1. Recap of: Chapter 1 Exploring Data AP Statistics Borrowed from Brian Hamilton & Jason Molesky

2. Chapter 1 Section 1 Displaying Distributions with Graphs
HW: 1.1(a, b), 1.2, 1.3, 1.4, 1. 5, 1.6
1.8, 1.9, 1.10, 1.11
1.13, 1.14, 1.16

3. Exploratory Data Analysis
Exploratory Data Analysis: Statistical practice of analyzing distributions of data through graphical displays and numerical summaries.
Distribution: Description of the values a variable takes on and how often the variable takes on those values.
An EDA allows us to identify patterns and
departures from patterns in distributions.

4. Categorical Data
Categorical Variable: Values are labels or categories. Distributions list the categories and either the count or percent of individuals in each.
Displays: BarGraphs & Pie chart(not done in AP)

5. Bar Graph

6. Quantitative Data Quantitative Variable: Displays:
Values are numeric - arithmetic computation makes sense (average, etc.)
Distributions list the values and # of times the variable takes on that value.
Displays:
Dotplots
Stemplots
Histograms
Boxplots

7. Only organized Data can Illuminate!
Your goal is to make neat, organized, labeled graphs that display the distribution of data effectively and provide an insight into patterns and departures from patterns.

8. Dotplots
Small datasets with a small range (max-min) can be easily displayed using a dotplot.
Draw and label a number line from min to max.
Place one dot per observation above its value.
Stack multiple observations evenly.
34 values ranging from 0 to8.

9. Stemplots
A stemplot gives a quick picture of the shape of a distribution while including the numerical values.
Separate each observation into a stem and a leaf.
eg. 14g -> 1| > 25|6 32.9oz -> 32|9
Write stems in a vertical column and draw a vertical line to the right of the column.
Write each leaf to the right of its stem.

10. Stemplots
Example1.4, pages 42-43
Literacy Rates in Islamic Nations

11. Stemplots Note: Stemplots do not work well for large data sets
Back-to-Back Stemplots: Compare datasets
Splitting Stems: Double the number of stems, writing 0-4 after the first and 5-9 after second.

12. Histograms
Histograms break the range of data values into classes and displays the count/% of observations that fall into that class.
Divide the range of data into equal-width classes.
Count the observations in each class - “frequency”
Draw bars to represent classes - height = frequency
Bars should touch (unlike bar graphs).

13. Histograms
Let’s look ex. 1.6, page 49 on 5th grade IQ scores

14. Histograms Describe the SOCS or SECS What do these data suggest?
TAKE THE DATA AND MAKE CLASSES WITH FREQUENCY

15. EDA Summary
The purpose of an Exploratory Data Analysis is to organize data and identify patterns/departures.
PLOT YOUR DATA - Choose an appropriate graph
Look for overall pattern and departures from pattern
Shape {mound, bimodal, skewed, uniform}
Outliers {points clearly away from body of data}
Center {What number “typifies” the data?}
Spread {How “variable” are the data values?}

16. Let’s Look/Investigate
Draw different “shaped” histograms

17. Cumulative Relative Frequency Graphs (OGIVE)
Histograms do a good job of displaying the distribution of values or a quantitative variable, but tell us little about the relative standing of an individual observation. If we want relative standing information, we use an ogive.

18. Cumulative Relative Frequency Graphs (OGIVE)

19. Cumulative Relative Frequency Graphs
Decide on class intervals and determine the frequency in each class (like with a histogram), but add some additional columns to your table.
Label and scale your axes and title your graph.
Plot a point corresponding to the relative cumulative frequency in each class interval at the left endpoint of the next class interval.

20. Using Ogives
Bill Clinton took office at the age of 46. Find his relative standing for the age he took office.
About 10% of all U.S. Presidents were the same age as or younger than Bill Clinton when they were inaugurated.

22. Average Price of Gasoline from 1996-2006

23. Chapter 1 Section 2 Describing Distributions with Numbers
HW: 1.27, , 1.33, 1.36,
1.37, 1.41, 1.43, 1.45, 1.46

24. Describing Distributions with Numbers
What three things do we always include when describing a distribution of numbers?
Shape
Center
Spread

25. Describing Distributions with Numbers
What three things do we always include when describing a distribution of numbers?
Shape- use a histogram
Center
Spread

26. Describing Distributions with Numbers
What three things do we always include when describing a distribution of numbers?
Shape- use a histogram
Center-
Spread-

27. Describing Center

28. Find the mean for the Fuel Economy for 2004 model motor vehicles
You can use a graphing calculator

29. How Do I write this?
Use your notation and begin the calculation (this tells me you understand the formula) and it will allow for partial credit versus no credit….
(show example)

30. The mean is sensitive to the influence of a few extreme observations.
Means & Outliers
Are there any values that appear to be outliers in our fuel economy data?
If so, find the mean without the outlier.
The single outlier adds 2 mpg onto the mean highway mileage of 2004 motor vehicles.
The mean is sensitive to the influence of a few extreme observations.
It is not a resistant measure of center.

