1. Chapter 1: Exploring Data
Sec. 1.2: Displaying Quantitative Data with Graphs

2. Displaying Quantitative Data with Graphs
MAKE and INTERPRET dotplots and stemplots of quantitative data
DESCRIBE the overall pattern of a distribution and IDENTIFY any outliers
IDENTIFY the shape of a distribution
MAKE and INTERPRET histograms of quantitative data
COMPARE distributions of quantitative data

3. Dotplots
Dotplot - Each data value is shown as a dot above its location on a number line.
Quantitative data
Values at the bottom
Each dot = 1 piece of data
Heights are how many at each value
How to make a dotplot:
Draw a horizontal axis (a number line) and label it with the variable name.
Scale the axis from the minimum to the maximum value.
Mark a dot above the location on the horizontal axis corresponding to each data value.

4. Number of Goals Scored Per Game by the 2012 US Women’s Soccer Team
Dotplots Example, p. 25
Number of Goals Scored Per Game by the 2012 US Women’s Soccer Team 2 1 5 3 4 13 14

5. Stemplots
Stemplots (a.k.a. stem-and-leaf plot) give us a quick picture of the distribution while including the actual numerical values.
Quantitative data
Each stem must have an equal # of possibilities
Around 5 stems is good minimum
May need to round data (5.3 ≈ 5)
May have to split (high/low)
Numbers listed in increasing order
Leaves can only be one digit
How to make a stemplot:
Separate each observation into a stem (all but the final digit) and a leaf (the final digit).
Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column.
Write each leaf in the row to the right of its stem.
Arrange the leaves in increasing order out from the stem.
Provide a key that explains in context what the stems and leaves represent.

6. Example, p. 31
These data represent the responses of 20 female AP Statistics students to the question, “How many pairs of shoes do you have?” Construct a stemplot. 50 26 31 57 19 24 22 23 38 13 34 30 49 15 51
Stems 1 2 3 4 5
Add leaves
4 9
Order leaves
4 9
Add a key
Key: 4|9 represents a female student who reported having 49 pairs of shoes.

7. Example, p. 32
When data values are “bunched up”, we can get a better picture of the distribution by splitting stems.
Females
Females
333 95
4332 66
410 8 9
100 7 1 2 3 4 5 50 26 31 57 19 24 22 23 38 13 34 30 49 15 51
Key: 4|9 represents a student who reported having 49 pairs of shoes.
***Splitting stems is also useful if you do not have the recommended minimum of 5.
“split stems”

8. Example, p. 32
Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems.
Males 14 7 6 5 12 38 8 10 11 4 22 35
***Note that the distribution on the left has the leaves decreasing in order (increasing going out from the stem)
Females
333 95
4332 66
410 8 9
100 7
Males
0 4 1
2 2 2 3
3 58 4 5
Key: 4|9 represents a student who reported having 49 pairs of shoes.

9. Examining the Distribution of a Quantitative Variable
To understand a graph ask, “What do I see?”
How to Examine the Distribution of a Quantitative Variable
Shape – Look at main features; look for rough symmetry or clear skewness. Use modifiers!!!! (roughly, approximately, somewhat, etc.)
Outliers – outcomes that deviate from the overall pattern.
Center - where the middle of the distribution falls.
Spread – (for right now) smallest to largest value.
To get full credit on the AP Exam, you must describe all four aspects. Many students lose points because they only comment on the shape. Use modifiers because distributions are very rarely exactly a certain shape.
***We will discuss Center & Spread more in Sec. 1.3.***

10. Describing Shape A distribution is:
roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. 2
467 3
00157 4 5
12468 6
2257 7 1

11. Describing Shape
skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. 2
112467 3 4
1468 5
12468 6 27 7 1

12. Describing Shape
skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. 5 2 6 79 7
002
55699 8 9 10 00

13. Remember …
Which way is the skier skiing?
Which side is the tail on?

14. Shape?
Symmetric
Skewed-left
Skewed-right

15. Other Shapes Unimodal Bimodal Uniform Unimodal – one peak, one “mode”
Bimodal – two peaks, two “modes”
Uniform – pretty much the same throughout

16. Comparing Distributions
Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable.
Not enough to just say what the center and spread are…compare them…Distribution A has a center that is larger than Distribution B.

17. Example, p. 30
Compare the distributions of household size for these two countries.
Don’t forget your SOCS!
Shape: The distribution of household sizes in South Africa is skewed right, while the distribution of household sizes in the U.K. is roughly symmetric.

18. Example, p. 30
Compare the distributions of household size for these two countries.
Don’t forget your SOCS!
Outliers: South Africa appears to have 1 maybe 2 outliers at 26 & 15. The U.K. does not appear to have any outliers.

19. Example, p. 30
Compare the distributions of household size for these two countries.
Don’t forget your SOCS!
Center: The center for South African household sizes seems to be larger than the U.K.. South Africa – 6, U.K. – 4

20. Example, p. 30
Compare the distributions of household size for these two countries.
Don’t forget your SOCS!
Spread: The distribution for households in South Africa has more variability 3 to 26 people) than that of the U.K. (2 to 6).

21. Homework – Due Friday
P. 41 #37, 41, 45, 50