Using the Standard Normal Distribution and z-Tables in Statistics

Standard Normal Distribution and z-Tables

Introduction to the Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is widely used in statistics to calculate probabilities and to standardize values from any normal distribution using z-scores.

• z-score: The number of standard deviations a value is from the mean. Calculated as $z = \frac{x - \mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

• Probability: For continuous random variables, probability is represented by the area under the curve of the probability density function.

Finding Standard Normal Probabilities Using the z-Table

Using the z-Table

The z-table provides cumulative probabilities (areas) to the left of a given z-score under the standard normal curve. These probabilities are used to answer questions about the likelihood of observing values less than, greater than, or between certain z-scores.

• To find $P(Z < z)$: Look up the z-score in the table; the value is the area to the left.

• To find $P(Z > z)$: Use $P(Z > z) = 1 - P(Z < z)$.

• To find $P(a < Z < b)$: Subtract the area to the left of $a$ from the area to the left of $b$: $P(a < Z < b) = P(Z < b) - P(Z < a)$.

Example Table: Types of Areas Under the Standard Normal Curve

Type	Expression	How to Find
Area to Left	$P(Z < z)$	Look up z in table
Area to Right	$P(Z > z)$	$1 - P(Z < z)$
Area Between Two Values	$P(a < Z < b)$	$P(Z < b) - P(Z < a)$

Examples and Applications

• Example 1: Find $P(Z < -0.64)$

  • Look up $z = -0.64$ in the z-table: $P(Z < -0.64) = 0.2611$

• Example 2: Find $P(Z > 2.27)$

  • Look up $z = 2.27$ in the z-table: $P(Z < 2.27) = 0.9884$

  • So, $P(Z > 2.27) = 1 - 0.9884 = 0.0116$

• Example 3: Find $P(-0.64 < Z < 2.27)$

  • $P(Z < 2.27) = 0.9884$

  • $P(Z < -0.64) = 0.2611$

  • $P(-0.64 < Z < 2.27) = 0.9884 - 0.2611 = 0.7273$

Finding z-Scores from Probabilities Using the z-Table

Reverse Lookup: Probability to z-Score

Sometimes, you are given a probability (area) and need to find the corresponding z-score. This is done by finding the closest probability in the z-table and reading the associated z-score.

• For area to the left: Find the probability in the table, then read the z-score.

• For area to the right: Subtract the given probability from 1 to get the area to the left, then look up the z-score.

Recall: $P(Z > z) = 1 - P(Z < z)$

• Example: Find the z-score such that $P(Z < z) = 0.6331$

  • Look up 0.6331 in the z-table; the corresponding z-score is approximately 0.34.

Using Calculators to Find Probabilities

TI-84 (CE) Calculator Methods

Modern calculators can compute probabilities for the standard normal distribution using built-in functions. For the TI-84, use normalcdf for probabilities and ShadeNorm for visual graphs.

• To find $P(Z < z)$: Use normalcdf with lower bound = -1E99, upper bound = z.

• To find $P(Z > z)$: Use normalcdf with lower bound = z, upper bound = 1E99.

• To find $P(a < Z < b)$: Use normalcdf with lower bound = a, upper bound = b.

Example: To find $P(Z < -0.81)$, enter normalcdf(-1E99, -0.81, 0, 1) on the calculator.

z-Table Reference

The z-table lists cumulative probabilities for z-scores from the standard normal distribution. Values are areas to the left of the given z-score.

z	Area to Left
-0.64	0.2611
0.00	0.5000
0.34	0.6331
1.21	0.8869
2.27	0.9884

Additional info: The full z-table contains many more values, typically ranging from about -3.4 to +3.4 in increments of 0.01.

Summary Table: Steps for Using the z-Table

Task	Step 1	Step 2	Step 3
Find $P(Z < z)$	Identify z-score	Look up in z-table	Read area to left
Find $P(Z > z)$	Identify z-score	Look up $P(Z < z)$	Subtract from 1
Find $P(a < Z < b)$	Identify a and b	Find $P(Z < b)$ and $P(Z < a)$	Subtract: $P(Z < b) - P(Z < a)$
Find z for given area	Identify area	Find closest value in z-table	Read corresponding z-score

Key Takeaways

• The standard normal distribution is a foundational concept in statistics for calculating probabilities.

• z-tables and calculators are essential tools for finding areas (probabilities) under the normal curve.

• Always sketch the region under the curve to visualize the probability being calculated.

• Remember to adjust for left, right, or between areas as needed using the properties of the normal distribution.