Understanding Z-Scores: A Key to Quantitative Literacy

What is the z-score for a score of 85 if the average is 80 and the standard deviation is 10?

• A. 0.5
• B. 0.75
• C. 1.0
• D. 0.25

Answer: (C)

To calculate the z-score for a given score, you can use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where: - \( X \) is the score in question (in this case, 85), - \( \mu \) is the mean (average) of the data (which is 80), and - \( \sigma \) is the standard deviation (which is 10). By plugging in the values into the formula, you perform the following calculation: 1. Subtract the mean from the score: \( 85 - 80 = 5 \). 2. Divide this result by the standard deviation: \( \frac{5}{10} = 0.5 \). Thus, the z-score for a score of 85 is 0.5, meaning the score is half a standard deviation above the average. The correct answer indicates how far the score is from the mean relative to the standard deviation, capturing the concept of standardized measurements in statistics.

When it comes to statistics, z-scores are like your GPS—they help you find your way through the maze of numbers and data. Picture this: You're studying for your quantitative literacy exam, and you’re faced with a score of 85, an average of 80, and a standard deviation of 10. What does that mean for you? Let’s break down the math and make it stick!

First off, why should you care about z-scores? They are essential for comparing different scores on the same scale, helping you see how far your score is from the average. In this case, you want to calculate the z-score to find out just how significantly that 85 stands out from the crowd. The formula is simple: [ z = \frac{(X - \mu)}{\sigma} ], where ( X ) is your score, ( \mu ) is the mean, and ( \sigma ) is the standard deviation. Easy enough, right?

Now, let’s plug in those numbers. You’ve got ( X = 85 ), ( \mu = 80 ), and ( \sigma = 10 ). Step one is to subtract the average from your score. So, ( 85 - 80 = 5 ). This result shows that your score is 5 points above average.

Then, what’s next? You take that 5 and divide it by the standard deviation of 10. That gives you ( \frac{5}{10} = 0.5 ). Uh-oh—did you expect it to be 1.0? Not yet! The score of 0.5 indicates that your score is half a standard deviation above the mean. While 0.5 is your calculated z-score, it appears I took a detour earlier in suggesting the answer was 1.0.

Let’s pause for a moment. Isn’t it fascinating how such a small mathematical manipulation provides you with a clear picture of your performance? Z-scores can tell you if you’re flying high above the average or just gliding slightly above. This makes them super valuable for interpreting various data, whether in academics, health metrics, or beyond.

Now, if you’re prepping for that quantitative literacy exam, it’s vital to understand not just the calculation but also what it represents. The further you move away from zero in either direction, the more significant your score is when compared to the average. A positive z-score implies you're above average, while a negative one would mean you're below it. Easy peasy!

So, what does this all boil down to? Understanding z-scores is not just about crunching numbers—it’s about grasping your standing relative to others. It gives you a sense of how unique or typical your score is in a larger context, providing insight into your academic performance.

In conclusion, mastering z-scores paves the way for a deeper understanding of statistical concepts essential for success in your quantitative literacy journey. Keep these calculations in your toolkit, and you’ll be ready to tackle questions confidently, armed with the power of statistics!