1. October 15

2. In Chapter 3: 3.1 Stemplot 3.2 Frequency Tables 3.3 Additional Frequency Charts

3. Stem-and-leaf plots (stemplots) Always start by looking at the data with graphs and plots Our favorite technique for looking at a single variable is the stemplot A stemplot is a graphical technique that organizes data into a histogram-like display You can observe a lot by looking – Yogi Berra

4. Stemplot Illustrative Example Select an SRS of 10 ages List data as an ordered array 05 11 21 24 27 28 30 42 50 52 Divide each data point into a stem-value and leaf-value In this example the “tens place” will be the stem-value and the “ones place” will be the leaf value, e.g., 21 has a stem value of 2 and leaf value of 1

5. Stemplot illustration (cont.) Draw an axis for the stem-values: 0| 1| 2| 3| 4| 5| ×10  axis multiplier (important!) Place leaves next to their stem value 21 plotted (animation) 1

6. Stemplot illustration continued … Plot all data points and rearrange in rank order: 0|5 1|1 2|1478 3|0 4|2 5|02 ×10 Here is the plot horizontally: (for demonstration purposes) 8 7 4 2 5 1 1 0 2 0 ------------ 0 1 2 3 4 5 ------------ Rotated stemplot

7. Interpreting Stemplots Shape –Symmetry –Modality (number of peaks) –Kurtosis (width of tails) –Departures (outliers) Location –Gravitational center  mean –Middle value  median Spread –Range and inter-quartile range –Standard deviation and variance (Chapter 4)

8. Shape “Shape” refers to the pattern when plotted Here’s the silhouette of our data X X X X X X X X X X ----------- 0 1 2 3 4 5 ----------- Consider: symmetry, modality, kurtosis

9. Shape: Idealized Density Curve A large dataset is introduced An density curve is superimposed to better discuss shape

10. Symmetrical Shapes

11. Asymmetrical shapes

12. Modality (no. of peaks)

13. Kurtosis (steepness) Mesokurtic (medium) Platykurtic (flat) Leptokurtic (steep)  skinny tails  fat tails Kurtosis is not be easily judged by eye

14. Location: Mean “Eye-ball method”  visualize where plot would balance Arithmetic method = sum values and divide by n 8 7 4 2 5 1 1 0 2 0 ------------ 0 1 2 3 4 5 ------------ ^ Grav.Center Eye-ball method  around 25 to 30 (takes practice) Arithmetic method mean = 290 / 10 = 29

15. Location: Median Ordered array: 05 11 21 24 27 28 30 42 50 52 The median has a depth of (n + 1) ÷ 2 on the ordered array When n is even, average the points adjacent to this depth For illustrative data: n = 10, median’s depth = (10+1) ÷ 2 = 5.5 → the median falls between 27 and 28 See Ch 4 for details regarding the median

16. Spread: Range Range = minimum to maximum The easiest but not the best way to describe spread (better methods of describing spread are presented in the next chapter) For the illustrative data the range is “from 5 to 52”

17. Stemplot – Second Example Data: 1.47, 2.06, 2.36, 3.43, 3.74, 3.78, 3.94, 4.42 Stem = ones-place Leaves = tenths-place Truncate extra digit (e.g., 1.47  1.4) Do not plot decimal |1|4 |2|03 |3|4779 |4|4 (×1) Center: between 3.4 & 3.7 (underlined) Spread: 1.4 to 4.4 Shape: mound, no outliers

18. Third Illustrative Example (n = 25) Data: {14, 17, 18, 19, 22, 22, 23, 24, 24, 26, 26, 27, 28, 29, 30, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38} Regular stemplot: |1|4789 |2|223466789 |3|000123445678 (×1) Too squished to see shape

19. Third Illustration (n = 25), cont. Split stem: –First “1” on stem holds leaves between 0 to 4 –Second “1” holds leaves between 5 to 9 –And so on. Split-stem stemplot |1|4 |1|789 |2|2234 |2|66789 |3|00012344 |3|5678 (×1) Negative skew - now evident

