ZScores
Zscores
Section1: Zscores in SPSS
Zscoresare type of descriptive statistics that presents the number of thestandard deviations between every data point and the mean. A sampleof 13 students was taken to act as a representative of their classand the Zscores calculated using Excel. The results are presented inthe tables below.
Descriptive Statistics 

N 
Mean 
Std. Deviation 

quiz1 
105 
7.47 
2.481 
quiz2 
105 
7.98 
1.623 
quiz3 
105 
8.05 
2.322 
quiz4 
105 
7.80 
2.280 
quiz5 
105 
7.87 
1.765 
Final 
105 
61.48 
7.943 
Total 
105 
100.57 
15.299 
Percent 
105 
80.34 
12.135 
Valid N (listwise) 
105 
Zscore: Final 
Zscore: Total 
Zscore: Percent 

Mean 
Mean 
Mean 

ID 
106484 
1.06707 
1.34458 
1.34672 
108642 
.94118 
.29880 
.27547 

127285 
.56351 
.16807 
.19306 

132931 
.82128 
.15874 
.13656 

140219 
.56950 
.48554 
.46617 

142630 
1.57663 
1.40061 
1.45502 

153964 
.19183 
.74699 
.79579 

154441 
1.19896 
1.26988 
1.29021 

157147 
1.57663 
1.46597 
1.45502 

164605 
1.70252 
1.53133 
1.53743 

164842 
.31173 
.23343 
.19306 

167664 
1.19896 
1.13916 
1.12540 

175325 
.94717 
.68163 
.71338 
Apositive zscore indicates that the data point considered is abovethe mean, while a negative zscore implies that the data point inquestion is below the mean.
Section2: Case Studies of Type I and Type II Errors
Thetype I Error result when the null hypothesis is rejected when it isin fact true. Formany disciplines, the classical hypothesis testing is critical inanalyzing the research data. In a typical condition, a small sampleof data is assumed to have an alpha significance level of .05.Normally, null hypothesis depicts lack of relationship between two ormultiple variables. To stimulate this, a situation can be assumed. Inthis case, considering an experimental drug supposed to reduce theblood pressure but suspected to induce cancer, the drug may be firstadministered to the rodents. When the tumor rate is known to be 10%among untreated animals, the null hypothesis can be “the rate oftumor in treated animals is less than or is equal to 10%”. In thiscase, the drug is safe and does not cause any tumors. On the otherhand, the alternative hypothesis may be “tumor rate in treatedanimals is over 10%”.
AType I error results when a true null hypothesis is rejected. In thecase provided, Type I error results when conclusions are made thatthe drug is unsafe when absolutely it is. On the other hand, a TypeII error results from failure of rejecting a false null hypothesis.In the case provided, Type II error results when it is concluded thatthe drug is safe when truly it is not.
Section3: Case Studies of Null Hypothesis Testing
Nullhypothesis refers to a statement that is considered to be truethroughout the entire analysis. A dubious judgment statistician wantsto determine whether the revolver with the capacity of holding sixcartridges is well loaded. Therefore, he proposes to run a nullhypothesis of the gun not being loaded through spinning of thecylinder in order to align one amongst the six chambers with barrelrandomly, then pulling the trigger two times consecutively. If gunwill go off when the trigger is pulled, then he will conclude thatwas loaded, hence reject the null hypothesis. On the other hand, ifthe gun will go off, he will be justified to conclude that the gunwas not loaded. Thereafter, the significance level of this test willbe determined.
Ifthe gun will not be loaded, it will not go off until the trigger ispulled, hence the test cannot erroneously reject the null hypothesis.This will result to a significance level of zero. Supposing only onechamber is loaded, the chance of the gun going off is 100 percentless the chance that it fails to go off.
Reference
Klein, J. (2014). Assessing University Students’ Achievements by Means of Standard Score (Z Score) and Its Effect on the Learning Climate. Studies in Educational Evaluation .