MATH 225N Discussion 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.

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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).

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?
1. What part of the country did your study take place in?
1. What are the age ranges of your participants?
1. How many of each gender did you have in your study?
1. 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.

And attached here Friends are the 10 heights provided by the Instructor. You also gather heights from 10 Folks too for a total sample size of n = 20 for the Week 5 Lab turn in assignment.

Since the Instructor provided heights are in units of inches, then you must please gather your 10 heights in inches as well – PLEASE do not use centimeters for the units for your 10 gathered heights !!!!! 😉

Thanks Friends and best wishes ! Be sure to earn 50 / 50 points on each of the Week 5 and Week 7 Lab turn in assignments ! That will give you a little more flexibility and “wiggle room” on Final Exam performance / results ! Be well Friends and THANK YOU for your hard work, effort, and sacrifice on behalf of the course and on behalf of your learning !!

🙂

Week 5 lab assignment Math 225 Ten Initial Heights from Instructor_JAN 2021.xlsx

The first thing to realize about normal distributions is that there is more than one normal distribution.

So there are many, many, many normal distributions.

This is also true of t Distributions which we will encounter some in Weeks 6-7 of the course.

And the same was true of binomial distributions which we encountered in Week 4 of the course.

Notice in the slides how a particular normal distribution is designated and specified by its mean and standard deviation.

Thanks Friends and Best Wishes and Good Luck during Week 5 !!

🙂

Introductory Business Statistics Holmes Illowsky Dean 2018-6b6e901f-fa55-48f6-9276-69a5a1360071.pdf

Comparing and contrasting the means of two normal distributions and comparing and contrasting the standard deviations of two normal distributions is tricky until you get the hang of it. And this Week 5 graded Posting area asks you to speculate about the standard deviations of two normally distributed or approximately normally distributed variables. Also in a couple of early Instructor Posts this Week 5 I have a few examples and questions also asking you all to compare and contrast the means of two normal distributions and to determine which mean is larger and which mean is smaller.

So please let the following slides help introduce you to and hopefully ease you into this type of thought process and analysis.

Thanks Friends and Best Wishes and be well too !!

🙂Introductory Business Statistics Holmes Illowsky Dean 2018-425924d3-fcc6-44d6-9b1c-199cc1db7b00.pdf

Remember earlier when I wrote that there are many, many, many normal distributions ??

Well that is true of course, but guess what – there is ONLY ONE Standard Normal Distribution.

So the so-called Standard Normal Distribution is ONE SPECIFIC example and realization of a normal distribution.

And to be completely precise about it, the Standard Normal Distribution is the particular normal distribution with mean 0 and standard deviation 1 .

The Standard Normal Distribution Table is on page 6 of 8 of the attached Formula and Table sheet here.

On page 7 of that attachment you see information about t Distributions.

On page 8 of that attachment you see information about Chi – Square distributions.

Thanks Friends and please notice the following slides which give you some good info about the one and only Standard Normal Distribution.

Be well and take good care Friends !! Enjoy Week 5 and learn a lot !!

🙂

Formula Card, Reference Sheet, and Standard Normal Distribution Table-2056ca79-c61e-488f-8a7e-12f83839bec6.pdf

Technology use is ESSENTIAL in Weeks 5-8 of this challenging and fast paced course !

Class members can please use either the provided Weekly Excel spread sheet calculators to do the problems and exercises in the course or can use the TI 84+ Graphing Calculator model ( or the 83 83+ 84 models ) to do the problems and exercises in the course.

The Weekly Tutorial and Demonstration Videos near the top of each regular Weekly Graded Posting area go through the Knewton Assignments for the Week emphasizing the use of the Weekly Excel spread sheet calculator.

The Weekly Recorded Office Hour Videos generally emphasize the use of the TI 84+ Graphing Calculator to work on and solve the problems and exercises in the Knewton Homework Assignments in the course.

So you all are covered either way !!

Thanks Friends and please see the slides here for a Great example of technology use in Week 5 of the course !

Thanks Friends and Good Luck !!

🙂Introductory Business Statistics Holmes Illowsky Dean 2018-5e97b0ac-bc13-4966-9c38-413ecbbe0222.pdf

The Central Limit Theorem is a topic that almost gets completely lost in a fast paced, fast moving course such as our course here.

However, if you are ever going to take additional and more advanced courses in statistics and quantitative research and data analysis then you please need the type of foundation offered in the following slides here so that the concept of the Central Limit Theorem is not unfamiliar to you when you undertake such future courses.

Thanks Friends and Best Wishes and Good Luck in Week 5 here !!

MATH 225N Discussion The Normal Distribution