Question

At Hogwarts School of Witchcraft and Wizardry, Professor Snape was concerned about grade inflation, and suggested that the school should issue standardized grades (or 2-scores), in addition to the regular grades. How might this work? Harry was in four classes, each with 20 students. Harry's score, the class mean, and the class standard deviation are given below. Compute his standardized grade in each class. If we judged by standardized grades, where did he do best? Where did he do worst?​

Answer 1

Answers:

The z scores are approximately:

• Care of Magical Creatures:  z = 0.333
• Defense Against the Dark Arts:  z = 0.583
• Transfiguration:  z = -0.263
• Potions: z = -0.533

From those scores, we can say:

• Best grade = Defense Against the Dark Arts
• Worst grade = Potions

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Further Explanation:

We'll need to convert each given score to a corresponding standardized z score.

The formula to use is

z = (x - mu)/sigma

where,

• x = given grade for each class
• mu = mean
• sigma = standard deviation

Let's find the z score for the Care of Magical Creatures class

z = (x - mu)/sigma

z = (3.80 - 3.75)/(0.15)

z = 0.333 approximately

Repeat this process for the Defense Against the Dark Arts score.

z = (x - mu)/sigma

z = (3.60 - 3.25)/(0.60)

z = 0.583 approximately

And for the Transfiguration class as well

z = (x - mu)/sigma

z = (3.10 - 3.20)/(0.38)

z = -0.263 approximately

The negative z score means his grade below the average, whereas earlier the other scores were above the average since he got positive z scores.

Now do the final class (Potions) to get this z score

z = (x - mu)/sigma

z = (2.50 - 2.90)/(0.75)

z = -0.533 approximately

This grade is below average as well.

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To summarize, we have these z scores

• Care of Magical Creatures:  z = 0.333
• Defense Against the Dark Arts:  z = 0.583
• Transfiguration:  z = -0.263
• Potions: z = -0.533

Harry did his best in Defense Against the Dark Arts because the z score of 0.583 (approximate) is the largest of the four z scores. On the other hand, his worst grade is in Potions because -0.533 is the lowest z score.