Z-Score and the Empirical Rule: Learn It 4

The Empirical Rule

If a distribution is bell shaped, unimodal, and symmetric, then we can estimate how many observations are within a certain number of standard deviations. The Empirical Rule (also known as the [latex]68-95-99.7[/latex] rule) is a guideline that predicts the percentage of observations within a certain number of standard deviations.

the empirical rule

[insert a video describing (but not using) the Empirical Rule]–>this video could be good, but she refers back to other lessons and writes on the diagram in a way that could be confusing (calculating half of 68% and not others, uses x bar and s instead of mu and sigma, writes 99.7% on the outside of the bell while the others are clearly written inside). She begins an example at 4:41.

The Empirical Rule states that:

  • about [latex]68[/latex]% of observations in a data set will be within one standard deviation of the mean.
  • about [latex]95[/latex]% of the observations in a data set will be within two standard deviations of the mean.
  • about [latex]99.7[/latex]% of the observations in a data set will be within three standard deviations of the mean.

Graphically, the Empirical Rule can be expressed like this:

A bar graph with the highest bars in the middle and lower bars to either side. In the center, the x-axis is labeled "mu." Three bars to the left, it is labeled "mu - sigma," three more bars to the left it is labeled "mu - 2 sigma," and three more to the left, it's labeled "mu - 3 sigma." Three to the right of the center, it is labeled "mu + sigma." Three more to the right and it is labeled "mu + 2 sigma" and three more to the right, it is labeled "mu + 3 sigma." The center six bars are all green and labeled as 68& collectively. The three leftmost center bars are labeled 34.1%, and the three rightmost center bars are also labeled 34.1%. The next three bars out on either side of the center six are each labeled 13.6% and the center 12 are all labeled 95% collectively. Lastly, the next three out on either side of the center twelve are each labeled 2.1% and all 18 are collectively labeled 99.7% ≈ 100%

Fill in the blank for each of Questions 10 – 12

question 10

question 11

question 12