These lessons, with videos, examples and step-by-step solutions, help Statistics students learn how to use z-scores to work out probability based on the normal distribution.

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Normal Distribution: Probability in a given range
Standard Deviation
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Every unique pair of μ and σ defines a different normal distribution. This characteristic of the normal curve (actually a family of curves) could make analysis by the normal distribution tedious because volumes of normal curve tables – one for each different combinations of μ and σ - would be required.

Fortunately, all normal distributions can be converted into a single distribution, the standardized normal distribution or the z distribution, which has mean 0 and standard deviation 1. We write Z ∼ N(0, 1).

The conversion formula for any x value of a given normal distribution is:


A z-score is the number of standard deviations that a value, x, is above or below the mean.
If the value of x is less than the mean, the z score is negative.
If the value of x is more than the mean, the z score is positive.
If the value of x equals the mean, the z score is zero.

This formula allows conversion of the distance of any x value form its mean into standard deviation units. A standard z score table can then be used to find probabilities for any normal distribution problem that has been converted to z scores.

Normal Distribution & Z-scores
This video shows how to calculate “inside areas” and “areas in the extreme” in a normal distribution using Z-scores.

The following video gives the definition of z score based on the bell curve.

The following video gives examples of calculating z-score.

The following video shows how to solve some z-score homework problems.
What is the z-score if our standard deviation was 2, mean of 49 and our specific observation is 47?
What is the probability an observation is more than 49?
What is the probability an observation is more than 47 and less than 49?
What is the probability an observation is more than 47?
What is the probability an observation is less than 47?

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