Question

Assume that adults have IQ scores that are normally distributed with a mean of 100.4 and a standard deviation 19.7. Find the first quartile Q1, which is the IQ scoreseparating the bottom 25% from the top 75%. (Hint: Draw a graph.)The first quartile is?

Answer 1

The first quartile is the number in between the lowest number of a data set and the median. To find it, I would use a Z table as shown below

The z score that corresponds to 0.25 is -0.67 on the table above.

Using the equation of z-score below

$\begin{gathered} z=\frac{x-\mu}{\sigma} \\ \text{Where}\mu=\operatorname{mean},\sigma=s\tan dard\text{ deviation} \end{gathered}$

Where x is the value we're looking for and σ and μ are the standard deviations and the mean given in the problem.

Given that the mean is 100.4 and the standard deviation is 19.7, make x the subject of the z-score formula and substitute for the mean and standard deviation as shown below

$\begin{gathered} z=\frac{x-\mu}{\sigma} \\ x-\mu=z\times\sigma \\ x=z\times\sigma+\mu \\ z=-0.67,\mu=100.4,\sigma=19.7 \end{gathered}$$\begin{gathered} x=-0.67\times19.7+100.4 \\ x=-13.199+100.4 \\ x=87.201 \end{gathered}$

Hence, the first quartile is 87.201