Z-SCORE

Z-score = Symmetry & Standardized measure

We use the z-scores to calculate the probability aka percentage of a value occurring in a normally distributed data set

A z-score, also called a standard score, is a measure of position derived from the mean and standard deviation of the data set.

The z-score is a measure of how many standard deviations an element falls above or below the mean of the data set.

The z-score has a positive value if the element lies above the mean and a negative value if the element lies below the mean.

A z-score associated with an element of a data set is calculated by subtracting the mean of the data set from the element and dividing the result by the standard deviation of the data set.

How do we use Z-score?

We use the z-scores to calculate the probability aka percentage of a value occurring in a normally distributed data set.

8 Steps to calculate Z-score

State the null and research hypotheses.

Set the level of risk associated with the null hypothesis.

Select the appropriate test statistic.

Compute the test statistic value (called the obtained value)

Determine the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic.

Compare the obtained value and the critical value.

Make your decision

Wonder why the result was the way it was and attempt to explain it.

Example

Suppose that scores on a mathematics exam follow a normal distribution with mean μ = 70 and standard deviation σ = 15. What proportion of the scores are above 80?

Use the z-score formula

Where Z = Standard (Normal) or Z score

x = mean sample of score

μ = mean (population mean)

σ = standard deviation

Then, consult the appropriate z-score table. For this example, you would use since the calculated z-score covers the area below the score of 80.

calculate z score

score above 80 (x > 80)

mean (μ = 70)

standard deviation (σ = 15)

z = (80 - 70)/ 15 = 10/15 = 0.67

Result

Finally, convert that to the percentage and answer.

0.7486 or 74.86% of the scores will be above 80.

Table 1: Z-Test table