Normal Probability Distributions: Finding Values and Applications

Normal Probability Distributions

Finding Values in Normal Distributions

The normal distribution is a fundamental concept in statistics, describing how data values are distributed around a mean in many natural and social phenomena. This section focuses on determining specific values (z-scores or x-values) given probabilities or percentiles, and vice versa, using the properties of the normal distribution.

Finding a z-Score Given an Area or Percentile

• z-score: The number of standard deviations a value is from the mean in a standard normal distribution (mean = 0, standard deviation = 1).

• To find a z-score for a given cumulative area (probability) to the left, locate the area in the body of the Standard Normal Table. The corresponding row and column values give the z-score.

• For an area to the right, subtract the area from 1 to get the cumulative area to the left, then proceed as above.

• Percentile: The value below which a given percentage of observations fall. The z-score for a percentile is found by locating the corresponding area in the Standard Normal Table.

Examples:

• Find the z-score for a cumulative area of 0.3632: Locate 0.3632 in the table; the z-score is the intersection of the corresponding row and column.

• Find the z-score with 10.75% of the area to the right: 1 - 0.1075 = 0.8925 (cumulative area to the left). Locate 0.8925 in the table; the z-score is 1.24.

• Find the z-score for the 90th percentile: Locate 0.9 in the table; the z-score is approximately 1.28.

Transforming a z-Score to an x-Value

To convert a z-score to a data value x in a normal distribution with mean $\mu$ and standard deviation $\sigma$, use the formula:

$x = \mu + z \sigma$

• Interpretation: This formula allows you to find the actual data value corresponding to a given z-score in any normal distribution.

Example: If cat weights are normally distributed with $\mu = 9$ pounds and $\sigma = 2$ pounds:

• For $z = 1.96$: $x = 9 + 1.96 \times 2 = 12.92$ pounds

• For $z = 0$: $x = 9 + 0 \times 2 = 9$ pounds

Finding a Specific Data Value for a Given Probability

• To find the data value x corresponding to a given probability (e.g., top 10%), first find the z-score for the cumulative area, then use the transformation formula above.

Example: Scores for a test are normally distributed with $\mu = 50$, $\sigma = 10$. To find the lowest score in the top 10% (above the 90th percentile):

• Find the z-score for area 0.9: $z = 1.28$

• Calculate $x = 50 + 1.28 \times 10 = 62.8$ (rounded to 63)

Application: Cholesterol Levels

• Given: Mean cholesterol $\mu = 187$ mg/dL, $\sigma = 41.1$ mg/dL. Find the highest cholesterol level in the bottom 1% (lowest 1%).

• Find the z-score for area 0.01: $z = -2.33$ (approximate value from the table).

• Calculate $x = 187 + (-2.33) \times 41.1 = 91$ mg/dL (rounded).

Summary Table: Steps for Finding Values in a Normal Distribution

Given	Find	Steps
Probability/Area	z-score	Locate area in Standard Normal Table; read corresponding z-score
z-score	x-value	Use $x = \mu + z \sigma$
Percentile	z-score, then x-value	Find area, locate z-score, then use $x = \mu + z \sigma$

Additional info: Technology (such as statistical calculators or software like Minitab or Excel) can be used to find z-scores and x-values, and may yield slightly different results due to rounding or interpolation.