the Normal Distribution: it is symmetric, is in the middle, half of the area (0.5) is the left and half of the area (0.5) is the right. No table is needed. This is fact that you simply have know. No table is needed. These are facts that you simply have know.
curve the left of. Area = 0.0217 Probability = 0.0217 or 2.17% curve the right of. Area = 0.9783 Probability = 0.9783 or 97.83% curve between and. Area = 0.2682 Probability = 0.2682 or 26.82% curve between and. Area = 0.9494 Probability = 0.9494 or 94.94% The table only has positive z-values. Because of symmetry, the area the left of z=-2.02 Is the same as the area the right of z=+2.02. Look in row 2.00 column 0.02 find 0.4783. Area between z=0 and z=2.02 (positive) is 0.4783. Therefore area the right of z=+2.02 (positive) is 0.5000 minus 0.4783 equals 0.0217. The table has only positive z-values. Because of symmetry, the area between z=0 and z=+2.02 Is the same as the area between z=-2.02 and z=0. Look in row 2.00 column 0.02 find 0.4783. The tal area between z=-2.02 and z=0 is 0.4783. Add 0.5000 for the right half of the area = 0.9783 tal. Because of symmetry, the area between z=-1.38 and z=-0.38 Is the same as the area between z=+0.38 and z=+1.38. Look in row 1.30 column 0.08 find 0.4162. Look in row 0.30 column 0.08 find 0.1480. Subtract 0.4162 minus 0.1480 get area 0.2682. Or, if you added the 0.5000 each, 0.9162 minus 0.6480 = 0.2682, the same answer again. The right half: Look in row 2.20 column 0.000 find 0.4861. The left half: Because of symmetry, the area between z=-1.79 and z=0 is the same as the area between z=0 and z=+1.79. Look in row 1.70 column 0.09 find 0.4633. Add 0.4861 plus 0.4633 = tal area 0.9494
50% of the area? 50% = 0.5000 shaded in middle 0.5000 divided by 2 = 0.2500 68% of the area? 68% = 0.6800 shaded in middle 0.6800 divided by 2 is 0.3400 Look in the body of the table find the value closest 0.2500. It is 0.2486. Read outward see you re in row 0.60 column 0.07, So the z-score is 0.67. By symmetry, the other end is at z=-0.67. Look in the body of the table find the value closest 0.3400. It is 0.3389. Read outward see you re in row 0.90 column 0.09. So the z-score is 0.99 By symmetry, the other end is at z=-0.99. This is why the Empirical Rule says In a Normal Distribution, about 68% of the data lies within 1 standard deviation of the mean.
95% of the area? 95% = 0.9500 shaded in middle 0.9500 divided by 2 is 0.4750 99.7% of the area? 99.7% = 0.9970 0.9970 divided by 2 is 0.4985 This is why the Empirical Rule says In a Normal Distribution, about 95% of the data lies within 2 standard deviations of the mean. Look in the body of the table find the value closest 0.4750. It is in the table, this is unusual! Read outward see you re in row 1.90 column 0.06. So the z-score is 1.96 By symmetry, the other end is at z=-1.96. Look in the body of the table find the value closest 0.4985. It is in the table twice. That happens with the extreme tails of the normal distribution. Read outward see you re in row 2.90 column 0.06 or column 0.07. So the z-score is 2.96 or 2.97 This is why the Empirical Rule says In a Normal Distribution, about 95% of the data lies within 2 standard deviations of the mean. By symmetry, the other end is at z=-2.96 or z=-2.97 It turns out that with the help of computers or calculars we can determine that the 2.97 version is really more accurate.
What z-score separates the p 10% of the data from the botm 90%? 90% = 0.9000 area is the left. 10% = 0.1000 area is the right What score separates the botm third of the data from the p two-thirds? 1/3 the left. You can use the fraction in TI-84 or in Excel. The table gives areas in the right half. The tal area the left is 0.9000. Subtract the 0.5000 left half get 0.4000 area in the right half. Look in the body of the table find the value closest 0.4000. It is 0.3997. Read outward see you re in row 1.20 column 0.08. The z-score of interest is therefore 1.28 By symmetry, the z-score that separates the botm 1/3 of the area is the negative of the z-score that separates the p 1/3 of the area. In the right half, we want the z-score that separates 0.1667 of the area the left from the 0.3333 of the area the right. Look in the body of the table find the value closest 0.1667. It is 0.1664. Read out find you re in row 0.40 column 0.03. Therefore the z-value is 0.43. But we want the mirror image of that, so our z-score is -0.43.