normal distribution

Z Table

A statistical table used to find the area or probability associated with a given Z score in a standard normal distribution.

Standard Normal Distribution

A normal distribution with a mean of $0$ and a standard deviation of $1$. This is the focus of Section 7.1.

Non-Standard Normal Distribution

A normal distribution with any mean ($\mu$) and standard deviation ($\sigma$). Problems involving these distributions require calculating a Z score before using the Z table, as discussed in Section 7.2.

Z Score Formula

The formula used to convert an individual data point ($x$) from a non-standard normal distribution to a Z score: $z = \frac{x - \mu}{\sigma}$.

Probability (in Z Table Context)

Interchangeable with 'area' and 'proportions,' representing the likelihood of a value falling within a certain range under the normal curve, as found using the Z table.

Complement Rule (for Z Scores)

Used to find probabilities greater than a Z score. If the Z table gives $P(Z < z)$, then $P(Z > z) = 1 - P(Z < z)$. Alternatively, one can switch the sign of Z and use the table directly for the area.

Between Problems (Z Score)

Problems requiring the probability of a value falling between two limits ($x_1$ and $x_2$). Solved by finding two Z scores ($z_1$, $z_2$) and subtracting their corresponding areas: $P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)$.

Calculating X from Z

When given an area or percentile, first find the corresponding Z value from the Z table, then use the formula: $x = \mu + z \cdot \sigma$.

Rounding Z Score

Z scores should always be rounded to two decimal places immediately after calculation to ensure correct Z table lookup.