Understanding Z-Scores: A Deep Dive for Quantitative Literacy Exams

If scores on a test have a mean of 75 and a standard deviation of 9, what is Anna's z-score if she scores 78?

• A. 0.5
• B. 0.33
• C. 1.0
• D. 1.5

Answer: (B)

To find Anna's z-score, you need to understand that the z-score is a measure of how many standard deviations an individual score is from the mean of the data set. The formula for calculating a z-score is given by: \[ z = \frac{(X - \mu)}{\sigma} \] where: - \( X \) is the individual score, - \( \mu \) is the mean score, - \( \sigma \) is the standard deviation. In this case, Anna’s score is 78, the mean is 75, and the standard deviation is 9. Plugging these values into the formula: 1. Subtract the mean from Anna's score: \( 78 - 75 = 3 \) 2. Divide this result by the standard deviation: \( \frac{3}{9} = 0.33 \) Therefore, Anna's z-score is 0.33, reflecting that her score is approximately one-third of a standard deviation above the mean. This correctly indicates her relative position within the distribution of scores. This is why the z-score of 0.33 is the correct answer, as it accurately represents Anna's performance in relation to the mean.

When it comes to understanding statistics, especially in the context of quantitative literacy exams, few concepts stand out quite like the z-score. So, what’s the deal with z-scores, and why should they matter to you? Well, if you're prepping for your exams, grasping how scores relate to one another can be a game changer for your confidence and overall score.

Let’s break it down with a simple example: Anna scores 78 on a test where the mean score is 75, and the standard deviation is 9. At first glance, it seems like a solid score. But how does it stack up against her peers? That's where the z-score struts in, waving its mathematical wand to unveil the truth.

You see, the z-score is like your personal performance badge in a sea of scores. It tells you how far away from the average your score is, expressed in terms of standard deviations. In Anna's case, her z-score can be calculated using the formula:

[ z = \frac{(X - \mu)}{\sigma} ]

Now, if you’ve squirmed at math before, hang tight! It's simpler than it looks. Here, (X) is Anna's score, (μ) is the average score, and (σ) is the standard deviation.

Okay, so let's plug in those numbers:

1. First, we find the difference between Anna’s score and the mean: (78 - 75 = 3).

2. Next, divide that difference by the standard deviation: ( \frac{3}{9} = 0.33).

Voilà! Anna's z-score is 0.33. This tells us that her performance is roughly one-third of a standard deviation above the mean. What does that really mean for her? Well, it signifies a competitive edge among her classmates, perhaps nudging her one step closer to her academic goals.

Now, isn’t that interesting? This little number carries quite a bit of weight. It can indicate whether you’re performing just average, better than most, or even at the top of the class. So, the next time you're staring down your exam score, don't just glance at the digits; take a moment to calculate your z-score.

But wait—here's a nifty fact: the beauty of z-scores lies in their universality. They’re applicable not just in academics but across various fields, from healthcare metrics to financial analyses. Who knew math could play such a versatile role?

In conclusion, z-scores are more than mere figures—they're your key to understanding where you stand within a data set. For students gearing up for their quantitative literacy exams, grasping this concept isn’t just useful—it's essential. So, remember Anna’s journey and the z-score formula next time you tackle statistics. With a little practice, you’ll be calculating z-scores with the finesse of a seasoned pro, and maybe, just maybe, feeling a bit more confident about that upcoming test.