• Topic: z-scores, standard deviation, mean, and percentages
• Example Used: IQ Scores
  • mean = 100
  • standard deviation = 15

The mean is the average of the data:

\[\mu=\frac{1}{n}\sum_{i=1}^{n}x_i\]

The standard deviation is the square root of the average squared distance from the mean:

\[\sigma=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\mu)^2}\]

For IQ: \(\mu=100\) and \(\sigma=15\).

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

\[z=\frac{x-\mu}{\sigma}\]

• Positive z: above the mean
• Negative z: below the mean
• Example: an IQ of 130 gives

\[z=\frac{130-100}{15}=2\]

For data that is roughly bell shaped:

\[P(\mu-\sigma\le X\le\mu+\sigma)\approx68.27\%\] \[P(\mu-2\sigma\le X\le\mu+2\sigma)\approx95.45\%\] \[P(\mu-3\sigma\le X\le\mu+3\sigma)\approx99.73\%\]

For IQ this means about 68% of people score between 85 and 115 or one standard deviation from the mean.

z=seq(-4,4,length.out=800)
df=data.frame(z=z,y=dnorm(z))

p1=ggplot(df,aes(z,y))+
  geom_area(data=subset(df,abs(z)<=3),fill="khaki")+
  geom_area(data=subset(df,abs(z)<=2),fill="gold")+
  geom_area(data=subset(df,abs(z)<=1),fill="darkred")+
  geom_line(linewidth=1)+
  annotate("text",x=0,y=0.18,label="68.27%",color="white")+
  annotate("text",x=1.5,y=0.06,label="95.45%")+
  annotate("text",x=2.6,y=0.02,label="99.73%")+
  scale_x_continuous(name="z-score",breaks=-3:3,sec.axis=sec_axis(~mu+sigma*.,name="IQ score",breaks=seq(55,145,15)))+
  labs(title="The Empirical Rule",y="Density")+
  theme_minimal()

\[z=\frac{130-100}{15}=2\qquad P(Z\le2)\approx0.9772\]