Understanding Normal Distribution and Z-Scores in Statistics

Exam Tip: Remember that symmetry alone does not make a distribution normal. A distribution can be symmetric but "bimodal" (having two peaks), which would not be a normal distribution.

Calculating and Interpreting Z Scores

A z-score is a statistical measure that expresses a value's distance from the mean in units of standard deviation. This allows us to compare scores from entirely different distributions—such as comparing a student's height in inches to their weight in pounds—by putting them on a common scale.

The formula for calculating a z-score is: $z = \frac{x - \mu}{\sigma}$

Where:

• $x$ is the raw score.
• $\mu$ is the population mean.
• $\sigma$ is the standard deviation.

Interpreting the Result

The resulting number tells you exactly where the value sits:

• $z = 0$: The value is exactly at the mean.
• $z = +1$: The value is one standard deviation above the mean.
• $z = -1$: The value is one standard deviation below the mean.

Worked Example: Comparing Two Classes

Imagine two different classes both have a mean grade of 80. A student in both classes receives an 85. Does the student perform better in one class than the other?

Class A (Small spread): $\mu = 80$, $\sigma = 5$, $x = 85$ $z = \frac{85 - 80}{5} = 1$

Class B (Large spread): $\mu = 80$, $\sigma = 10$, $x = 85$ $z = \frac{85 - 80}{10} = 0.5$

Even though the raw score (85) is the same, the student in Class A has a higher z-score. This means their grade is more "unusual" or higher in relative terms compared to their peers than the student in Class B.

Probabilities and Percentiles

When a distribution is approximately normal, we can use z-scores to determine the probability of a value occurring or to find a percentile. A percentile tells us the percentage of values that fall below a specific score.

For example, a z-score of $+1$ corresponds to approximately the 84th percentile, meaning about 84.1% of the population scores lower than that value.

One-Tailed vs. Two-Tailed Probabilities

When researchers look for "unusual" values, they must decide which direction to look:

1. One-tailed probability: This measures the probability in a single direction (e.g., "What is the probability of scoring above a certain value?"). For $z = +1$, the one-tailed probability of being above that score is approximately 15.9%.
2. Two-tailed probability: This measures the probability of being "extreme" in either direction (e.g., "What is the probability of scoring more than one standard deviation away from the mean, either much higher or much lower?"). For $z = \pm 1$, the two-tailed probability is approximately 31.7%.

The Standard Normal Distribution

The standard normal distribution is a special case where the mean is always $0$ and the standard deviation is always $1$. When we convert any normal distribution into this format using z-scores, we create a "standardized" scale. This is what allows us to use Z-tables to quickly look up probabilities and make statistical inferences.

By mastering z-scores, you move from simply describing data to understanding the relative significance of every individual value within a larger population.