Summary of Z-Score and the Empirical Rule

Monitoring Your Readiness

To effectively plan and use your time wisely, it helps to think about what you know and do not know. For each of the following, rate how confident you are that you can successfully do that skill. Use the following descriptions to rate yourself:

5—I am extremely confident I can do this task.

4—I am somewhat confident I can do this task.

3—I am not sure how confident I am.

2—I am not very confident I can do this task.

1—I am definitely not confident I can do this task.

Skills Needed for Z-Score and the Empirical Rule: Forming Connections

Skill or Concept: I can . . . Questions to check your understanding Rating from 1 to 5
Convert values into standardized scores. 1-9
Use a value’s standardized score to determine whether the value is above, below, or equal to the mean. 6-9
Explain the Empirical Rule. 10-12

Now use the ratings to get ready for your next in-class activity. If your rating is a 3 or below, you should get help with the material before class. Remember, your instructor is going to assume that you are confident with the material and will not take class time to answer questions about it.

Ways to get help:

  • See your instructor before class for help.
  • Ask your instructor for on-campus resources.
  • Set up a study group with classmates so you can help each other.
  • Work with a tutor.

Essential Concepts

  • Standardizing the value includes finding the difference between the given value and the mean, and dividing that distance by the standard deviation. The resulting value is a number of standard deviations, and has no units associated with it.
  • Standardized scores can result in positive and negative values. A negative can be thought of as indicating a value that lies to the left of the mean, and a positive indicates a value that lies to the right of the mean.
  • An estimate of how many observations are within a certain number of standard deviations can be made if a distribution is bell shaped, unimodal, and symmetric.
  • The Empirical Rule states that:
    • about 68% of observations in a data set will be within one standard deviation of the mean
    • about 95% of observations in a data set will be within two standard deviations of the mean
    • about 99.7% of the observations in a data set will be within three standard deviations of the mean

Study Tips: Evidence-based strategies for learning

  • When doing practice problems of calculations, mix in different types of questions to each practice session. For example, practice calculating a value’s standardized score by hand along with other types of problems from recent sections that could appear on an upcoming test.
  • Explain to a friend, real or imaginary, why order of operations is important to keep in mind when calculating standardized scores.
  • Create mnemonics, symbols, or diagrams to help you keep track of various formulas that look very similar.
    • Standardized Score, z-score, [latex]z[/latex], is for a single data value. It’s the distance from the mean divided by standard deviation.
    • Standard Deviation, [latex]s[/latex] for samples or [latex]\sigma[/latex] for populations, is for data sets. It’s a measure of the “typical” distance of an observation in the data set from the mean of the data set.
    • Variance, [latex]\sigma^{2}[/latex] or [latex]s^{2}[/latex] is the square of the standard deviation.
  • Write out, from memory, by hand, the formulas for z-score (standardized score), standard deviation, and variance.
  • Pretend you are teaching a young person what the Empirical Rule is and how to use it to determine if an observation is unusual. What do you understand well about it? What is still challenging to understand?
  • In your study group take turns explaining how to compare two observations using z-scores (or use the video feature on your phone).
    • What information must you have in order to calculate a z-score?
    • Are there any units associated with a z-score — why or why not?
    • Refer to Questions 5, 6, and 7 in Forming Connections to verify your explanation.

Foundational Knowledge

Key Equations

  • Converting values into standardized scores

[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.

Glossary

Empirical Rule
a guideline that predicts the percentage of observations within a certain number of standard deviations. Also known as the 68-95-99.7 Rule which states that in a bell-shaped, unimodal distribution, almost all of the observed data values, [latex]x[/latex], lie within three standard deviations, [latex]\sigma[/latex], to either side of the mean, [latex]\mu[/latex]. More specifically, about 68% of observations in a data set will be within one standard deviation of the mean [latex]\left(\mu\pm\sigma\right)[/latex], about 95% of the observations in a data set will be within two standard deviations of the mean [latex]\left(\mu\pm2\sigma\right)[/latex], and about 99.7% of the observations in a data set will be within three standard deviations of the mean [latex]\left(\mu\pm3\sigma\right)[/latex].
standardized value
the number of standard deviations an observation is away from the mean. Also referred to as a z-score.

 

My Skills Checklist:

  • I can utilize standardized scores and the Empirical Rule to determine if an observation is usual.
  • I can utilize standardized scores and the Empirical Rule to determine if an observation is unusual.
  • I can compare two observations by calculating and comparing the standardized score.

Check mark list on clipboard

Topic Complete – now test your understanding in the Self-Check.