4.2 Normal Distribution

Normal Distribution

  • Half the data is on the left of the peak, half on the right
  • Mean, median and mode are all the same value
  • Can't find value $P(X=N)$ for any $N$ as you can't integrate to an absolute value
  • Curve has points of inflection one standard deviation from the mean
    • This means the curve changes concavity
  • Standard Deviations:
    • $1\times\sigma$ = 68% of the data
    • $2\times\sigma$ = 95% of the data
    • $3\times\sigma$ = 99.7% of the data

Syntax

$X \sim B(n,p)$ $n$ is the mean, $p$ is the square of the standard deviation $X \sim N(\mu, \sigma^2)$ $\mu$ represents the center $\sigma$ represents one standard deviation

Example

  1. Diameters of a rivet modelled by $X \sim N(8, 0.2^2)$ a) Find $P(X>8)$ 50% b) Find $P(7.8 < X < 8.2)$ 1 sd so 68%
  1. Criteria for joining Mensa is an IQ of at least 131. Assuming that IQ has the distribution $X \sim N(100, 15^2)$ for a population a) What percentage of people are eligible? $P(X>=131)$ = 1.9%