1. Displaying Distributions – Quantitative Variables
Lecture 15
Secs – 4.4.3
Mon, Sep 17, 2007

2. Frequency Plots
Frequency Plot

3. Drawing Frequency Plots
Draw the real line.
Choose a resolution, e.g., 0.1.
Mark the minimum and maximum values.
Label the values on the scale, as on a ruler.
Mark at regular intervals.
For each data value, draw an X over that value on the scale.

4. Example Make a frequency plot of the following GPAs. 2.946 2.335 3.418
1.890
2.731
3.855
1.344
2.126
2.881
2.542
2.504
3.367
1.950
2.392
2.443
3.053

5. Frequency Plots
What information is conveyed by a frequency plot?

6. Shapes of Distributions
Symmetric – The left side is a mirror image of the right side.
Unimodal – A single peak, showing the most common values.
Bimodal – Two peaks.
Uniform – All values have equal frequency.
Skewed – Stretched out more on one side than the other.

7. Stem-and-Leaf Displays
Each value is split into two parts: a stem and a leaf.
For example, the value 1.23 could be split as
stem = 123, leaf = 0, or
stem = 12, leaf = 3, or
stem = 1, leaf = 2, or
stem = 0, leaf = 1.

8. Stem-and-Leaf Displays
The stem consists of the leftmost digits of the value, as many as deemed appropriate.
The leaf consists of the next digit (one digit).
A note should be added indicating how to interpret the numbers.
Note: 12|3 means 1.23.

9. Stem-and-Leaf Displays
A note should be added indicating how to interpret the numbers.
Note: 12|3 means 1.23.

10. Stem-and-Leaf Displays
A note should be added indicating how to interpret the numbers.
Note: 12|3 means 1.23.
stem
leaf
actual value

11. Splitting the Numbers
We choose where to split the numbers in order to avoid
Too many stems, each with too few leaves.
Too few stems, each with too many leaves.

12. Splitting the Numbers
We choose where to split the numbers in order to avoid
Too many stems, each with too few leaves.
Too few stems, each with too many leaves.

13. Example Draw a stem and leaf display of the following GPAs. 2.946
2.335
3.418
1.890
2.731
3.855
1.344
2.126
2.881
2.542
2.504
3.367
1.950
2.392
2.443
3.053

14. Example We may split the values at the decimal point:
Note: 1|2 means 1.2. 1 2 3
3 8 9

15. Example We may split the values at the decimal point:
Note: 1|2 means 1.2. 1 2 3
3 8 9

16. Example Or we may split the values after the first decimal place:13 14 15 4 16 17 18 19 20 : 9 5
Note: 12|3 means
1.23.

17. Example Or we may split the values after the first decimal place:13 14 15 4 16 17 18 19 20 : 9 5
Note: 12|3 means
1.23.

18. Example
Which is better?
Is either one particularly good?

19. Stem Splitting
We can obtain a good compromise (in this examle) by splitting the stems.
Each stems appears twice.
The first time for leaves 0 – 4.
The second time for leaves 5 – 9.

20. Stem Splitting Note: 1|2 means 1.2. 1 2 3 8 9 1 3 3 4 5 5 7 8 9 0 3 4

21. Stem Splitting Note: 1|2 means 1.2. 1 2 3 8 9 1 3 3 4 5 5 7 8 9 0 3 4

22. Shapes of Distributions
If the distribution of household incomes were skewed to the right, what would that tell us?
If a grade distribution were skewed to the left, what would that tell us?