1. 2: Frequency distributions Stemplot, frequency tables, histograms Frequency Distributions

2. Stem-and-leaf plots (stemplots) • Analyses start by exploring data with pictures • My favorite technique is the stemplot: a histogram-like display of data points You can observe a lot by looking – Yogi Berra Frequency Distributions

3. Illustrative example: sample.sav • A SRS of AGE (in years) • Data as an ordered array (n = 10): 05 11 21 24 27 28 30 42 50 52 • Divide each data point into • Stem values  first one or two digits • Leaf values  next digit • In this example • Stem values  tens place • Leaf values  ones place • e.g., 21 has a stem value of 2 and leaf value of 1 Frequency Distributions

4. Stemplot (cont.) • Draw stem-like axis from lowest to highest stem 0| 1| 2| 3| 4| 5| ×10  axis multiplier (important!) • Place leaves next to stem • 21 plotted (animation) 1 Frequency Distributions

5. Continue plotting … • Rearrange leaves in rank order: 0|5 1|1 2|1478 3|0 4|2 5|02 ×10 • For discussion, let’s rotate the plot 8 7 4 25 1 1 0 2 0------------0 1 2 3 4 5 (x10) ------------Rotated stemplot Frequency Distributions

6. Interpreting frequency distributions • Central Location • Gravitational center  mean • Middle value  median • Spread • Range and inter-quartile range • Standard deviation and variance (next week) • Shape • Symmetry • Modality • Kurtosis Frequency Distributions

7. Mean = arithmetic average “Eye-ball method”  visualize where plot would balance Arithmetic method = total divided by n 8 7 4 25 1 1 0 2 0------------0 1 2 3 4 5 ------------ ^ Grav.Center Eye-ball method  balances around 25 to 30 Actual arithmetic average = 29.0 Frequency Distributions

8. Middle point  median • Count from top to depth of (n + 1) ÷ 2 • For illustrative data: • n = 10 • Depth of median = (10+1) ÷ 2 = 5.5 Frequency Distributions

9. Spread  variability • Easiest way to describe spread is by stating its range, e.g., “from 5 to 52” (not the best way) • A better way is to divide the data into low groups and high groups • Quartile 1 = median of low group • Quartile 3 = median of high group Frequency Distributions

10. Shape  visual pattern • Skyline silhouette of plot • Symmetry • Mounds • Outliers (if any) • When n is small, it’s too difficult to describe shape accurately X X X XX X X X X X------------0 1 2 3 4 5 ------------ Frequency Distributions

11. What to look for in shape • Idealized shape = density curve • Look for: • General pattern • Symmetry • Outliers Frequency Distributions

12. Symmetrical shapes Frequency Distributions

13. Asymmetrical shapes Frequency Distributions

14. Modality (no. of peaks) Frequency Distributions

15. Kurtosis (steepness of peak)  fat tails Mesokurtic (medium) Platykurtic (flat)  skinny tails Leptokurtic (steep) Kurtosis can NOT be easily judged by eye Frequency Distributions

16. Second example (n = 8) • Data: 1.47, 2.06, 2.36, 3.43, 3.74, 3.78, 3.94, 4.42 • Truncate extra digit (e.g., 1.47  1.4) • Stem = ones-place • Leaves = tenths-place • 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 Frequency Distributions

17. Third example (pollution.sav) Regular stem: |1|4789|2|223466789|3|000123445678(×1) • Regular stemplot (top)  too squished • Split-stem (bottom) • First 1 on stem  leaves 0 to 4 • Second 1 on stem  leaves 5 to 9 Split-stem: |1|4|1|789|2|2234|2|66789|3|00012344|3|5678(×1) Note negative skew Frequency Distributions

18. How many stem-values? • Start with between 4 and 12 stem- values • Then, trial and error to draw out shape for the most informative plot (use judgment) Frequency Distributions

19. Body weight (n = 53) Data range from 100 to 260 lbs.  100 lb. multiplier seems too broad (only two stem values) 100 lb. multiplier w/ split stem-values still too broad (only 4 stem values) Try 10 pound stem multiplier Frequency Distributions

20. Body weight (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) 10|0 means “100” Shape: Positive skew, high outlier (260) Location: median = 165 (underlined) Spread: from 100 to 260 Frequency Distributions

21. Quintuple split:Body weight data (n = 53) 1*|0000111 1t|222222233333 1f|4455555 1s|666777777 1.|888888888999 2*|0111 2t|2 2f| 2s|6 (×100) • Codes: • * for leaves 0 and 1 t for leaves two and threef for leaves four and fives for leaves six and seven. for leaves eight and nine • Example: • 2t| 2 means a value of 222 (×100) Frequency Distributions

22. Frequency counts (SPSS plot) Age of participants SPSS provides frequency counts w/ 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) 3 . 0 means 3.0 years Because of large n, each leaf represents 2 observations Frequency Distributions

23. Frequency tables 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% • Frequency = count • Relative frequency = proportion or % • Cumulative frequency  % less than or equal to current value Frequency Distributions

24. Class intervals • When data sparse  group data into class intervals • Classes can be uniform or non-uniform Frequency Distributions

25. Uniform class intervals • Create 4 to 12 class intervals • Set end-point convention - include left boundary and exclude right boundary • e.g., first class interval includes 0 and excludes 10 (0 to 9.99 years of age) • Talley frequencies • Calculate relative frequency • Calculate cumulative frequency (demo) Frequency Distributions

26. Here’s age data in sample.sav… Frequency Distributions

27. Histogram – for quantitative data Bars are contiguous Frequency Distributions

28. Bar chart – for categorical data Bars are discrete Frequency Distributions