The Normal Distribution

Starting point: Years 9 and 10

Bell-shaped data

Data that is symmetric and bell-shaped has most values near the middle and fewer further out. For this shape, the mean, median and mode are all about equal.

z-scores

A z-score tells you how many standard deviations a value is from the mean:

$$z=\frac{x-\mu }{\sigma }$$

• $z=0$: exactly at the mean.
• $z=2$: two standard deviations above the mean.
• $z=-1$: one standard deviation below.

Quick summary

• Normal data is bell-shaped and symmetric. Mean, median and mode are equal.
• $z=\frac{x-\mu }{\sigma }$ counts standard deviations from the mean.
• 68% within 1 sd, 95% within 2 sd, 99.7% within 3 sd.
• Convert to $Z\sim N(0,1)$ and use $P(Z\le z)$ tables, with symmetry and $1-$ for the other side.
• Work backwards from a probability to $z$, then $x=\mu +z\sigma$.

Quiz