Question

A test has a mean of $153$ and a standard deviation of$12$, how do you find the test scores that correspond to
A. $z = 1.2$
B. $z = 2.4$?

Answer 1

Hint: In this question we need to find the test score which corresponds to a given standard score. Mean is basically the central value of any set of numbers which can be calculated by dividing the number of terms by their sum. Mean are generally of three types - Arithmetic Mean, Geometric Mean and Harmonic Mean. Standard variation tells us about how much dispersion is present in the values. It can be calculated in three ways, which are- Actual mean method, Assumed mean method and Step deviation method. Standard score gives us a glimpse of how distant from mean a data point is.

Complete step by step solution:
We are given,
Mean$(\mu )$ =$153$
Standard deviation $(\sigma )$= $12$
We know the formula to find $z$is,
$ \Rightarrow z = \dfrac{{x - \mu }}{\sigma }$
Finding $x$when,
$z = 1.2$
$ \Rightarrow 1.2 = \dfrac{{x - 153}}{{12}}$
$ \Rightarrow 14.4 = x - 153$
$ \Rightarrow x = 167.4$
$z = 2.4$
$ \Rightarrow 2.4 = \dfrac{{x - 153}}{{12}}$
$ \Rightarrow 28.8 = x - 153$
$ \Rightarrow x = 181.8$

Note: It is important to calculate standard score because-
it helps researchers in calculating the probability of a score occurring within a standard normal distribution;
It also enables us to compare two scores that are from different set of numbers (which may have different means and standard deviations).
What can be understood from the standard score?
If a standard score is equal to 0, it is on the mean.
A positive standard score indicates the raw score is higher than the mean average.
A negative standard score reveals the raw score is below the mean average.