(Lecture 14) Normal Distributions and the 68-95-99.7 Rule

Section 6.2: Probabilities for Bell-Shaped Distributions

The Normal Distribution

The normal distribution is a fundamental probability distribution in statistics, characterized by its symmetric, bell-shaped curve. It is defined by two parameters: the mean ($\mu$), which determines the center of the distribution, and the standard deviation ($\sigma$), which measures the spread or variability.

• Definition: A continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

• Notation: $X \sim N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance.

• Key Property: The probability of observing values within a given number of standard deviations from the mean is the same for all normal distributions, regardless of the specific values of $\mu$ and $\sigma$.

The 68-95-99.7 Rule (Empirical Rule)

The 68-95-99.7 Rule (also known as the Empirical Rule) describes the approximate percentage of data that falls within one, two, and three standard deviations of the mean in a normal distribution.

• 68% Rule: About 68% of the data falls within one standard deviation of the mean ($\mu \pm \sigma$).

• 95% Rule: About 95% of the data falls within two standard deviations of the mean ($\mu \pm 2\sigma$).

• 99.7% Rule: About 99.7% of the data falls within three standard deviations of the mean ($\mu \pm 3\sigma$).

Example: For adult women's heights, if $\mu = 65$ inches and $\sigma = 3.5$ inches:

• 68% are between $65 - 3.5 = 61.5$ and $65 + 3.5 = 68.5$ inches.

• 95% are between $65 - 7 = 58.0$ and $65 + 7 = 72.0$ inches.

• 99.7% are between $65 - 10.5 = 54.5$ and $65 + 10.5 = 75.5$ inches.

Z-Scores and the Standard Normal Distribution

A z-score indicates how many standard deviations a value $x$ is from the mean $\mu$ of a normal distribution. The standard normal distribution is a special case of the normal distribution with $\mu = 0$ and $\sigma = 1$.

• Z-Score Formula:

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

• Interpretation: A positive z-score means the value is above the mean; a negative z-score means it is below the mean.

• Standard Normal Distribution: The distribution of z-scores from any normal distribution is itself a normal distribution with mean 0 and standard deviation 1.

Using the Standard Normal Table (Table A)

The standard normal table (Table A) provides cumulative probabilities for z-scores, i.e., the probability that a standard normal random variable is less than or equal to a given value.

• To find the probability that $Z < z$:

  • Find the z-score to the first decimal place in the leftmost column.

  • Find the second decimal place in the top row.

  • The intersection gives the cumulative probability.

• Example: For $z = 1.43$, the table gives $P(Z < 1.43) = 0.9236$.

• The probability above $z$ is $1 - P(Z < z)$.

Finding Probabilities and Values Using Z-Scores

There are two main types of problems involving the normal distribution:

1. Given a value $x$, find the probability:

   • Convert $x$ to a z-score using $z = \frac{x - \mu}{\sigma}$.

   • Use the standard normal table to find the cumulative probability.

2. Given a probability, find the value $x$:

   • Find the z-score corresponding to the cumulative probability in the table.

   • Convert the z-score to $x$ using $x = \mu + z\sigma$.

Example: To find the value of $z$ for a cumulative probability of 0.025, look up 0.025 in the table body. The corresponding $z$ is approximately -1.96. This means 2.5% of the distribution lies below $z = -1.96$.

Comparing Scores from Different Normal Distributions

Z-scores allow for the comparison of values from different normal distributions by standardizing them.

• Example: Comparing SAT and ACT scores:

  • SAT: $\mu = 500$, $\sigma = 100$, score = 650

  • ACT: $\mu = 21$, $\sigma = 7$, score = 30

  • SAT z-score: $z = \frac{650 - 500}{100} = 1.50$

  • ACT z-score: $z = \frac{30 - 21}{7} = 1.29$

  • The higher z-score indicates a better relative performance.

Summary Table: The 68-95-99.7 Rule

Interval	Percentage of Data	Formula
$\mu \pm 1\sigma$	68%	$[\mu - \sigma, \mu + \sigma]$
$\mu \pm 2\sigma$	95%	$[\mu - 2\sigma, \mu + 2\sigma]$
$\mu \pm 3\sigma$	99.7%	$[\mu - 3\sigma, \mu + 3\sigma]$

Additional info:

• The normal distribution is widely used in inferential statistics, including hypothesis testing and confidence intervals.

• Many real-world phenomena (e.g., heights, test scores) are approximately normally distributed.

• The empirical rule is an approximation; exact probabilities can be found using the standard normal table.