Displaying Distributions – Quantitative Variables

1. Displaying Distributions – Quantitative Variables Lecture 15 Secs. 4.4.1 – 4.4.3 Wed, Feb 6, 2008

2. Example • The following data represent GPAs. • How might we display these data graphically?

3. Frequency Plots • Frequency Plot

4. 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.

5. Example • Make a frequency plot of the following GPAs.

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

7. 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.

8. 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.

9. 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.

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

11. 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

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. 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.

14. Example • Draw a stem and leaf display of the following GPAs.

15. 1 3 8 9 2 1 3 3 4 5 5 7 8 9 3 0 3 4 8 Example • We may split the values at the decimal point: Note: 1|2 means 1.2.

16. 1 3 8 9 2 1 3 3 4 5 5 7 8 9 3 0 3 4 8 Example • We may split the values at the decimal point: Note: 1|2 means 1.2.

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

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

19. Example • Which is better? • Is either one particularly good?

20. 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.

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

22. 1 3 1 8 9 2 1 3 3 4 2 5 5 7 8 9 3 0 3 4 3 8 Stem Splitting Note: 1|2 means 1.2.

23. 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?