6.4 Normal Distribution—Pinkie Length

Normal Distribution and Pinkie Length

The normal distribution gives you a way to model how continuous measurements (like pinkie length) spread out around an average. Because so many real-world variables follow this pattern, learning to work with it lets you calculate probabilities, compare individuals to a population, and estimate population parameters with confidence intervals.

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Probabilities Using Normal Distribution

A normal distribution is a continuous, symmetric, bell-shaped probability distribution. Two numbers completely define its shape:

• Mean ($\mu$): the center of the distribution
• Standard deviation ($\sigma$): how spread out the data is around the mean

The empirical rule (68-95-99.7 rule) gives you quick benchmarks: about 68% of observations fall within $\pm 1\sigma$ of the mean, 95% within $\pm 2\sigma$, and 99.7% within $\pm 3\sigma$.

To find the probability that a randomly selected pinkie length falls above or below some value, you first convert that value to a z-score, which tells you how many standard deviations it sits from the mean:

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

A positive z-score means the observation is above the mean; a negative z-score means it's below. Once you have the z-score, you can look up the cumulative probability (area under the curve to the left) using a standard normal table or calculator.

Example walkthrough: Suppose pinkie lengths are normally distributed with $\mu = 6.5$ cm and $\sigma = 0.5$ cm. What's the probability someone's pinkie is shorter than 5.8 cm?

1. Calculate the z-score: $z = \frac{5.8 - 6.5}{0.5} = -1.4$

2. Look up $z = -1.4$ in the standard normal table (or use normalcdf): the area to the left is approximately 0.0808.

3. Interpretation: about 8.08% of the population has a pinkie length shorter than 5.8 cm.

For "greater than" probabilities, subtract the table value from 1. For "between" probabilities, find the area to the left of each z-score and subtract.

Interpretation of Percentiles and Z-Scores

A percentile tells you the percentage of observations that fall below a given value. If a pinkie length is at the 75th percentile, that means 75% of the population has a shorter pinkie.

Finding a percentile from a measurement:

1. Calculate the z-score for the pinkie length.
2. Use a z-table or calculator to find the area to the left of that z-score.
3. That area, expressed as a percentage, is the percentile rank.

Going the other direction (finding a measurement from a percentile):

1. Look up the desired percentile in the body of the z-table to find the corresponding z-score. For example, the 90th percentile corresponds to $z \approx 1.28$.
2. Solve for $x$: $x = \mu + z \cdot \sigma$

Z-scores also let you compare across different distributions. A z-score of 0 means the observation equals the mean. The magnitude of the z-score tells you how far from typical the observation is, regardless of the original units.

Confidence Intervals for Population Parameters

A confidence interval gives a range of plausible values for a population parameter, such as the true mean pinkie length. The formula for a confidence interval when $\sigma$ is known:

$\bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}$

• $\bar{x}$ = sample mean
• $z^*$ = critical z-value for your confidence level
• $\sigma$ = population standard deviation
• $n$ = sample size

Common critical values: $z^* = 1.645$ for 90% confidence, $z^* = 1.96$ for 95%, and $z^* = 2.576$ for 99%.

How to interpret it correctly: A 95% confidence interval does not mean there's a 95% chance the true mean is inside your specific interval. It means that if you repeated the sampling process many times, about 95% of the resulting intervals would capture the true population mean.

Three factors control the width of the interval:

• Sample size ($n$): Larger samples produce narrower intervals because $\sqrt{n}$ is in the denominator.
• Variability ($\sigma$): More spread in the data widens the interval.
• Confidence level: Higher confidence (say 99% vs. 95%) requires a larger $z^*$, which widens the interval. You're trading precision for greater certainty.

Additional Normal Distribution Concepts

• The standard normal distribution is the special case where $\mu = 0$ and $\sigma = 1$. Every z-score calculation converts your data into this distribution.
• The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as sample size increases, even if the underlying population isn't normal. This is why the normal distribution shows up so often in inference.
• Skewness measures asymmetry. A perfectly normal distribution has skewness of 0. Positive skew means a longer right tail; negative skew means a longer left tail.
• Kurtosis measures how heavy the tails are compared to a normal distribution. High kurtosis means more extreme outliers are likely.