Question

How do you find the area between two z-scores? Is there a formula you can go by if you have a set of data or are you supposed to refer to a chart?

Answer 1

In this question, we have to find the area between two z scores. As we know, z-score is a fractional number where the numerator is equal to the addition of the data point and the negative of mean, and the denominator is equal to the standard deviation. Thus, z score is a measure of the exact number of times the standard deviation is above or below the mean data point. So, we start solving this problem, by letting two data points and then find the z-score of each data point, to get the required solution for the problem.

Complete step by step solution:
According to the question, we have to find the area between two z-scores.
Thus, we will use the statistical method to get the solution.
Let us suppose there are two data points say ‘a’ and ‘b’, such that we have to find the area between these two data points, we get
$P\left( a\le z\le b \right)$ --------- (1)
So, we will find the z-score of each data point. As we know, z-score is a measure of the exact number of times the standard deviation is above or below the mean data point, therefore the formula to find z-score is,
$z-\text{score}=\dfrac{\text{data point-mean}}{\text{standard deviation}}$
Now, let us say for the same data set, mean is equal to $\mu$ , standard deviation is equal to $\sigma$ , and data point is ‘a’ and ‘b’ respectively, thus z-score is equal to
$z-\text{score}=\dfrac{\text{data point-}\mu }{\sigma }$
Now, we will find the z-score when the data point is ‘a’ , by substituting the value in the above equation, we get
$z-\text{score}=\dfrac{a-\mu }{\sigma }$
Similarly, the z-score when the data point is ‘b’ , by substituting the value in the z-score equation, we get
$z-\text{score}=\dfrac{b-\mu }{\sigma }$
Thus, the area between two z-scores, we get
$P\left( \dfrac{a-\mu }{\sigma }\le z\le \dfrac{b-\mu }{\sigma } \right)$ --------- (from (1))
Also, after putting the value of mean, standard deviation, and the data points, we get the exact area with the help of z-chart.
Therefore, the area between the two z-scores is equal to $P\left( \dfrac{a-\mu }{\sigma }\le z\le \dfrac{b-\mu }{\sigma } \right)$ . We have to find the value of the area using the z-chart.

Note: In such statistical problems, the mean, standard deviation and the data points are always given to us, we simply have to find the z-score of each point and then look upon the z-table to get the area between them. Also, this a two z-score formula, it is different from one z-score formula.