Mean, standard deviation and the Normal distribution | Vose Software

# Mean, standard deviation and the Normal distribution

For a Normal distribution only, the areas bounded 1, 2 and 3 standard deviations either side of the mean contain approximately 68.27%, 95.45% and 99.73% of the distribution:

Since a lot of distributions look similar to a Normal under certain conditions, people often think of 70% of a distribution being reasonably contained within one standard deviation either side of the mean, but this rule of thumb must be used with care. If it is applied to a distribution that is significantly non-Normal, like an Exponential distribution, the error can be quite large.

### Example

Panes of bullet proof glass manufactured at a factory have a mean thickness over a pane that is Normally distributed with mean 25mm and variance 0.04mm2. If 10 panes are purchased, what is the probability that all the panes will have a mean thickness between 24.8mm and 25.4mm2

The distribution of the mean thickness of a randomly selected pane is Normal(25,0.2)mm, since the variance is the square of the standard deviation. 24.8mm is 1 standard deviation below the mean, 25.4mm is 2 standard deviations above the mean. The probability p that a pane lies between 24.8mm and 25.4mm is then half the probability of lying +/- 1 standard deviation from the mean plus half the probability of lying +/- 2 standard deviations from the mean, i.e. p » (68.27% + 95.45%)/2 = 81.86%. The probability that all panes will have a mean thickness between 24.8mm and 25.4mm, provided that they are independent of each other, is therefore » (81.86%)10 = 13.51%.