4.1 Some rules of thumb
The Normal distribution contains some very useful properties which we can use as rules of thumb. This is best explained via an example. Let us again consider a continuous random variable \(X\) that denotes the height in cm of university students. For argument's sake, let us further assume that \(X\) is Normally distributed with:
- \(\text{E}(X) = \mu = 172.38\)
- \(\text{SD}(X) = \sigma = 9.85\)
Recall that we can express the distribution of an arbitrary normally distributed random variable \(X\) as \(X \sim N(\mu, \sigma^2)\). Therefore, we can succinctly write down the distribution of the height variable \(X\) as:
- \(X \sim N(172.38, 9.85^2)\)
Three very useful rules of thumb are as follows:
Some rules of thumb for an approximately normal data set
- About 68% of values are within 1 standard deviation of the mean
- About 95% of values are within 2 standard deviations of the mean
- About 99.7% of values are within 3 standard deviations of the mean
Considering here that the mean is \(172.38\) and the standard deviation is \(9.85\), we can apply these rules of thumb as follows:
- About 68% of students' heights are within the range \(172.38 \pm 1\times9.85 = (172.38 - 9.85, 172.38 + 9.85) = (162.53, 182.23)\)
- About 95% of students' heights are within the range \(172.38 \pm 2\times 9.85 = (172.38 - 19.7, 172.38 + 19.7) = (152.68, 192.08)\)
- About 99.7% of students' heights are within the range \(172.38 \pm 3\times 9.85 = (172.38 - 29.55, 172.38 + 29.55) = (142.83, 201.93)\)
This is visually represented in the below plot:
