Interpret data using the bell curve and standardize with z-scores.
The normal distribution is a symmetric, bell-shaped probability distribution completely described by its mean Îŧ and standard deviation Ī. The empirical rule (68-95-99.7 rule) describes how data clusters around the mean.
The Empirical Rule
Approximately symmetric â 34% on each side within one standard deviation.
A z-score measures how many standard deviations a data point lies above (positive) or below (negative) the mean. The standard normal distribution has mean 0 and standard deviation 1.
Exam scores are normally distributed with Îŧ = 75 and Ī = 10. A student scored 92. z = (92 â 75)/10 = 1.7. This student scored 1.7 standard deviations above the mean.
Remember This!
A z-score of 0 means the value equals the mean. Positive z: above the mean. Negative z: below the mean. Most real-world data falls between z = â3 and z = 3.
Heights are normally distributed with Îŧ = 68 in and Ī = 3 in. What is the z-score for a person who is 74 inches tall?