6.2 Using the Normal Distribution

Normal Distribution Calculations and Interpretation

Normal distributions let you calculate the probability of observing values within specific ranges. By converting raw data to z-scores, you can use a single standardized table (or calculator function) to find probabilities for any normal distribution, regardless of its original mean and standard deviation.

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Probability Calculations in Normal Distributions

Z-scores measure how many standard deviations a value sits from the mean. They're the bridge between your raw data and the standard normal table.

$z = \frac{x - \mu}{\sigma}$

• $x$ = the observed value
• $\mu$ = the population mean
• $\sigma$ = the population standard deviation

A z-score of 1.5 means the value is 1.5 standard deviations above the mean. A z-score of -2 means it's 2 standard deviations below.

Standardizing a normal distribution means converting it from $N(\mu, \sigma)$ to $N(0, 1)$, the standard normal distribution. Once standardized, every problem uses the same table or calculator function.

Finding probabilities with the standard normal table:

The table gives you the cumulative probability to the left of a z-score. That's the proportion of data at or below that value.

• "Less than" problems: Look up the z-score directly. The table value is your answer.
• "Greater than" problems: Subtract the table value from 1.

$P(X > x) = 1 - P(X \leq x)$

• "Between two values" problems: Find the cumulative probability for each boundary, then subtract.

$P(a < X < b) = P(X < b) - P(X < a)$

Interpretation of Normal Distribution Graphs

Normal distributions have a few defining properties worth memorizing:

• The curve is symmetric and bell-shaped, centered on the mean.
• The mean, median, and mode are all equal, sitting at the center.
• The Empirical Rule (68-95-99.7 Rule) describes how data clusters:
  • ~68% of data falls within $\pm 1\sigma$ of the mean
  • ~95% falls within $\pm 2\sigma$
  • ~99.7% falls within $\pm 3\sigma$
• The total area under the curve equals 1 (representing 100% probability).

The Empirical Rule is useful for quick estimates. If someone asks "is a value of 98 unusual?" and the distribution has $\mu = 74$ and $\sigma = 8$, you can note that 98 is 3 standard deviations above the mean, placing it in the outer 0.3% of the distribution. That's quite rare.

When reading a normal curve graph, the shaded area between any two points represents the probability of a randomly selected value falling in that range. A wider shaded region means a higher probability; a narrow region in the tails means a lower one.

Technology for Normal Distribution Analysis

Graphing calculators and statistical software handle these calculations without needing a z-table. The two main functions you'll use:

Normal CDF (cumulative distribution function):

1. Enter the lower bound and upper bound of your range.
2. Enter the mean ($\mu$) and standard deviation ($\sigma$).
3. The output is the probability of a value falling within that range.

For "less than" problems, use $-10^{99}$ (or a very large negative number) as the lower bound. For "greater than" problems, use $10^{99}$ as the upper bound.

Inverse Normal (quantile function):

This works in reverse: you provide a probability and get back the corresponding value.

1. Enter the cumulative probability (area to the left).
2. Enter the mean and standard deviation.
3. The output is the value in the original distribution that corresponds to that percentile.

Theoretical Foundations and Applications

The Central Limit Theorem explains why normal distributions appear so often: when you average many independent random variables, the result tends toward a normal distribution regardless of the original shape. This is why heights, test scores, measurement errors, and many biological traits approximate a bell curve.

Normal distributions are foundational for statistical inference. Confidence intervals and hypothesis tests rely heavily on normal (and related) distributions. The skills you're building here with z-scores and probability calculations carry directly into those later topics.