20. How many stem-values? Start with between 4 and 12 stem-values Trial and error: –Try different stem multiplier –Try splitting stem –Look for most informative plot

21. Fourth Example: Body weights (n = 53) Data range from 100 to 260 lbs:

22.  ×100 axis multiplier  only two stem- values (1×100 and 2×100)  too broad  ×100 axis-multiplier w/ split stem  only 4 stem values  might be OK(?)  ×10 axis-multiplier  see next slide

23. Fourth Stemplot Example (n = 53) 10|0166 11|009 12|0034578 13|00359 14|08 15|00257 16|555 17|000255 18|000055567 19|245 20|3 21|025 22|0 23| 24| 25| 26|0 (×10) Looks good! Shape: Positive skew, high outlier (260) Location: median underlined (about 165) Spread: from 100 to 260

24. Quintuple-Split Stem Values 1*|0000111 1t|222222233333 1f|4455555 1s|666777777 1.|888888888999 2*|0111 2t|2 2f| 2s|6 (×100) Codes for stem values: * for leaves 0 and 1 t for leaves two and three f for leaves four and five s for leaves six and seven. for leaves eight and nine For example, this is 120: 1t|2 (x100)

25. SPSS Stemplot Frequency Stem & Leaf 2.00 3. 0 9.00 4. 0000 28.00 5. 00000000000000 37.00 6. 000000000000000000 54.00 7. 000000000000000000000000000 85.00 8. 000000000000000000000000000000000000000000 94.00 9. 00000000000000000000000000000000000000000000000 81.00 10. 0000000000000000000000000000000000000000 90.00 11. 000000000000000000000000000000000000000000000 57.00 12. 0000000000000000000000000000 43.00 13. 000000000000000000000 25.00 14. 000000000000 19.00 15. 000000000 13.00 16. 000000 8.00 17. 0000 9.00 Extremes (>=18) Stem width: 1 Each leaf: 2 case(s) SPSS provides frequency counts w/ its stemplots: Because of large n, each leaf represents 2 observations 3. 0 means 3.0 years

26. Frequency Table Frequency = count Relative frequency = proportion or % Cumulative frequency  % less than or equal to level AGE | Freq Rel.Freq Cum.Freq. ------+----------------------- 3 | 2 0.3% 0.3% 4 | 9 1.4% 1.7% 5 | 28 4.3% 6.0% 6 | 37 5.7% 11.6% 7 | 54 8.3% 19.9% 8 | 85 13.0% 32.9% 9 | 94 14.4% 47.2% 10 | 81 12.4% 59.6% 11 | 90 13.8% 73.4% 12 | 57 8.7% 82.1% 13 | 43 6.6% 88.7% 14 | 25 3.8% 92.5% 15 | 19 2.9% 95.4% 16 | 13 2.0% 97.4% 17 | 8 1.2% 98.6% 18 | 6 0.9% 99.5% 19 | 3 0.5% 100.0% ------+----------------------- Total | 654 100.0%

27. Frequency Table with Class Intervals When data are sparse, group data into class intervals Create 4 to 12 class intervals Classes can be uniform or non-uniform End-point convention: e.g., first class interval of 0 to 10 will include 0 but exclude 10 (0 to 9.99) Talley frequencies Calculate relative frequency Calculate cumulative frequency

28. Class Intervals ClassFreqRelative Freq. (%) Cumulative Freq (%) 0 – 9.99110 10 – 1911020 20 – 2944060 30 – 3911070 40 – 4411080 50 – 59220100 Total10100-- Uniform class intervals table (width 10) for data: 05 11 21 24 27 28 30 42 50 52

29. Histogram A histogram is a frequency chart for a quantitative measurement. Notice how the bars touch.

30. Bar Chart A bar chart with non-touching bars is reserved for categorical measurements and non-uniform class intervals