32. Finding the median Median highway mileage for two-seaters
Without the Honda insight?
The median is a more resistant measure of center than the mean.

33. In a right skewed distribution?
Mean verses Median
How do you think the mean and median compare in a symmetric distribution?
In a right skewed distribution?
Don't confuse the “average” value (mean) with the “typical” value (median).

34. Mean vs. Median
The mean and the median are the most common measures of center.
If a distribution is perfectly symmetric, the mean and the median are the same.
The mean is not resistant to outliers.
You must decide which number is the most appropriate description of the center...
MeanMedian Applet

35. Mean verses Median
Generally, when you encounter outliers, correct them if wrongly recorded, delete them for good reason, or otherwise give them individual attention.
If correctly recorded outliers cannot be discarded, consider using resistant measures.

36. Measuring Spread

37. Quartiles Quartiles Q1 and Q3 represent the 25th and 75th percentiles.
To find them, order data from min to max.
Determine the median - average if necessary.
The first quartile is the middle of the ‘bottom half’.
The third quartile is the middle of the ‘top half’. 19 22 23 26 27 28 29 30 31 32
Q1=23
med
Q3=29.5 45 68 74 75 76 82 91 93 98
med=79 Q1 Q3

39. Finding the Quartiles
Find the Quartiles, and IQR.

40. Computer Software Output

42. 5-Number Summary, Boxplots
We can visualize the 5 Number Summary with a boxplot.
min=45
Q1=74
med=79
Q3=91
max=98 45 50 55 60 65 70 75 80 85 90 95
100
Quiz Scores

43. Measures of Spread
Variability is the key to Statistics. Without variability, there would be no need for the subject.
When describing data, never rely on center alone.
Measures of Spread:
Range - {rarely used...why?}
Quartiles - InterQuartile Range {IQR=Q3-Q1}
Variance and Standard Deviation {var and sx}
Like Measures of Center, you must choose the most appropriate measure of spread.

44. What about OUTLIERS?
THIS HELPS SETS UP “FENCES” TO SEE IF THERE ARE ANY DATA THAT FALLS OUTSIDE..IF SO THEY ARE OUTLIERS

45. 1.5 • IQR Rule To determine outliers: Find 5 Number Summary
Determine IQR
Multiply 1.5xIQR
Set up “fences” Q1-(1.5IQR) and Q3+(1.5IQR)
Observations “outside” the fences are outliers.

46. Outlier Example } { 10 20 30 40 50 60 70 80 90 100 Spending ($)10 20 30 40 50 60 70 80 90
100
Spending ($)
All data on p. 48.
IQR=
IQR=26.66
1.5IQR=1.5(26.66)
1.5IQR=39.99 }
fence:
= 85.71
fence: = {
outliers

47. Let’s CREATE a Box-Plot

48. What do we need?

49. Standard Deviation Another common measure of spread
is the Standard Deviation: a measure of
the “average” deviation of all observations from the mean.
To calculate Standard Deviation:
Calculate the mean.
Determine each observation’s deviation (x - xbar).
“Average” the squared-deviations by dividing the total squared deviation by (n-1).
This quantity is the Variance.
Square root the result to determine the Standard Deviation.

50. Standard Deviation Variance: Standard Deviation:
Example 1.16 (p.85): Metabolic Rates
1792
1666
1362
1614
1460
1867
1439

51. Total Squared Deviation
Standard Deviation
1792
1666
1362
1614
1460
1867
1439
Metabolic Rates: mean=1600 x
(x - x)
(x - x)2
1792
192
36864
1666 66
4356
1362
-238
56644
1614 14
196
1460
-140
19600
1867
267
71289
1439
-161
25921
Totals:
214870
Total Squared Deviation
214870
Variance
var=214870/6
var=
Standard Deviation
s=√
s= cal
What does this value, s, mean?

52. Linear Transformations
Variables can be measured in different units (feet vs meters, pounds vs kilograms, etc)
When converting units, the measures of center and spread will change.
Linear Transformations (xnew=a+bx) do not change the shape of a distribution.
Multiplying each observation by b multiplies both the measure of center and spread by b.
Adding a to each observation adds a to the measure of center, but does not affect spread.

53. Data Analysis Toolbox To answer a statistical question of interest:
Data: Organize and Examine
Who are the individuals described?
What are the variables?
Why were the data gathered?
When,Where,How,By Whom were data gathered?
Graph: Construct an appropriate graphical display
Describe SOCS
Numerical Summary: Calculate appropriate center and spread (mean and s or 5 number summary)
Interpretation: Answer question in context!

54. Chapter 1 Summary
Data Analysis is the art of describing data in context using graphs and numerical summaries. The purpose is to describe the most important features of a dataset.