Normal Distribution Properties to Know for Intro to Probability for Business

Normal distribution is key in probability for business, showing how data behaves in a predictable, bell-shaped pattern. Understanding its properties helps in analyzing data, making informed decisions, and applying statistical methods effectively in real-world scenarios.

1. Bell-shaped symmetry

   • The normal distribution graph is symmetric around the mean.
   • The left and right halves of the curve are mirror images.
   • This symmetry implies that probabilities are evenly distributed around the mean.

2. Mean, median, and mode are equal

   • In a normal distribution, the mean, median, and mode all occur at the center of the distribution.
   • This equality simplifies analysis and interpretation of data.
   • It indicates that the data is evenly distributed around the central value.

3. 68-95-99.7 rule (empirical rule)

   • Approximately 68% of data falls within one standard deviation of the mean.
   • About 95% of data lies within two standard deviations.
   • Nearly 99.7% of data is found within three standard deviations, providing a quick way to assess data spread.

4. Standard normal distribution (z-score)

   • The standard normal distribution has a mean of 0 and a standard deviation of 1.
   • Z-scores measure how many standard deviations an element is from the mean.
   • Z-scores allow for comparison between different normal distributions.

5. Area under the curve equals 1

   • The total area under the normal distribution curve represents the total probability, which equals 1.
   • This property ensures that all possible outcomes are accounted for.
   • It is fundamental for calculating probabilities and understanding distribution behavior.

6. Defined by mean and standard deviation

   • The shape and position of the normal distribution are determined by its mean (center) and standard deviation (spread).
   • A higher standard deviation results in a wider curve, while a lower standard deviation produces a steeper curve.
   • This relationship is crucial for modeling real-world data.

7. Probability density function

   • The probability density function (PDF) describes the likelihood of a random variable taking on a specific value.
   • The area under the PDF curve over an interval gives the probability of the variable falling within that interval.
   • The PDF is essential for understanding the distribution of continuous random variables.

8. Cumulative distribution function

   • The cumulative distribution function (CDF) represents the probability that a random variable is less than or equal to a certain value.
   • The CDF is a non-decreasing function that approaches 1 as the variable approaches infinity.
   • It is useful for determining probabilities over intervals and for statistical inference.

9. Linear combinations of normal distributions

   • The sum of independent normal random variables is also normally distributed.
   • This property allows for the modeling of complex systems using simpler normal distributions.
   • It is important for risk assessment and decision-making in business contexts.

10. Central Limit Theorem

    • The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as sample size increases, regardless of the original distribution.
    • This theorem is fundamental for making inferences about population parameters based on sample statistics.
    • It underpins many statistical methods and is crucial for hypothesis testing and confidence intervals.