Similar to the quintile and percentile ranks, this rating is often used in the school system to quantify the quality of a student's academic record.
The |\boldsymbol{Z}|-score, also called standard score, measures an object's position relative to the group’s mean. Specifically, it measures the distance between a data point and the average, expressed in number of standard deviations.
By considering that there are several concepts present in the very definition of the standard score, the associated formula may appear imposing.
||Z=\dfrac{x_{i}-\overline{x}}{\sigma}||
|x_i\!:| data value for which we want to calculate the |Z|-score
|\overline{x}\!:| mean of the distribution
|\sigma\!:| standard deviation of the distribution
To determine a |Z|-score, first determine the mean and the standard deviation of the data set. Then, use the appropriate formula to determine the value of the |Z|-score. Regardless of the situation, the results can be categorized into three groups.
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If the |x_i| value is greater than the mean, the numerator of the equation will be positive. In this case, the |Z|-score will be positive.
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If the |x_i| value is less than the mean, the numerator of the equation will be negative. In this case, the |Z|-score will be negative.
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If the |x_i| value is equal to the mean, it will have a deviation from the mean, which will be zero. For this reason, a data with the same value as the mean has a score |Z = 0.|
As with other formulas in mathematics, this formula can be used to calculate a result, to compare results, or to find missing data.
In Samuel's math group, the average on the second-term exam is |75\%| and the standard deviation equals |7\%.|
If Samuel got |82\%,| what is the |Z|-score associated with his result?
All the necessary values for the calculation are known. Next, replace them in the formula and complete the calculation:||Z=\dfrac{82-75}{7}=1||The |Z|-score associated with Samuel's result is equal to |1.|
In general, it is unnecessary to calculate the |Z|-score for one student. Remember that the |Z|-score is mainly used to qualify students' academic records.
Catherine and Gabriel, two students in Secondary 3, attend separate schools and take different exams.
Catherine received a final mark of |77\%.| The class average was |65\%| and the standard deviation was |6\%.|
Gabriel finished the year with a mark of |80\%| and a |Z|-score of |1.23.| Who performed better between Catherine and Gabriel?
Since they did their evaluations in different settings, it is appropriate to use the standard score (|Z|-score) to compare their achievements.
In Catherine's case, the standard deviation and the mean are known. Thus, we can calculate her |Z|-score directly:||Z=\dfrac{77-65}{6}=2||Even though Gabriel obtained a better final mark, Catherine performed better since her |Z|-score is greater than Gabriel’s.
When qualifying student achievement, it is important to consider many factors, such as class composition, teacher requirements, school requirements, and so on. By calculating the |Z|-score with the mean and the standard deviation, the role of these variables is taken into account and the comparison becomes fairer.
Maxime took an exam that has an average of |75\%| and a standard deviation of |7\%.| Knowing the |Z|-score associated with his result is |–1,| what grade did he get?
All the data other than Maxime’s result is known. Simply replace these values in the equation and isolate the missing value.||\begin{align}-1&=\dfrac{x_{i}-75}{7}\\-7&=x_i-75\\68&=x_i\end{align}||Therefore, Maxime got |68\%| on the exam.
|Z|-scores play an important role in granting admission to various post-secondary education programs. In fact, some programs require a certain |Z|-score to establish a fair and equitable admission ranking for prospective students.