Calculating Z-Scores for Normal Distribution

Finding z-scores using Chart

OBJECTIVE Find the z-score of a standard normal distribution.

RELEVANCE Find probabilities and values of populations whose data can be represented with a normal distribution.

Sometimes, you must find a specific z-value for a given area under the curve. The procedure is to work backward.

Find the z-score such that the area under the normal distribution curve between 0 and z is 0.2123. Draw and Shade. Find the area closest to 0.2123 in the chart and work backwards. Answer: z=0.56 Example…….

Find the z for an area of .4066 between 0 and z. Answer: z = 1.32 Example……

Find the z for an area of .0239 to the right of z. First find the area between 0 and z: 0.5 - .0239 = 0.4761. Next, find the z-score closest to 0.4761. Answer: z=1.98 Example……

Find the z for an area of 0.9671 to the left of z. First, find the area between 0 and z: 0.9671 – 0.5000 = 0.4671. Next, find the z closest to 0.4671. Answer: z = 1.84 Example……

Example…… • Find the z-value to the right of the mean so that a. 53.98% of the area lies to the left of it. b. 71.90% of the area lies to the left of it.

Answer…… • 53.98% to the left. Area needed: 0.5398 – 0.5000 = 0.0398. z-score = 0.10

71.90% to the left. Area needed: 0.7190 – 0.5000 = 0.2190. Answer: z = 0.58

Find the z-score for an area of 0.05 to the right of the z. NOTE: If an area is exactly in between 2 areas, use the larger z-score. Area needed: 0.5000 – 0.0500 = 0.4500. Answer: Z = 1.65 Example……

Find the z-scores for an area of 40% that falls between two z-scores. Area needed: 0.4000/2 = 0.2000 Answer: z = -0.52 and +0.52 Example……

Calculating Z-Scores for Normal Distribution