Part 1: Converting Pulse Rates to Z-scores

The formula for converting a raw score $x$ to a z-score is:

$z = \frac{(x - \mu)}{\sigma}$

Where:

• $x$ is the raw score,
• $\mu$ is the mean, and
• $\sigma$ is the standard deviation.

When you convert all pulse rates to z-scores:

• The mean $\mu$ of the z-scores will always be 0. This is because the mean in the z-score formula cancels out the shift in data.
• The standard deviation $\sigma$ of the z-scores will always be 1. This is because the standard deviation is scaled to standardize the data.

Thus, the answers for this part are:

• Mean ( $\mu$ ) = 0
• Standard deviation ( $\sigma$ ) = 1

Part 2: Units of the Z-scores

Z-scores are dimensionless, meaning they are simply numbers that indicate how many standard deviations a value is from the mean. The raw data had units of "beats per minute," but after conversion to z-scores, the data no longer carries units because it's a standardized score.

Thus, the correct choice is:

• D. The z-scores are numbers without units of measurement.

Would you like more details or have any questions?

5 Related Questions:

1. What does a positive or negative z-score tell you about the data point?
2. How can you use z-scores to determine outliers in a dataset?
3. What is the significance of a z-score of 1, 2, or 3 in terms of standard deviations?
4. How can z-scores be used to compare data from different distributions?
5. Can z-scores be used for data that is not normally distributed?

Tip:

Z-scores are useful for comparing data points from different distributions, as they standardize the data and remove the units.