Study Notes on Normal Distribution and Standard Normal Distribution

Normal Distribution

Key Terms

  • Normal distribution:

    • Often described as a bell curve.
    • Defined by two parameters: mean and standard deviation.
    • As a continuous distribution, the total area under the curve is equal to 1.

  • Standard deviation:

    • A measure of how far data values are from their mean (denoted as $\mu$).

Understanding Standard Deviation

  • Standard deviation indicates the spread of the normal distribution.
    • A high standard deviation:
    • Results in a graph that is short and spread out.
    • A low standard deviation:
    • Results in a graph that is tall and skinny.
  • Example: Distribution A has a larger standard deviation than Distribution B.

Characteristics of Normal Distribution

  • Normal distributions exhibit symmetry around their mean.
  • The mean, median, and mode of a normal distribution are all equal.
  • The area under the normal curve equals 1.
  • A normal distribution is defined by:
    • Mean ($\mu$)
    • Standard deviation ($\sigma$)
  • The standard normal distribution is a normal distribution where:
    • Mean = 0
    • Standard deviation = 1

The Standard Normal Distribution

  • The Standard Normal Distribution consists of standardized values known as z-scores.
    • Z-scores are measured in units of standard deviation.
    • Example Z-scores:
    • A z-score of 1.25 indicates a value 1.25 standard deviations to the right of the mean.
    • A z-score of -3.14 indicates a value 3.14 standard deviations to the left of the mean.

Standardizing a Normally Distributed Random Variable

  • Example with Jerome:
    • Average points per game: 16
    • Standard deviation: 4
    • Let X = points per game, then $X \sim N(16, 4)$.
    • If Jerome scores 10 points in a game, calculate the z-score:
    • Calculation:
      z=X−μσz = \frac{X - \mu}{\sigma}
      =10−164= \frac{10 - 16}{4}
      =−64=−1.5= \frac{-6}{4} = -1.5
    • Interpretation: The z-score of -1.5 indicates that 10 points is 1.5 standard deviations to the left of the mean.

Mean, Standard Deviation, and Z-scores

  • Formula to find the z-score:
    • z=X−μσz = \frac{X - \mu}{\sigma}
  • Using the z-score, if we know any three quantities (z-score, value, mean, or standard deviation), we can find the fourth.
    • Example where z = 4:
    • Given: $z = 4$, $X = -3$, and standard deviation = 2.5.
      • To find the mean ($\mu$):
        4=−3−μ2.54 = \frac{-3 - \mu}{2.5}
      • Multiplying both sides by 2.5:
        10=−3−μ10 = -3 - \mu
      • Rearranging gives:
        μ=−3−10=−13\mu = -3 - 10 = -13

Further Example: Karen's Points per Game

  • Example with Karen:
    • Average points per game: 18
    • Z-score for scoring 12 points: -3.
    • Calculation to find standard deviation:
    • Formula:
      −3=12−18σ-3 = \frac{12 - 18}{\sigma}
    • Rearranging gives:
      −3σ=12−18-3 \sigma = 12 - 18
      −3σ=−6⇒σ=2-3 \sigma = -6 \Rightarrow \sigma = 2