📉Intro to Business 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 while standard deviation controls the spread
• Total area under the curve equals 1, representing all possible outcomes
• Most common values cluster around the mean, with decreasing probability for values further away
  • For example, in a normal distribution of heights, most people will be close to the average height, with fewer people being very short or very tall
• Widely used in statistics due to its well-defined properties and natural occurrence in many real-world phenomena (test scores, measurement errors)

Key Features and Properties

• Symmetry: the left and right halves of the distribution are mirror images of each other
• Unimodal: only one peak, located at the mean
• Mean, median, and mode are all equal and located at the center of the distribution
• Inflection points (where the curve changes from concave to convex) are located at $\mu \pm \sigma$
• 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 one standard deviation of the mean ($\mu \pm \sigma$)
  • Approximately 95% of data falls within two standard deviations of the mean ($\mu \pm 2\sigma$)
  • Approximately 99.7% of data falls within three standard deviations of the mean ($\mu \pm 3\sigma$)
• Skewness and kurtosis are both 0 for a perfect normal distribution

The Standard Normal Distribution

• Special case of the normal distribution with a mean of 0 and a standard deviation of 1
• Denoted as $Z \sim N(0,1)$
• Allows for standardization of any normal distribution using z-scores
• Z-score represents the number of standard deviations a value is from the mean
  • Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean
• Standard normal distribution tables provide probabilities for z-scores, eliminating the need for integration
• Enables comparison of values from different normal distributions on a common scale

Z-Scores and Probability

• Z-score formula: $z = \frac{x - \mu}{\sigma}$, where $x$ is the value of interest, $\mu$ is the mean, and $\sigma$ is the standard deviation
• Z-scores allow for the calculation of probabilities using standard normal distribution tables or software
• Probability of a value being less than, greater than, or between specific z-scores can be determined
  • For example, $P(Z < 1.5)$ represents the probability of a value being less than 1.5 standard deviations above the mean
• Percentiles can be found using z-scores and the standard normal distribution
  • For instance, a z-score of 1.28 corresponds to the 90th percentile, meaning 90% of the data falls below this value

Applications in Business

• Quality control: identifying products that fall outside acceptable limits (usually $\mu \pm 3\sigma$)
• Financial analysis: modeling stock returns, portfolio risk, and option pricing (Black-Scholes model)
• Marketing research: analyzing customer satisfaction scores or product ratings
• Human resources: setting performance benchmarks and evaluating employee performance
• Forecasting: predicting demand, sales, or revenue using historical data and assuming normality
• Operations management: determining optimal inventory levels and reorder points based on lead time and demand variability

Common Misconceptions

• Not all data follows a normal distribution; it's essential to check assumptions before applying normal distribution techniques
• The normal distribution is a continuous distribution, not discrete; it's an approximation for large sample sizes
• The empirical rule (68-95-99.7) is a guideline, not an exact rule; actual percentages may vary slightly
• Z-scores do not indicate the probability directly; they need to be converted using the standard normal distribution
• The mean and standard deviation are sensitive to outliers; robust measures like the median and interquartile range may be more appropriate for skewed or heavy-tailed distributions

Calculating with Normal Distributions

• Finding probabilities: use z-scores and standard normal distribution tables or software (e.g., Excel's

  NORM.DIST

  or

  NORM.S.DIST

  functions)
  • Example: given $X \sim N(100, 15)$, find $P(X < 90)$ by calculating the z-score and using the standard normal distribution
• Finding values: use inverse z-scores and standard normal distribution tables or software (e.g., Excel's

  NORM.INV

  or

  NORM.S.INV

  functions)
  • Example: find the value that corresponds to the 25th percentile in a distribution with $\mu = 50$ and $\sigma = 10$
• Linear transformations: if $X \sim N(\mu, \sigma)$, then $aX + b \sim N(a\mu + b, |a|\sigma)$
  • Example: if test scores follow $N(70, 5)$ and are scaled by a factor of 1.5, the new distribution is $N(105, 7.5)$

Beyond the Basics: Related Concepts

• Central Limit Theorem: the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution
• Confidence intervals: ranges of values that are likely to contain the true population parameter, based on the sample mean and standard error
• Hypothesis testing: using the normal distribution to determine the likelihood of observing a sample statistic under the null hypothesis
• Analysis of Variance (ANOVA): comparing means of multiple groups, assuming normality and equal variances
• Regression analysis: modeling the relationship between variables, with residuals often assumed to follow a normal distribution