1. Math 153 Stats Starts Here Mr. Anthony Calise

2. Course/Grading Overview
Website:
3 Exams (15% each for top two and 10% for low)
2 Projects (10% Each)
3 HW/Quizzes (10% Total)
Cumulative Final Exam 30%

3. 4 Units Covered
Unit 1: Graphs, Describing Data, Normal Distributions (Z-Scores)
Unit 2: Basic Probability, Geometric and Binomial Probabilities
Unit 3: Sampling Distribution and Confidence Intervals
Unit 4: Hypothesis Testing, Types of Errors

4. What to get out of the class….
How to better understand and interpret graphs.
How to understand probabilities in outcomes.
How to understand reports and findings.
How to not get mislead by advertisements and reports.
How to write and conduct an appropriate hypothesis test.

5. Stats Starts Here
According to 100% of people surveyed, this is the greatest class ever offered in college.
List of people surveyed: ME

6. What Is (Are?) Statistics?
Statistics (the discipline) is a way of reasoning, a collection of tools and methods, designed to help us understand the world.
Statistics (plural) are particular calculations made from a sample of data (Means, Medians, Standard Deviations, etc…)
Data are values with a context.
What is Statistics About?
Statistics is about variation. All measurements are imperfect, since there is variation that we cannot see.
Statistics helps us to understand the real, imperfect world in which we live.

7. Think, Show, Tell
There are three simple steps to doing Statistics right:
first. Know where you’re headed and why.
is about the mechanics of calculating statistics and graphical displays, which are important (but are not the most important part of Statistics).
what you’ve learned. You must explain your results so that someone else can understand your conclusions.

8. What Are Data? Data can be numbers, record names, or other labels.
Not all data represented by numbers are numerical data (e.g., 1=male, 2=female).
Data are useless without their context…

9. The Exam (FICTIONAL DATA)
The class average last semester was a 94 on the final exam.
It was out of 500 points!!!!
A group of individuals averaged a 81% on an algebra exam……
The group of individuals were 7 year olds…
Or…the group of individuals were Algebra teachers!

10. Dream Job
You are going to be hired by a company that you love, doing a job that you would love to do.
The average salary at the company that has 25 employees is $8,500,000 per year.
You will work 40 hours a week and have 4 weeks paid vacation.
Would you want this job?
The CEO makes
$212,400,000 per year…
The other 24 employees
Average $4, per/year (About $2.00/hr)

11. The “W’s” To provide context we need the W’s Who
What (and in what units)
When
Where
Why (if possible)
and How
of the data.
Note: the answers to “who” and “what” are essential. This will appear throughout the course when we discuss “Population” and “Parameters”.

12. What Variables are characteristics recorded about each Individual.
A categorical (or qualitative) variable names categories and answers questions about how cases fall into those categories.
Categorical examples: sex, race, ethnicity
A quantitative variable is a measured variable (with units) that answers questions about the quantity of what is being measured.
Quantitative examples: income ($), height (inches), weight (pounds)

13. What (cont.)
Example: In a student evaluation of instruction at a large university, one question asks students to evaluate the statement “The instructor was generally interested in teaching” on the following scale: = Disagree Strongly; 2 = Disagree; 3 = Neutral; = Agree; 5 = Agree Strongly.
Question: Is interest in teaching categorical or quantitative?

14. Examples Q C Time it takes to get to school in minutes
Height in inches
Number of shoes owned
Gender
Hair color
Age of Oscar winners in years
Temperature of a cup of coffee in Co
Type of pain medication
Jellybean flavors
Hours spent on Social Media Q C

15. Displaying and Describing Categorical Data
Copyright © 2009 Pearson Education, Inc.

16. Types of Graphs for Categorical Data
Bar Charts
Pie Charts
Frequency Table (Two-Way Table)

17. Frequency Tables: Making Piles
We can “pile” the data by counting the number of data values in each category of interest.
We can organize these counts into a frequency table, which records the totals and the category names.

18. Frequency Tables: Making Piles
A relative frequency table is similar, but gives the percentages (instead of counts) for each category.

19. Bar Charts
A bar chart displays the distribution of a categorical variable, showing the counts for each category next to each other for easy comparison.
A bar chart stays true to the area principle.
Thus, a better display for the ship data is:

20. Bar Charts (cont.)
A relative frequency bar chart displays the relative proportion of counts for each category.
A relative frequency bar chart also stays true to the area principle.
Replacing counts with percentages in the ship data:

21. Pie Charts
When you are interested in parts of the whole, a pie chart might be your display of choice.
Pie charts show the whole group of cases as a circle.
They slice the circle into pieces whose size is proportional to the fraction of the whole in each category.

