Normal Probability Distributions: Finding Data Values and Percentiles

Normal Probability Distributions

Finding an x-Value Corresponding to a z-Score

In statistics, the normal distribution is a fundamental concept used to describe data that clusters around a mean. The relationship between a z-score and a data value (x) allows us to translate standardized scores into actual measurements.

• z-score: A measure of how many standard deviations a data point is from the mean.

• Formula: To find the x-value corresponding to a z-score, use: $x = \mu + z \sigma$ where $\mu$ is the mean and $\sigma$ is the standard deviation.

• Example: For cat weights with $\mu = 9$ pounds and $\sigma = 2$ pounds:

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

  • For $z = -0.44$: $x = 9 + (-0.44) \times 2 = 8.12$ pounds

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

• Interpretation: Values to the right of the mean (positive z) are above average; values to the left (negative z) are below average; $z = 0$ is exactly at the mean.

Finding a Specific Data Value for a Given Probability (Percentile)

The normal distribution can be used to find the data value (x) that corresponds to a specific percentile or probability. This is useful for determining cutoffs, eligibility, or thresholds in real-world scenarios.

• Percentile: The value below which a given percentage of data falls.

• Procedure:

  1. Find the cumulative probability (area under the curve) for the desired percentile.

  2. Use the Standard Normal Table to find the corresponding z-score.

  3. Convert the z-score to an x-value using $x = \mu + z \sigma$.

• Example: California Peace Officer Standards and Training test scores:

  • Mean $\mu = 50$, standard deviation $\sigma = 10$

  • Top 10%: Find the score at the 90th percentile

  • From the table, $z \approx 1.28$ for area 0.9

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

• Technology: Calculators and software (e.g., TI-84, Excel) can compute these values using functions like invNorm or NORM.INV.

Applications: Mutual Funds, Cholesterol Levels, and Seniority

Normal distribution methods are widely used in finance, health, and employment to determine thresholds and eligibility based on percentiles.

• Mutual Funds: The mean annual rate of return for large growth mutual funds is $\mu = 21.2\%$, $\sigma = 3.5\%$. The middle 90% of data lies between two values found using the 5th and 95th percentiles.

• Cholesterol Levels: For U.S. adults, $\mu = 187$ mg/dL, $\sigma = 41.1$ mg/dL. The lowest 1% corresponds to $z \approx -2.33$: $x = 187 + (-2.33) \times 41.1 \approx 91$ mg/dL

• Seniority: Employee tenure is normally distributed with $\mu = 11.2$ years, $\sigma = 2.1$ years. The lowest 10% (laid off) corresponds to the 10th percentile.

Summary Table: Finding x-Values for Given z-Scores or Percentiles

Scenario	Mean ($\mu$)	Std. Dev. ($\sigma$)	Percentile/Probability	z-score	x-value
Cat Weights	9	2	z = 1.96	1.96	12.92
Peace Officer Test	50	10	90th percentile	1.28	62.8
Cholesterol Levels	187	41.1	1st percentile	-2.33	91
Mutual Funds	21.2%	3.5%	Middle 90%	z = -1.645, 1.645	Values between 15.44% and 26.96%
Employee Seniority	11.2	2.1	10th percentile	-1.28	8.5

Key Formulas

• z-score formula: $z = \frac{x - \mu}{\sigma}$

• x-value formula: $x = \mu + z \sigma$

Additional info:

• Percentiles are often used in admissions, hiring, and risk assessment to set thresholds.

• Technology tools (TI-84, Excel) can automate calculations for percentiles and probabilities in normal distributions.