Question

What does a Z score tell you?

Answer 1

We explain this question by using the standard definition of Z score and formula. The Z score gives the idea of how far the data point is, from the mean point.
The formula of Z score is given as,
${{z}_{i}}=\dfrac{{{x}_{i}}-\bar{x}}{\sigma }$
Where, ${{x}_{i}}$ is the data point we need to find the Z score, $\bar{x}$ is the mean of the data and $\sigma$ is the standard deviation of the data.
We take one example to find Z score and compare the definition to get the best idea about the Z score.

Complete step-by-step answer:
We are asked to explain about the Z score.
We know that the Z score is used to get a brief idea of how far from the mean point the required data point lies.
We can say that it is the measure of the number of standard deviations above or below the raw score.
Let us take an example that a student got a score of 1100 marks in a test in which that mean score is 1026 and the standard deviation is 209.
Now, let us find how well that student scored compared to the mean scorer which is the Z score.
We know that the formula of Z score is given as,
${{z}_{i}}=\dfrac{{{x}_{i}}-\bar{x}}{\sigma }$
Where, ${{x}_{i}}$ is the data point we need to find the Z score, $\bar{x}$ is the mean of the data and $\sigma$ is the standard deviation of the data.
By using the above formula the Z score of the position of the student can be calculated as,
$\begin{aligned} & \Rightarrow Z=\dfrac{1100-1026}{209} \\\ & \Rightarrow Z=\dfrac{74}{209}=0.354 \\\ \end{aligned}$
Here, we can see that the value of Z score is positive which says that he is 0.354 standard deviations above the mean scorer.
So, we can conclude that the Z score is used to calculate the number of deviations above or below the mean scorer the data point is. This will be used to help to understand the position and capacity of a data point with respect to other data points or mean points.

Note: We need to note that this Z score is used to mention the position of data point form mean point in terms of standard deviations. It cannot directly mention the percentage of the data point that we require.
We need to use the Z score table to find the area under the graph that represents the required Z score which will be the percentage of the data point.
Here, we took an example of the student marks and obtained a Z score as 0.354 which doesn’t means that the student got 35.4%. If we want to find the percentage then we need to use the standard Z score table.