The normal distribution The normal curve Shape defined

The normal distribution The normal curve: Shape defined by the standard deviation and the mean

The normal distribution FREQUENCY Various weights of a group of female students 20 30 40 50 WEIGHT (kg) 60 70 80

The normal distribution The equation of the normal curve Allows us to compute the height of the curve y for a given value of x Y

The normal distribution As the term (x- ) becomes smaller, y becomes larger and is at a maximum when x = . s=1 s = 1. 5 s=2 PROBABILITY 0. 4 0. 3 0. 2 0. 1 0. 0 -5 -4 -3 -2 -1 0 X 1 2 3 4 5

The normal distribution As the term (x- ) becomes smaller, y becomes larger and is at a maximum when x = . =0 =1 =2 PROBABILITY 0. 4 0. 3 0. 2 0. 1 0. 0 -5 -4 -3 -2 -1 0 X 1 2 3 4 5

The normal distribution A normal curve is symmetrical Axis of symmetry passes through the baseline where x = (one of the parameters of the curve) Theoretically, two tails never touch the horizontal axis.

The normal distribution Vertical axis of the distribution re-scaled by dividing by the number of observations - becomes a probability distribution The total probability encompassed by the density is 1.

The normal distribution Total area under the curve is 100%: The area bounded by one standard deviation on either side of the central axis is approximately 68. 26% of the total area.

Normal distribution Probability determination example • A normal distribution of values • Mean = 50 • Standard deviation = 15 • What is the probability of finding a value greater than 75?

Normal distribution Z (standard) distribution To convert a data point (Xi) to a Z score (Zi): Zi = Xi - X S

Normal distribution Z (standard) distribution • • Xi = 75, mean = 50, standard deviation = 15 Z score in example is 1. 67 Table value = 0. 0475 Probability = 0. 0475

Normal distribution Z (standard) distribution Area under the two tails: (0. 0475 x 2) = 0. 095 1 – 0. 095 = 0. 905 0. 0475