This page would contain resource information like a glossary of terms from the section, key equations, and a reminder of concepts that were covered.
Make this more relevant to what students want — help them to build their processes, study guides, mnemonics, and memory dump material.
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 dataset will be within one standard deviation of the mean
- about 95% of observations in a dataset will be within two standard deviations of the mean
- about 99.7% of the observations in a dataset will be within three standard deviations of the mean
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
- standardized value
- the number of standard deviations an observation is away from the mean. Also referred to as a z-score.
- 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 dataset will be within one standard deviation of the mean [latex]\left(\mu\pm\sigma\right)[/latex], about 95% of the observations in a dataset 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 dataset will be within three standard deviations of the mean [latex]\left(\mu\pm3\sigma\right)[/latex].
Put formal DCMP I Can statements to prepare for the self-check.
These I Can Statements are new (the first one is the “you will understand” rephrased as an I Can):
- 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.
