Lesson 5.7 – Normal Distribution Calculations
- Calculate the probability that a value falls within a given interval in a normal distribution.
- Find a value corresponding to a given probability (area) in a normal distribution.
Activity: What is a good SAT math score?
For this Activity, we used question #1 to model the expectation for any normal distribution calculation. Each solution must include (1) Picture of the normal curve. Must be labeled N(mean, SD), the center labeled with the mean, and then the shaded area of interest. (2) Formula for a z-score, then numbers plugged in. (3) Find correct z-score and then use Table A to turn it into an area.
For questions where students have to calculate a value with a given area, there are two general approaches. (1) Have students write the formula for a z-score, plug in the values that they know (z-score, mean, SD) and solve for what they don’t know (value). The other option (2) is to have students first find the z-score. Then have students think about the interpretation of the z-score to arrive at the answer. For example, if your z-score is 1.65, you know that you are looking for the value that is 2.48 standard deviations above the mean. Easy. Mean + 1.65SD will get you your answer (no algebra necessary). Of course these two approaches are equivalent. If you have developed a strong understanding of the z-score, approach (2) might be better.
When debriefing the Activity, be sure to have students notice that #1 is area to the left, #2 is area to the right, #3 is area between, and #4 is to find a value given an area. Check out the cool flip chart we made later to help students remember how to find these areas.
Let’s talk about why we have these ridiculous requirements for each normal distribution calculation.
(1) Carefully labeled and shaded normal curve. The N(mean,SD) helps students keep track of the correct mean and standard deviation. Also, later in the course we will have to check that our distributions are Normal and this notation will remind us to do that check. Labeling the mean at the center of the curve helps students to identify whether the given value is less than or greater than the mean. Students will also start to think about how far away the given value might be from the mean. The shaded region will help students to check their final answers. If the shaded region is clearly less than 50% of the curve and they have an answer of Area = 0.8943 then they know they have made a mistake.
(2) Formula for a z-score and then numbers plugged in. We require this step to try and set up students for success later in the course. When we get to calculating test statistics for significance tests (Chapter 8,9,10), the formulas for calculating a z-score will get much more complex. We want students in the habit or writing formulas. This step also reminds students that a z-score tells us how many standard deviations away from the mean a value is.
(3) Find a correct z-score and turn it into an area. Later in the course, we will be doing significance testing. The z-score will be our test statistic and the area will be our P-value. The P-value is the holy grail of introductory statistics.
Bonus: If you have extra time, try the online only Bonus Lesson 5.7A: Assessing Normality, Lesson 5.7B: Transforming Random Variables, and Lesson 5.7C: Combining Random Variables.