Question

The first Stat exam had a mean of 80 and a standard deviation of 4 points; the second had a mean of 70 and a standard deviation of 15 points. Reginald scored an 80 on the first test and an 85 on the second. Sara scored an 88 on the first but only a 65 on the second. Although Reginald's total score is higher, Sara feels she should get the higher grade. Explain her point of view.

Short answer

Sara argues her higher performance relative to class standards on the first exam with a Z-score of 2.

Step-by-step solution

Understanding the Z-Score

The Z-score measures how many standard deviations an element is from the mean. It is calculated using the formula $Z=\frac{X-\mu }{\sigma }$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Calculating Reginald's Z-Scores

For the first exam, Reginald's Z-score is calculated as $Z_1=\frac{80-80}{4}=0$. For the second exam, his Z-score is $Z_2=\frac{85-70}{15}=1$.

Calculating Sara's Z-Scores

For the first exam, Sara's Z-score is $Z_1=\frac{88-80}{4}=2$. For the second exam, her Z-score is $Z_2=\frac{65-70}{15}=-0.33$.

Comparing Their Performance

Reginald's Z-scores are 0 and 1 for the exams, whereas Sara's are 2 and -0.33. Although Reginald has a higher total score, Sara has more consistently excelled compared to the class in the first exam, with a higher Z-score of 2.

Key concepts

Understanding the Standard Deviation

The standard deviation is a crucial concept in statistics, representing the amount of variation or dispersion in a set of values. To put it simply, it tells us how much the values in a data set deviate from the mean or average.

• Low Standard Deviation: This indicates that the data points tend to be close to the mean. It shows a more consistent or stable set of numbers.
• High Standard Deviation: This suggests that the data points are spread out over a wider range. It signals more variability within the data set.

Understanding standard deviation is essential for comparing individual scores to the mean of a group. For instance, if we take the example of exam scores, knowing the standard deviation helps us evaluate how a student performed compared to the average performance of the class.

The Concept of Mean in Statistics

The mean, also known as the average, is a central point of a data set and is calculated by summing all the values and dividing by the number of values. It serves as a useful metric to indicate the general trend of a data set.
To calculate the mean:

• Add up all the values in your data set.
• Divide the sum by the total number of values.

For instance, in the first exam scenario where the mean is 80, it implies that the average score of all students was 80. Understanding the mean allows us to determine how individual scores compare with this average value.
Sara and Reginald's performances are evaluated against these means, and their Z-scores show how far their scores are from these average scores, taking the standard deviation into account.

Statistical Performance Comparison using Z-Scores

Comparing statistical performance using Z-scores provides insight into how an individual score stands out within a group. The Z-score tells us how many standard deviations away a particular score is from the mean, making it a powerful tool in comparing performances across different sets of data with different scales and distributions. In our example, Reginald and Sara have different performances across the two exams:

• Reginald’s scores:
  • First exam Z-score: 0 (meaning his score is exactly at the mean)
  • Second exam Z-score: 1 (meaning his score is one standard deviation above the mean)
• Sara’s scores:
  • First exam Z-score: 2 (indicating she outperformed the average significantly, two standard deviations above the mean)
  • Second exam Z-score: -0.33 (slightly below the average)

In this scenario, although Reginald’s overall raw score is higher, Sara’s Z-scores show her exceptional performance in the first exam, which is why she might feel her performance merits a higher grade. Understanding and using Z-scores can effectively illuminate strengths and differences in performance that raw scores might overlook.