Standardizing a Score
At this point, students will be presented with two datasets. They will be able to choose which one they would like to use to answer example questions.
Now that you have obtained the standard deviation of the data set Runtimes using technology, you can calculate any observation’s z-score to locate it in the data set relative to the mean.
calculating z-scores
[Worked example video – a 3-instructor video that works through an example like questions 6 – 9]
interactive Example
Recall, to calculate a z-score given an observation, use the formula [latex]z=\dfrac{x-\mu}{\sigma}[/latex], where [latex]x[/latex] represents the value of the observation, [latex]\mu[/latex] represents the population mean, [latex]\sigma[/latex] represents the population standard deviation, and [latex]z[/latex] represents the standardized value, or z-score.
We’ll use data set Sleep Study: Average Sleep, you saw in Comparing Variability of Data Sets: What to Know to demonstrate how to calculate z-scores for individual observations in the data set. The mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex] in the formula represent the population the sample came from. Since we don’t know these, we’ll use the sample mean and standard deviation in our calculations.
The mean of the data is 7.97 hours with a standard deviation of 0.965. Calculate the z-scores for each of the following observations and indicate if the given value lies above or below the mean. Round your calculations to two decimal places.
- [latex]6.93\text{ hours}[/latex]
- [latex]9.87\text{ hours}[/latex]
- [latex]7.97\text{ hours}[/latex]
- [latex]4.95\text{ hours}[/latex]
Use the mean and the standard deviation you calculated in Questions 2 and 3 to answer Questions 6 – 9.
