Meaning of Z-score
Z-SCORE
Z-SCORE
Z-score = Symmetry & Standardized measure
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
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.
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 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.
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.
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?
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.
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
8 Steps to calculate Z-score
State the null and research hypotheses.
State the null and research hypotheses.
Set the level of risk associated with the null hypothesis.
Set the level of risk associated with the null hypothesis.
Select the appropriate test statistic.
Select the appropriate test statistic.
Compute the test statistic value (called the obtained value)
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.
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.
Compare the obtained value and the critical value.
Make your decision
Make your decision
Wonder why the result was the way it was and attempt to explain it.
Wonder why the result was the way it was and attempt to explain it.
Example
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?
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
Use the z-score formula
Where Z = Standard (Normal) or Z score
Where Z = Standard (Normal) or Z score
x = mean sample of score
x = mean sample of score
μ = mean (population mean)
μ = mean (population mean)
σ = standard deviation
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.
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
calculate z score
score above 80 (x > 80)
score above 80 (x > 80)
mean (μ = 70)
mean (μ = 70)
standard deviation (σ = 15)
standard deviation (σ = 15)
z = (80 - 70)/ 15 = 10/15 = 0.67
z = (80 - 70)/ 15 = 10/15 = 0.67
Result
Result
Finally, convert that to the percentage and answer.
Finally, convert that to the percentage and answer.
0.7486 or 74.86% of the scores will be above 80.
0.7486 or 74.86% of the scores will be above 80.
Table 1: Z-Test table
Table 1: Z-Test table
