Question

How do you find the z-score for an IQ test score of 142 when the mean is 100 and the standard deviation is 15?

Answer 1

Hint: To obtain the z-score of the given data use the standardizing formula of it. Firstly we will write down all the values given in the question. Then we will write down the formula for measuring z-score and according to the initial of the formula put the values in the formula. Finally we will solve them to get the desired answer.

Complete step by step answer:
The values given are:
The score $=142$
Mean $=100$
Standard Deviation $=15$
We know that the standard formula to calculate z-score is given as below:
$z=\dfrac{X-\mu }{\sigma }$…….$\left( 1 \right)$
Where,
$X=142$ (The score)
$\mu =100$ (Mean)
$\sigma =15$ (Standard Deviation)
Here, we have the Z-score as the difference between the Mean of the population and the raw score which is divided by the standard deviation of that population.
On substituting above values in equation (1) and simplifying it we get,
$\begin{align}
  & z=\dfrac{142-100}{15} \\
 & \Rightarrow z=\dfrac{42}{15} \\
 & \therefore z=2.8 \\
\end{align}$

Hence the z-score of given IQ test is $2.8$.

Note: Z-score usually tells how many standard deviations away the raw score is from the mean value. If the z-score is positive that means the score is higher than the mean average and if a z-score is negative it means the raw score is below the mean average. We also refer to the Z-score as standard score as it is used to compare the score on different variables by the mean of standardizing the distribution given. The z-score is calculated by subtracting the raw score from the population mean and the divide it by the population standard deviation.