*MATH 225N The Normal Distribution*

MATH 225N The Normal Distribution

Many variables in medicine follow a normal distribution where there are approximately an equal number of values below the mean as above the mean. Describe two variables that would probably follow a normal distribution. Also note which of the two variables would be likely to have a larger standard deviation and why.

Greetings Friends !!

One of the big initial questions this Week 5 involves which variable do you think has a larger standard deviation and why.

So I thought it would be good to point out and emphasize some of the implications of having a larger or smaller standard deviation.

So I copied a couple graphics from page 235 from another statistics text book for another Chamberlain statistics course and posted them here.

In the first graphic, we see that the distribution with the A designation has the smaller standard deviation because most of the data values occur over a smaller “range” or “segment” so to speak. In that first graphic, the data values for the B distribution are more “spread out” or have larger “variation” or “dispersion.”

So take in the example and then in the second graphic do the Try It Yourself exercise and Post your answers in response to that graphic. Then double check each other and make sure you all agree on the correct answers to the Try It Yourself exercise. In the example and in the Try It Yourself, pay attention to the questions involving means as well as standard deviations.

Thanks Friends and I am fairly convinced from my career that pictures and graphics are essential for better understanding in an introductory Statistics course.

Be well and have a Super day !!

**Reference**:

Larson, R. & Farber, B. (2015). * Elementary statistics: Picturing the world *(6th ed.). Boston, MA: Pearson.

### Manage Discussion Entry

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Here is another Quiz for everyone:

In this Graphic which distribution has the largest mean ? The smallest mean ??

In this Graphic which distribution has the largest standard deviation ? The smallest standard deviation ??

Try not to cheat by looking at the numbers in the legend ! Try looking just at the green / red / blue / orange distributions !!

Please Post your answers here and please be sure you all agree on what the correct answers are for these questions that have been asked here ! Thanks !! 🙂

Only a few class members need to please answer these questions here – NOT everyone !! LOL 😉

** Reference** ( where the Graphic was taken from ):

The blue, red and yellow curves all seem to peak at the same point along the x axis, so they all share bragging rights

to the largest mean. The green peaks to the left on the x axis, so that has the smallest mean.

The yellow curve is most spread out, so that has the largest standard deviation. The blue curve is narrowest, so that has the smallest standard deviation.

And your answers here are confirmed by the Legend in the figure. 😉

What I would say about the orange / yellow curve is that it is the “flattest” curve and so that is why it has the largest standard deviation.

But what you wrote about “most spread out” for the orange / yellow curve and what you wrote about “narrowest” for the blue curve is right on target too.

Thanks Elaine and best wishes Elaine and Everyone ! Enjoy Week 5 Elaine and Everyone Enjoy Week 5 and work hard and Learn a lot !

I really appreciate it Everyone !!

I agree with Elaine. Based on the graphic alone I could see that the yellow curve was more spread out, meaning it had a higher range of numbers with different values. Yellow, blue and red had peaks on or around the same value which means they have a similar and larger average. While my vocabulary does not include the specific terms such as standard deviation and mean, I essentially got the same idea from the graph. I appreciate how Elaine is able to look at the data and automatically apply the statistical vocabulary associated with the image.

In this Graphic which distribution has the largest mean? The smallest mean ??

Answer: The yellow, red, and blue curves all have the same mean= 0, which is also the largest mean. The green curve has the smallest mean=-2.

In this Graphic which distribution has the largest standard deviation? The smallest standard deviation ??

The yellow curve is the widest with the largest standard deviation=. The blue curve is the narrowest curve with the smallest standard deviation=

**Required Resources**

Read/review the following resources for this activity:

- OpenStax Textbook: Chapter 2, 6, and 7
- Lesson
- Weeks 3 and 5 Excel Spreadsheets

**Scenario/Summary**

This week’s lab highlights the use of probability and normal distribution.

Follow the directions below to gather data, calculate using Excel spreadsheets, and interpret the results.

**Deliverables**

The deliverable is a Word document with your answers to the questions posed below based on the data you find.

**Required Software**

- Microsoft Word
- Microsoft Excel

**Demonstrate**

**Steps to Complete Week 5 Lab**

Use the Weeks 3 and 5 spreadsheets **from the Weeks 3 and 5 Lessons **to help you answer the questions below.

Step 1: __Your instructor will provide you with 10 values to use for this lab.__

**Gather 10 MORE of your own to add to the 10 provided by your instructor. Do the following:**

Survey or measure 10 people to find their heights. Determine the mean and standard deviation for **the 20 values** by using the Week 3 Excel spreadsheet. Post a screen shot of the portion of the spreadsheet that helped you determine these values. How does your height compare to the mean (average) height of the **20 **values? Is your height taller, shorter, or the same as the mean of the sample?

Data Example of 10 people with different heights

- Data Example of 10 people with different heights

**(your spreadsheet will have **__20 values__—**10 from your instructor and 10 from your own data gathering).**

__20 values__

Step 2: Give some background information on the group of people you used in your study. You might consider using the following questions to guide your answer.

- How did you choose the participants for your study? What was the sampling method:
**systematic, convenience, cluster, stratified, simple random**? - What part of the country did your study take place in?
- What are the age ranges of your participants?
- How many of each gender did you have in your study?
- What are other interesting factors about your group?

Step 3: Use the Week 5 Excel spreadsheet for the following.

- (Use the
**Empirical Rule**tab from the spreadsheet). Determine the 68%, 95%, and 99.7% values of the Empirical Rule in terms of the**20 heights**in your height study. - What do these values tell you?
- Post a screen shot of your work from the Week 5 Excel spreadsheet.

Week 5 Spreadsheet Example

- (Use the normal probability tab from the spreadsheet). Based on your study results, what percent of the study participants are shorter than you? What percent are taller than you?
- Post a screen shot of your work from the Week 5 Excel spreadsheet.
**Example:**If my height is 73 inches, then 20.86% of the relevant population is shorter. The other 79.14%, of course, is taller.

### Week 5 Spreadsheet Example

Step 4: Be sure your name is on the Word document, save it, and then submit it under “Assignments” and “Week 5: Lab”.

**Grading**

This activity will be graded based on the Week 5 Lab Rubric.