22. Contingency Tables
A contingency table allows us to look at two categorical variables together.
It shows how individuals are distributed along each variable, contingent on the value of the other variable.
Example: we can examine the class of ticket and whether a person survived the Titanic:

23. Contingency Tables (cont.)
The margins of the table, both on the right and on the bottom, give totals and the frequency distributions for each of the variables.
Each frequency distribution is called a marginal distribution of its respective variable.

24. Marginal Distribution in %’s
If we wanted to find the marginal distribution in %’s for Class, we would look at the Totals for class. (i.e. 325, 285, 706, etc.)
We would then convert those numbers into percents by dividing by the totals and multiplying the decimal by 100.

26. Conditional Distributions
A conditional distribution shows the distribution of one variable for just the individuals who satisfy some condition on another variable.
The following is the conditional distribution of ticket Class, given the person survived:

27. Conditional Distributions (cont.)
The following is the conditional distribution of ticket Class, conditional on having perished:

28. Conditional Distributions (cont.)
The conditional distributions tell us that there is a difference in class for those who survived and those who perished.
This is better shown with pie charts of the two distributions:

29. Conditional Distributions (cont.)
We see that the distribution of Class for the survivors is different from that of the non-survivors.
The variables would be considered independent when the distribution of one variable in a contingency table is the same for all categories of the other variable.
Summary: If percent's are different then they are NOT independent.

30. Segmented Bar Charts (Not on Exam)
A segmented bar chart displays the same information as a pie chart, but in the form of bars instead of circles.
Here is the segmented bar chart for ticket Class by Survival status:

31. What Can Go Wrong? Don’t violate the area principle.
While some people might like the pie chart on the left better, it is harder to compare fractions of the whole, which a well-done pie chart does.

32. What Can Go Wrong? (cont.)
Keep it honest—make sure your display shows what it says it shows.
This plot of the percentage of high-school students who engage in specified dangerous behaviors has a problem. Can you see it?

36. Displaying and Summarizing Quantitative Data
Chapter 4
Displaying and Summarizing Quantitative Data
Copyright © 2009 Pearson Education, Inc.

37. Types of Graphs (Quantitative Data)
Histograms
Stem Plots (Stem and Leaf)
Box Plots (Box and Whisker)
Dot Plots
Scatter Plots (Two variables x and y)

38. Histograms: Earthquake Magnitudes
A histogram breaks up the entire span of values covered by the quantitative variable into equal-width piles called bins.
A histogram plots the bin counts as the heights of bars. (like a bar chart)
Here is a histogram of earthquake magnitudes

39. Histograms: Earthquake magnitudes
A relative frequency histogram displays the percentage of cases in each bin instead of the count.
In this way, relative
frequency histograms are faithful to the area principle.
Here is a relative frequency histogram of earthquake magnitudes:

40. Stem Plots (Stem and Leaf)
Stem-and-leaf displays show the distribution of a quantitative variable, like histograms do, while preserving the individual values.

41. Constructing a Stem-and-Leaf Display
First, cut each data value into leading digits (“stems”) and trailing digits (“leaves”).
Use the stems to label the bins.
Use only one digit for each leaf—either round or truncate the data values to one decimal place after the stem.

42. Dotplots Winning Time (sec)
A dotplot is a simple display. It just places a dot along an axis for each case in the data.
The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot.
You might see a dotplot displayed horizontally or vertically. 30 20 10 # O F R A C E S
Winning Time (sec)

43. Think Before You Draw, Again
Remember before you decide what type of graph to use check what kind of variable you are graphing. (Quantitative or Categorical)
Shape, Center, and Spread
When describing a distribution, make sure to always tell about three things: Shape, Outliers Center, and Spread…
Remember your SOCS!

44. What is the Shape of the Distribution?
Is the histogram symmetric or skewed?
Is the histogram symmetric?
If you can fold the histogram along a vertical line through the middle and have the edges match pretty closely, the histogram is symmetric.

45. Symmetry (cont.)
The (usually) thinner ends of a distribution are called the tails. If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail.
In the figure below, the histogram on the left is said to be skewed left, while the histogram on the right is said to be skewed right.

46. Anything Unusual? 2. Do any unusual features stick out?
Sometimes it’s the unusual features that tell us something interesting or exciting about the data.
You should always mention any stragglers, or outliers, that stand off away from the body of the distribution.
Are there any gaps in the distribution? If so, we might have data from more than one group.

47. Anything Unusual? (cont.)
The following histogram appears to have outliers —there are three cities in the leftmost bar:

48. Center of a Distribution:The Mean
If we want to calculate a number, we can average the data.
We use the Greek letter sigma to mean “sum” and write:
The formula says that to find the mean, we add up the numbers and divide by n.

49. Center of a Distribution: Median
The median is the value with exactly half the data values below it and half above it.
It is the middle data value (once the data values have been ordered) that divides the histogram into two equal areas.
It has the same units as the data.

