📊Honors Statistics Unit 6 – The Normal Distribution

What's the Normal Distribution?

• Continuous probability distribution that is symmetrical and bell-shaped
• Defined by two parameters: mean ($\mu$) and standard deviation ($\sigma$)
• Mean determines the center of the distribution
• Standard deviation determines the spread or width of the distribution
• Larger standard deviation results in a wider, flatter distribution
• Smaller standard deviation results in a narrower, taller distribution
• Total area under the curve is equal to 1 or 100%

Key Characteristics

• Symmetrical about the mean with 50% of the data on each side
• Unimodal with a single peak at the mean
• Bell-shaped curve with tails approaching but never touching the x-axis
• Mean, median, and mode are all equal and located at the center of the distribution
• Skewness is zero, indicating perfect symmetry
• Kurtosis is a measure of the thickness of the tails relative to a normal distribution
  • Positive kurtosis has thicker tails (leptokurtic)
  • Negative kurtosis has thinner tails (platykurtic)

Standard Normal Distribution (Z-scores)

• Special case of the normal distribution with a mean of 0 and a standard deviation of 1
• Z-scores represent the number of standard deviations an observation is from the mean
• Formula for calculating Z-scores: $Z = \frac{X - \mu}{\sigma}$
  • $X$ is the individual value
  • $\mu$ is the population mean
  • $\sigma$ is the population standard deviation
• Positive Z-scores indicate values above the mean
• Negative Z-scores indicate values below the mean
• Z-scores allow for comparison of values from different normal distributions

Probability and Area Under the Curve

• Probability is represented by the area under the normal curve
• Total area under the curve is always equal to 1 or 100%
• Probability of a specific value is 0 since the area of a single point is 0
• Probability is calculated for intervals or ranges of values
• Standard normal distribution tables (Z-tables) are used to find probabilities
  • Z-tables provide the area to the left of a given Z-score
  • Subtract areas to find the probability between two Z-scores
• Probability density function (PDF) gives the height of the curve at any point

Empirical Rule (68-95-99.7 Rule)

• Describes the percentage of data within 1, 2, and 3 standard deviations of the mean
• Approximately 68% of data falls within 1 standard deviation of the mean ($\mu \pm \sigma$)
• Approximately 95% of data falls within 2 standard deviations of the mean ($\mu \pm 2\sigma$)
• Approximately 99.7% of data falls within 3 standard deviations of the mean ($\mu \pm 3\sigma$)
• Useful for quickly estimating the proportion of data in a given range
• Only applies to normal distributions

Applications in Real Life

• Heights and weights of a population often follow a normal distribution
• Scores on standardized tests (SAT, ACT, GRE) are designed to follow a normal distribution
• Measurement errors in manufacturing processes often follow a normal distribution
• Financial markets and stock prices can be modeled using normal distributions
• Quality control in manufacturing relies on the properties of the normal distribution
  • Six Sigma methodology aims to have 99.99966% of products within specification limits
• Insurance companies use normal distributions to model claim amounts and set premiums

Common Misconceptions

• Not all data follows a normal distribution; it's important to check assumptions
• The mean and standard deviation are not resistant to outliers; use median and IQR for skewed data
• The normal distribution is a continuous distribution, not discrete
• The normal distribution extends infinitely in both directions, but extreme values are highly unlikely
• The empirical rule is an approximation and may not hold for small sample sizes
• Z-scores do not change the shape of the distribution, only the scale and location
• Skewness and kurtosis are not directly related to the mean and standard deviation

Practice Problems and Examples

1. Given a normal distribution with $\mu=70$ and $\sigma=5$, find the probability of randomly selecting a value between 65 and 75.
2. A manufacturing process produces bolts with lengths that follow a normal distribution with $\mu=10$ cm and $\sigma=0.5$ cm. What proportion of bolts will have lengths between 9 cm and 11 cm?
3. The weights of a certain species of fish follow a normal distribution with $\mu=2.5$ kg and $\sigma=0.3$ kg. If a fish is randomly selected, what is the probability that it weighs more than 3.1 kg?
4. A company administers an aptitude test that follows a normal distribution with $\mu=500$ and $\sigma=100$. If an applicant scores 650, what is their Z-score?
5. The heights of adult males in a population follow a normal distribution with $\mu=175$ cm and $\sigma=8$ cm. What is the probability that a randomly selected adult male is shorter than 160 cm?