Finding Z-Scores and Data Values in the Normal Distribution

Normal Probability Distributions

Finding a Z-Score Given the Area Under the Normal Curve

The z-score is a standardized value that indicates how many standard deviations a data point is from the mean in a normal distribution. To find a z-score given a cumulative area (probability) under the normal curve, we use the standard normal table (z-table), which provides cumulative probabilities from the far left up to a given z-value.

• Step 1: Locate the cumulative probability in the z-table.

• Step 2: Read the corresponding z-score from the leftmost column (for the first two digits) and the top row (for the second decimal place).

• Note: The z-table typically gives the area to the left of the z-score (cumulative probability).

Example: Find the z-score for a cumulative area of 0.3632.

• Locate 0.3632 in the z-table. The corresponding z-score is -0.35.

• This means 36.32% of the distribution lies below z = -0.35.

Example: Find the z-score with 10.75% of the area to its right.

• Area to the right: 0.1075. Cumulative area to the left: 1 - 0.1075 = 0.8925.

• Locate 0.8925 in the z-table. The corresponding z-score is 1.24.

• Thus, z = 1.24 has 10.75% of the area to its right.

Percentiles: The nth percentile is the value below which n% of the data fall. For example, the 10th percentile (P10) is the value with 10% of the area below it.

• To find the z-score for the 10th percentile, look for a cumulative probability of 0.1000 in the z-table or use statistical software.

• Using Excel: NORM.INV(0.10, 0, 1) returns approximately -1.282.

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:

$x = \mu + z \sigma$

• x: The data value

• \mu: The population mean

• \sigma: The population standard deviation

• z: The z-score

Example: The weights of dogs are normally distributed with $\mu = 52$ pounds and $\sigma = 15$ pounds. Find the weight corresponding to each z-score:

• For $z = -2.33$:

  • $x = 52 + (-2.33) \times 15 = 17.05$ pounds

  • This dog weighs 2.33 standard deviations below the mean.

• For $z = 3$:

  • $x = 52 + 3 \times 15 = 97$ pounds

  • This dog weighs 3 standard deviations above the mean.

• For $z = 0.58$:

  • $x = 52 + 0.58 \times 15 = 60.7$ pounds

  • This dog weighs 0.58 standard deviations above the mean.

Finding a Specific Data Value Given a Probability

To find a data value corresponding to a given probability (e.g., the lowest 10%), follow these steps:

1. Find the z-score corresponding to the cumulative probability (e.g., for the lowest 10%, cumulative probability = 0.10).

2. Use the formula $x = \mu + z \sigma$ to find the data value.

Example: Employee tenure at a company is normally distributed with $\mu = 11.2$ years and $\sigma = 2.1$ years. What is the maximum tenure to be in the lowest 10%?

• Find the z-score for the 10th percentile: $z = -1.282$.

• Calculate $x$:

  • $x = 11.2 + (-1.282) \times 2.1 = 8.509$ years

  • Employees with less than approximately 8.5 years of service are in the lowest 10%.

Summary Table: Key Steps for Normal Distribution Calculations

Task	Step 1	Step 2	Step 3
Find z-score for a given area	Identify cumulative probability	Locate in z-table	Read corresponding z-score
Find x for a given z-score	Identify mean ($\mu$) and standard deviation ($\sigma$)	Plug z, $\mu$, $\sigma$ into $x = \mu + z \sigma$	Interpret result
Find x for a given probability	Find z-score for cumulative probability	Use $x = \mu + z \sigma$	Interpret result

Additional info: In practice, statistical software (such as Excel's NORM.INV function) can be used to find z-scores or data values for given probabilities, especially when the exact value is not found in standard tables.