Lesson 27 of 1524
The normal distribution
A z-score counts standard deviations from the mean.
Practice this chapterThe normal curve is the symmetric bell centered at the mean. The standard normal curve has mean 0 and standard deviation 1.
The empirical rule: about 68% of the values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. A negative z-score sits below the mean.
z-score
z = (x − μ) / σ
How many standard deviations x is from the mean.
Worked example
μ = 100, σ = 15, and x = 130. What is z?
- 1A z-score counts standard deviations from the mean: z = (x − μ) / σ.
- 2First find how far 130 is from 100: 130 − 100 = 30. The value is 30 units above the mean.
- 3One standard deviation is 15, so the number of standard deviations is 30 / 15 = 2.
- 4The sign is positive because 130 is above the mean. A value of 70 would have given z = (70 − 100) / 15 = −2.
Result: z = 2
Why. 130 sits 30 units above the mean, and each standard deviation is 15 units, so 30 / 15 = 2 standard deviations above the mean. The denominator is σ, not the variance σ².
Do not divide by the variance. The denominator of a z-score is the standard deviation.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.