Normal Distribution

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math

Is a continuous probability distribution defined as a bell-curve with the following properties:

1. The distribution is symmetrical about the mean \( \mu \).
2. The mode, median and mean are all equal.
3. About 68% of values lie within 1 standard deviation of the mean. About 95% of values lie within 2 standard deviations. About 100% of values lie within 3 standard deviations.
4. It has 2 parameters the mean \( \mu \) and the standard deviation \( \sigma \).

Note: Because the normal distribution is a probability distribution, the total area under the curve is 1.

\begin{figure}
  \centering
  \begin{tikzpicture}
    \begin{axis}[
      no markers, domain=0:8, samples=100,
      axis lines*=left, xlabel=$x$, ylabel=$y$,
      every axis y label/.style={at=(current axis.above origin),anchor=south},
      every axis x label/.style={at=(current axis.right of origin),anchor=west},
      height=5cm, width=12cm,
      xtick={4}, ytick=\empty,
      enlargelimits=false, clip=false, axis on top,
      grid = major
      ]
      \addplot [very thick,black] {gauss(4,1)};
    \end{axis}

  \end{tikzpicture}
  \caption{A normal distribution with a standard-deviation of 1 and a mean of 4.}
\end{figure}

We denote a normal distribution using \( X~N(\mu, \sigma^2) \). \( \mu \) defines where the centre of the curve is. \( \sigma \) alters the spread of the curve. Probabilities with a normal distribution are found by finding the area of the curve between two points. For example what's the probability of a that a male student selected at random has a height between 160cm and 175cm? To find this we'd a draw a line on the distribution at \( x = 160 \) and at \( x = 175 \) and then use the area under that region as the probability.

Standardised Normal Distribution

Is a normal distribution with mean 0 and standrad deviation 1. It's denoted by the random variable \(Z\), meaning \( Z~N(0, 1^2) \).

We can standardise, convert from, any normal distribution to the standardised normal distribution using the formula: \[ z = \frac{x - \mu}{\sigma} \]