2. Last chapter... Four Corners: Go to your corner based on if your birthday falls in the Winter, Spring, Summer, or Fall; 1 minute In your group, come to a consensus about the three most important topics we learned and list them on the board. 5 minutes.

3. Last chapter, we learned... Appropriate graphical representations (numerical & categorical data) Always graph the data; always. Always embed context. Always. Describing numerical distributions/data sets via SOCS (the basics; we will get more sophisticated with our descriptions soon); do we use SOCS to describe categorical data distributions? Why or why not?

4. SOCS... Shape, Outlier(s), Center, Spread We loosely defined ‘center’ and ‘spread’ Now we will be much more specific & detailed... And remember, always embed context Here we go...

5. Word association time... When I say a word, you immediately write down what you think it means; don’t think, just write. Don’t talk; don’t say anything to anyone. Ready?

6. Word association time... Average

7. Bill Gates walks into a diner... The annual salaries of 7 patrons in a diner are listed below. Find the mean and the median using Minitab Are the mean and the median similar? Would they represent a ‘typical’ or ‘average’ customer’s salary? Should we use the mean or the median in this case? Graph the data (let’s practice a histogram; then a box plot) using Minitab. What shape is the distribution? $45,000$48,000 $52,000$40,000 $35,000$58,000 $46,000

8. Now, Bill Gates walks into the diner... Find the mean and the median using Minitab Are the mean and the median similar? Would they represent a ‘typical’ or ‘average’ customer’s salary? Should we use the mean or the median in this case? Graph the data (histogram; box plot) using Minitab. What shape is the distribution? $45,000$48,000 $52,000$40,000 $35,000$58,000 $46,000$3,710,000,000

9. What’s the moral of this story? Means are excellent measures of central tendency if the data is (fairly) symmetric However, means are highly influenced by outlier(s) So, if the data has an outlier(s), then a better measure of central tendency is the median, which is not influenced by outliers; this is called ‘resistant’ So, consider the shape of your data/distribution, then wisely choose an appropriate measure of central tendency

10. Which measure of central tendency should we use?.

11. For this distribution, which is larger: mean or median?

12. Left skewed; how does mean compare to median?

13. The ‘C’ in SOCS So, when we are analyzing a numerical distribution (like looking at a histogram, stem plot, box plot, etc.), we need to wisely choose which ‘C’ to use... mean or median Generally, if symmetric use mean as a measure of central tendency If skewed use median as a measure of central tendency

14. Measures of Spread Calculate the mean & the median of each of the following data sets (you can use Minitab if you need to): (4, 4, 5, 6, 6)(5, 5, 5, 5, 5) Are they the same distribution/data set? Another characteristic that is helpful in describing distributions/data sets is the measure of spread (or the typical distance from the center)

15. Spread... The second ‘S’ in SOCS Another characteristic that is helpful in describing distributions/data sets is the measure of spread (or the typical distance from the center) Two measures of spread that we will focus on in this course are the standard deviation & inter-quartile range

16. Standard Deviation is... a typical distance of the observations from their mean is a number that measures how far away the typical observation is from the center of the distribution

17. Let’s play the standard deviation game... Your team’s task: Create a data set of four whole numbers (from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) with the lowest standard deviation value possible Input your four numbers (again use numbers from 0 to 10 only) into Minitab, then calculate the standard deviation Change a value or values until you get the lowest possible standard deviation you can. 3 minutes. Go. Now create a data set (again only from 0 to 10) with the largest possible standard deviation.

18. Which has the largest SD?

19. Calculating the standard deviation...

20. Variance... Another measure of spread Not used very often; usually, if we use a mean as a measure of central tendency, we use the standard deviation as our measure of spread Variance is related to standard deviation variance = (standard deviation) 2 standard deviation =

21. Practice time... With a partner, go to my website, open Math 140 data sheet, find ‘how many units are you planning to enroll in this semester’ column; copy and past into Minitab Create a histogram. Based on the histogram, which measure of central tendency (mean or median) do you believe is the best to use to describe this data set? Why? Calculate the measure of central tendency you chose. Then calculate the standard deviation. Print all of the above; put both your names on it.

