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Z scores and determining normal distribution
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- Mcdaniel College
Overview of z scores and determining a normal distribution, with examples.
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Tuesday, 10/22➢ Z score gives number of standard deviations between original measurement x and mean µ
(𝑥−µ)
➢ 𝑧= 𝛿
➢ Score close to zero means that the measurement is near the mean
➢ Positive score means the measurement is above the mean
➢ Negative score means the measurement is below the mean
Example
➢ Mean amount of cheese on a large pizza is 8 ounces. Standard deviation(δ) = 0.5. One
pizza is picked randomly and is found to have 6.9 ounces of cheese. If the amount of
cheese on a pizza falls out of 3 standard deviations from the mean, the pizza parlor will
go out of business.
➢ How many standard deviations is 6.9 from the mean? Will the parlor go out of business
(𝑥−µ) (6.9−8) 1.1
→ 𝑧= = = − 0.5 = -2.2; within 3 standard deviations of the mean, so the
𝛿 0.5
parlor will stay in business
➢ Raw score x: 𝑥 = 𝑧𝛿 + µ
→ Corresponds to z score
Guided Exercise 1
➢ Ezra finds that it takes on average(mean) 17 minutes with a standard deviation(δ) of 3
minutes to drive, park, and get to her morning class.
➢ One time it takes her 21 minutes. How many standard deviations is this away from the
mean? Is this a positive or negative z value?
(21−17)
→ 𝑧= = 4/3 = 1.33; positive value
3
➢ What time corresponds to z = -2.5? Could Ezra make it to class in this time?
→ 𝑥 = 𝑧𝛿 + µ -> 𝑥 = −2.5(3) + 17 --> 𝑥 = −7.5 + 17 = 9.5 minutes; She will not
make it to class on time
➢ Standard normal distribution has µ = 0 and δ = 1
Example
, ➢ Use Table 5 Appendix II to find described areas
➢ Find area under curve to the left of z = -1
→ 0.1587
➢ Find area to the left of z = 1.18
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