Eric has computed that it takes an average (mean) of 17 minutes with a standard deviation of 3 minutes to drive from home, park the car, and walk to his job. One day it took Eric 21 minutes to get to work. You would use the formula for transforming a raw score in a sample into a z-score to determine how many standard deviations the raw score represents. Since his "score" is 21, you would subtract the mean of 17 from 21 and divide that result (4) by the standard deviation of 3. The z-score of 1.33 tells you that Eric’s time to get to work is 1.33 standard deviations from the mean
Questions
Is the z value positive or negative? Explain why it should be positive or negative.
Another day, it took Eric only 12 minutes to get to work. Using the same formula, determine the z value. Is it positive or negative? Explain why it should be positive or negative.
On a different day, it took Eric 17 minutes to get from home to work. What is the z value? Why should you expect this result even before you did the calculation?
Based on your study of z-scores, explain the three uses of z-scores. How might they each relate to Eric's trip between home and work?
What is the relationship between z-scores and the standard normal curve?
Since Z = (score-mean)/SD, scores below the mean will have a negative value, and those above the mean will have a positive value.
Since you did the calculation of the first Z score, you should be able to do the same for the other Z scores.
For a normal distribution, the Z scores express raw scores in terms of SDs from the mean. The proportions cut off are the same for any normal distribution. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to your Z scores.
The z-value represents the number of standard deviations a raw score is from the mean. It can be positive or negative, depending on whether the raw score is above or below the mean, respectively.
In the case of Eric's trip taking 21 minutes, his z-value is positive. This is because 21 is greater than the mean of 17, indicating that his trip took longer than average.
On the other hand, if Eric's trip took only 12 minutes, his z-value would be negative. This is because 12 is less than the mean of 17, indicating that his trip was shorter than average.
When Eric's trip took exactly 17 minutes, the z-value would be 0. This is because the raw score matches the mean, so there is no deviation from it. Before calculating it, we would expect a z-value of 0 because the time is exactly at the mean.
Z-scores have three main uses: comparison, standardization, and probability calculation.
1. Comparison: Z-scores allow us to compare individual data points to the mean of a distribution. In the context of Eric's trips, we can compare his travel times to the average and see if they are above or below the mean.
2. Standardization: Z-scores also standardize data by transforming them into a common scale. This allows us to compare data from different distributions or variables. In the case of Eric's trips, we can standardize his travel times with the z-score formula to see how they compare to other variables or distributions.
3. Probability calculation: Z-scores are used to calculate probabilities by relating them to the standard normal distribution. The standard normal curve is a bell-shaped curve with a mean of 0 and a standard deviation of 1. By converting raw scores into z-scores, we can determine the likelihood of an event occurring based on its position on the standard normal curve.
The relationship between z-scores and the standard normal curve is that z-scores represent the distance of a raw score from the mean in terms of standard deviations. The standard normal curve represents a distribution with a mean of 0 and a standard deviation of 1. The z-value tells us where a raw score falls on this standard normal curve, allowing us to compare it to the overall distribution of scores.