Calculate and Compare The Z-Scores for The Following Pair

What is a Z-Score?

A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a dataset. It's calculated using the formula:

Z-scores are useful because they allow you to compare data points from different datasets that may have different means and standard deviations. A z-score of 0 means the data point is exactly at the mean, while positive z-scores indicate values above the mean and negative z-scores indicate values below the mean.

How to Calculate Z-Scores

To calculate a z-score, you'll need three pieces of information:

1. The value of the data point you're interested in (X)
2. The mean of the dataset (μ)
3. The standard deviation of the dataset (σ)

Once you have these values, you can plug them into the z-score formula. Here's a step-by-step example:

1. Calculate the difference between the data point and the mean: (X - μ)
2. Divide this difference by the standard deviation: (X - μ) / σ
3. The result is your z-score

For example, if you have a data point of 72, a mean of 65, and a standard deviation of 8, the z-score would be calculated as:

Comparing Z-Scores

Once you've calculated z-scores for multiple data points, you can compare them to understand their relative positions in the dataset. Here are some key interpretations:

• Z-scores greater than 0 indicate values above the mean
• Z-scores less than 0 indicate values below the mean
• The absolute value of the z-score indicates how many standard deviations the data point is from the mean
• Z-scores can be positive or negative, but the magnitude is what matters for comparison

For example, if you have two z-scores of 1.2 and -0.8, you can see that the first data point is 1.2 standard deviations above the mean, while the second is 0.8 standard deviations below the mean.

Example Calculation

Let's walk through a complete example of calculating and comparing z-scores for two data points.

Scenario

You have a dataset of test scores with the following statistics:

• Mean (μ) = 75
• Standard deviation (σ) = 10

You want to compare two students' scores:

• Student A: 85
• Student B: 60

Calculations

For Student A (85):

For Student B (60):

Interpretation

Student A's score of 85 has a z-score of 1.0, which means it's 1 standard deviation above the mean. Student B's score of 60 has a z-score of -1.5, meaning it's 1.5 standard deviations below the mean.

Comparing the two z-scores, we can see that Student B's performance is more extreme relative to the class average than Student A's, as indicated by the larger absolute value of the z-score.