22. The Empirical Rule... When distributions are uni-modal, ≈ symmetric, & mean ≈ median, then... life is beautiful Distribution is said to be ≈ Normal 68% of data within 1 standard deviation of mean 95% of data within 2 standard deviations of mean 99.7% of data within 3 standard deviations of mean

23. 68-95-99.7 Rule (Empirical Rule) For (≈)Normal Distributions Only

24. Empirical ‘Model’...

25. New topic.. (not female heights anymore)... We will get back to that in a few minutes... Is 120 big or small? Think – Pair - Share

26. TPS... Is 120 big or small? Big if... day’s temperature in LA in degrees Fahrenheit or # units a student takes during a semester (really big!) Small if... monthly rent paid for an apartment in LA Usual or ‘average’ if... weight in pounds for a 15-year-old girl or systolic blood pressure Nearly impossible to answer how unusual 120 is unless we know what we are comparing 120 to.

27. Something else to consider... A student’s ACT score was 25.9; their SAT score was 1172. Which is a better score? ACT scores’ (national) mean = 21, standard deviation 4.7 SAT (national) mean (critical reading & math) = 1010, standard deviation = 163

28. When we have a Normal distribution, then... Z- Scores, standardizing... When we have a Normal distribution, we can calculate z-scores, or standardizing data, convert raw data into # of SD’s away from mean

29. Let’s practice with some of our heights...

30. Can we calculate a z-score using a skewed distribution?

31. Remember Bill Gates? Median, ‘center’ value when data is organized from smallest to largest values Consider the distribution: 0, 0, 0, 0, 1, 1, 1, 2, 3 By the way, what could be the context? Median = Consider the distribution: 0, 0, 0, 0, 1, 1, 1, 2, 999 Median =

32. Data gathering time again... # siblings you have on board & enter into Minitab Numerical analysis (descriptive statistics in Minitab) and graphical representation Describe the distribution

33. Skewed? Shouldn’t use mean & SD But we still need to describe the center and the spread of the distribution Use median and IQR (Inter-quartile Range) Median & IQR are not effected by outlier(s) (resistant) IQR = Q3 – Q1 IQR is amount of space the middle 50% of the data occupy

34. Range of data... Another measure of variability (used with any distribution) is range Range = maximum value – minimum value Range for our data =

35. Boxplots...based on 5-number summary

36. Boxplots...

37. Modified boxplot – shows outlier(s)

38. Two modified boxplots...

39. What are outliers? Boxplots are the only graphical representation where we specifically define an outlier Potential outliers are values that are more than 1.5 IQRs from Q1 or Q3 IQR x 1.5; add that product to Q3; any value(s) beyond that point is an outlier to the right Q1; any value(s) beyond that point is an outlier to the left

40. Go back to our Siblings data... Using Minitab, calculate descriptive statistics Let’s calculate (by hand) to see if we have any outliers Q3 – Q1 = IQR IQR x 1.5; add this product to Q3; are there any values in our data set beyond this point to the right? IQR x 1.5; subtract product from Q1; are there any values in our data set beyond this point to the left? Now use Minitab to create a boxplot; are our calculations confirmed with our boxplot?

41. Be careful with outliers... Are they really an outlier? Is your data correct? Was it input accurately? COC’s recent 99-year-old graduate Don’t automatically throw out an unusual piece of data; investigate

42. Be careful... one more thing...

43. Four Corners... When were you born? Spring, Fall, Summer, Winter? Come to a consensus; what are your ‘top five’ concepts for Chapter 3 Write them on the board Class discussion

44. Exit Ticket...

45. Chapters 1, 2, 3... Review Exam