Question

How do you find the $z$ score of a percentile?

Answer 1

In the given question, we have been given that there is any percentile. We have to calculate the $z$ score of that percentile. To solve this question, we need to know what does $z$ score mean. The word called ‘$z$ score’ is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Above the mean value, a raw score has positive standard scores, while the standard scores are negative for a raw score below the mean value.

Formula Used:
We are going to use the formula of Standard Score:
Let $Z$ be standard score, $x$ be observed value, $\mu$ be the mean, $\sigma$ be the standard deviation, then:
$Z = \dfrac{{x - \mu }}{\sigma }$

Complete step by step answer:
In the given question, we have to find the $z$ score of a percentile.
Let $Z$ be the standard score of the sample, $x$ be the observed value of the sample, $\mu$ be the mean of the sample, and $\sigma$ be the standard deviation of the sample. Let the given percentile be $T\%$.
Then we can write the formula as:
$P\left( {X < {x_0}} \right) = T \Rightarrow P\left( {Z < \dfrac{{{x_0} - \mu }}{\sigma }} \right) = T$

Hence, z = \dfrac{{{x_0} - \mu }}{\sigma } = InvNorm\left( T \right)$$$$z

Note:
So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. In the given question, we had to write an expression for the score of a percentile. For doing that, it totally relied on our knowledge of the concept of the standard score. So, it is very important that we know the definition, the basic idea of the concept which is in the question, without which it is impossible to solve the given problem.