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In statistics, a normal distribution is a type of continuous probability distribution for a real-valued random variable. When graphed, it forms a bell-shaped curve that is symmetric about the mean. These are key characteristics of a normal distribution:

• Symmetric around the mean.
• Mean, median, and mode are all equal.
• Follows the 68-95-99.7 rule:

This means that approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
In the context of the ACT and SAT scores, both are assumed to be normally distributed, allowing us to apply statistical methods to compare scores. For the ACT, the mean score is 21.1 with a standard deviation of 5.1, while the SAT has a mean score of 1026 and a standard deviation of 210. This normality assumption is crucial for calculating Z-scores.

A Z-score, or standard score, indicates how many standard deviations an element is from the mean. It is a way to compare scores from different distributions by standardizing them. The formula for calculating a Z-score is:

\[Z = \frac{X - \mu}{\sigma}\]

where:

• \(X\) is the value you are standardizing,
• \(\mu\) is the mean of the distribution,
• \(\sigma\) is the standard deviation of the distribution.

Let's apply this to the scores:

• For the ACT: \[Z_{ACT} = \frac{26 - 21.1}{5.1} = 0.96\]
• For the SAT: \[Z_{SAT} = \frac{1240 - 1026}{210} = 1.02\]

A higher Z-score implies the score is farther from the mean in the positive direction, indicating better performance compared to peers.

Once we calculate the Z-scores for both tests, we can use them to make a statistical comparison about the performance relative to the respective test populations. Z-scores help us understand whether a given score is above or below the average and by how much compared to the spread of the data (standard deviation).
In this problem, the Z-scores are:

• ACT: 0.96
• SAT: 1.02

Since 1.02 > 0.96, the SAT score of 1240 is better in terms of deviation from the mean than the ACT score of 26.
Therefore, you performed relatively better on the SAT compared to the ACT. Converting raw scores to Z-scores has allowed us to make a fair comparison between the two different distributions.