50. Best Measure of Center (Mean vs. Median)
Because the median considers only the order of values, it is resistant to values that are extraordinarily large or small; it simply notes that they are one of the “big ones” or “small ones” and ignores their distance from center.
To choose between the mean and median, start by looking at the data. If the histogram is symmetric and there are no outliers, use the mean.
However, if the histogram is skewed or with outliers, you are better off with the median.

51. Spread: The Interquartile Range
The interquartile range (IQR) lets us ignore extreme data values and concentrate on the middle of the data.
To find the IQR, we first need to know what quartiles are…

52. 5-Number Summary
The 5-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum)
The 5-number summary for the recent tsunami earthquake Magnitudes looks like this:
We will use the 5-Number
summary later on to create
a box plot.

53. Spread: The Interquartile Range
Quartiles divide the data into four equal sections.
One quarter of the data lies below the lower quartile, Q1
One quarter of the data lies above the upper quartile, Q3.
The difference between the quartiles is the interquartile range (IQR), so
IQR = upper quartile – lower quartile
IQR = Q – Q1

54. Spread: The Interquartile Range (cont.)
The lower and upper quartiles are the 25th and 75th percentiles of the data, so…
The IQR contains the middle 50% of the values of the distribution, as shown in figure:

55. What About Spread? The Standard Deviation
A more powerful measure of spread than the IQR is the standard deviation, which takes into account how far each data value is from the mean.
A deviation is the distance that a data value is from the mean.
Since adding all deviations together would total zero, we square each deviation and find an average of sorts for the deviations.

56. What About Spread? The Standard Deviation
The variance, notated by s2, is found by summing the squared deviations and (almost) averaging them:
The variance will play a role later in our study, but it is problematic as a measure of spread—it is measured in squared units!

57. What About Spread? The Standard Deviation
The standard deviation, s, is just the square root of the variance and is measured in the same units as the original data.

58. Thinking About Variation
Since Statistics is about variation, spread is an important fundamental concept of Statistics.
Measures of spread help us talk about what we don’t know.
When the data values are tightly clustered around the center of the distribution, the IQR and standard deviation will be small.
When the data values are scattered far from the center, the IQR and standard deviation will be large.

59. Summary: Shape, Center, and Spread
Shape: Symmetric vs. Skewed (left or right)
Center: Mean vs. Median
Spread: Standard Deviation vs. IQR
Mean and Standard Deviation always go together
Median and IQR always go together.

60. The Five-Number Summary
The five-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum).
Example: The five-number summary for the daily wind speed is:
Max
8.67 Q3
2.93
Median
1.90 Q1
1.15
Min
0.20

61. Daily Wind Speed: Making Boxplots
A boxplot is a graphical display of the five-number summary.
Boxplots are particularly useful when comparing groups.

62. Constructing Boxplots (Outlier Check)
Draw a single vertical axis spanning the range of the data. Draw short horizontal lines at the lower and upper quartiles and at the median. Then connect them with vertical lines to form a box.

63. Constructing Boxplots (Outlier Check)
Erect “fences” around the main part of the data.
The upper fence is 1.5 IQRs above the upper quartile.
The lower fence is 1.5 IQRs below the lower quartile.
Note: the fences only help with constructing the boxplot and should not appear in the final display.

64. Constructing Boxplots (cont.)
Use the fences to grow “whiskers.”
Draw lines from the ends of the box up and down to the most extreme data values found within the fences.
If a data value falls outside one of the fences, we do not connect it with a whisker.

65. Constructing Boxplots (cont.)
Add the outliers by displaying any data values beyond the fences with special symbols.
We often use a different symbol for “far outliers” that are farther than 3 IQRs from the quartiles.

66. Formula: (Q1 – 1.5(IQR), Q3 + 1.5(IQR)) Needs to be memorized…
Check For Outliers
First we find the IQR
Then we multiply the IQR by 1.5
We then subtract that number from Q1
We also add that number to Q3
Any value falling outside of that range of values is considered an outlier.
Formula: (Q1 – 1.5(IQR), Q (IQR))
Needs to be memorized…

67. Summary (Test Topics) Categorical vs. Quantitative Variables
Types of Graphs for Categorical and Quantitative Data
Describe a Distribution: (Shape, Center, Spread)
Shape: Symmetric vs. Skewed
Center: Mean vs. Median
Spread: IQR vs. Standard Deviation
Checking for Outliers: Q1 – 1.5(IQR)
Q (IQR)

68. Summary: Shape, Center, and Spread
Mean and Standard Deviation always go together
Median and IQR always go together.
Use MEAN and STANDARD DEVIATION when the distribution is symmetric and there are NO Outliers!
Otherwise use Median and IQR.