Normal Distribution

Normal Distribution

Introduction to Normal Distribution

  • The binomial probability distribution, when plotted, often shows a pattern where probabilities are low on the outside and high in the middle.
  • This pattern closely resembles a continuous curve known as the normal distribution, modeled by Carl Gauss.
  • The normal distribution is also known as a bell-shaped curve due to its visual appearance.

Key Concepts

Definition

  • Normal Distribution: A 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.

Parameters

  • To define a normal distribution, only the mean and standard deviation are needed.

    • Mean: The average value of the data set, located at the center of the curve.
    • Standard Deviation: A measure of the spread or dispersion of the data around the mean.

Example

  • Consider the math portion of the SAT test given to a large group of students.

    • Let's say the mean score is 500 and the standard deviation is 100.

Properties of the Normal Curve

Symmetry

  • The normal curve is symmetric around the mean.
  • For large datasets, the mean and median are approximately the same.

Spread

  • The curve almost reaches the x-axis at three standard deviations to the right and left of the mean.

    • These points serve as horizontal asymptotes, meaning the curve approaches but never touches the x-axis.

Universality

  • The normal distribution applies to almost any set of numbers, whether it's heights, IQ scores, or SAT test scores.
  • Most of the data cluster around the middle, with extremes on both ends.

Percentages within Standard Deviations

  • 68% of the data falls within one standard deviation of the mean.
  • 95% of the data falls within two standard deviations of the mean.
  • 99.7% of the data falls within three standard deviations of the mean.

Calculating Probabilities

Area Under the Curve

  • The area under the entire curve represents 100% probability.
  • The area between any two points on the curve represents the probability of a value falling within that range.

Example Problem

  • What is the probability that a student scored between 600 and 700 on the SAT test (mean 500, standard deviation 100)?

    • 68% of students score between 400 and 600 (one standard deviation from the mean).
    • 34% score between 500 and 600.
    • 95% of students score between 300 and 700 (two standard deviations from the mean).
    • Subtracting 68% from 95% leaves 27%, which means 13.5% score between 600 and 700, and 13.5% score between 300 and 400.

General Rule

  • For any normally distributed data, the 68-95-99.7 rule can be used to answer probability questions.

Detailed Breakdown of Percentages

  • x̄ (x-bar), the mean, is in the middle.

  • Six standard deviations wide (three to the right and three to the left).

  • Between one standard deviation from the mean:

    • 34% on each side, totaling 68%.

  • Between one and two standard deviations from the mean:

    • 13.5% on each side.

  • Between two and three standard deviations from the mean:

    • 2.35% on each side.

  • Outside three standard deviations from the mean:

    • 0.15% on each side (since 99.7% is within three standard deviations).

Application Example: Height of Adult Females

Scenario

  • The average height of adult females in the United States is 5 foot 4 inches (64 inches).
  • The standard deviation is 2 inches.

Calculations

  • Create a normal curve with 64 inches in the middle.

  • Mark standard deviations:

    • 62 inches (one standard deviation below the mean)
    • 66 inches (one standard deviation above the mean)
    • 60 inches (two standard deviations below the mean)
    • 68 inches (two standard deviations above the mean)
    • 58 inches (three standard deviations below the mean)
    • 70 inches (three standard deviations above the mean)

Probability Question

  • What is the probability that a randomly chosen female is between 68 and 70 inches tall?

    • This corresponds to the area between two and three standard deviations above the mean, which is 2.35%.

Generalization

  • As long as the numbers align with standard deviations, probabilities can be easily calculated.
  • For numbers in between standard deviations, a calculator or a table can be used to find the area (probability).