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Chapter 3: Frequency Distributions

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

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Chapter 3: Frequency Distributions

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  1. Chapter 3: Frequency Distributions

  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 25 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)  fat tails Mesokurtic (medium) Platykurtic (flat)  skinny tails Leptokurtic (steep) 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 Eye-ball method  around 25 to 30 (takes practice) Arithmetic method mean = 290 / 10 = 29 8 7 4 25 1 1 0 2 0------------0 1 2 3 4 5 ------------ ^ Grav.Center

  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. Data range from 100 to 260 lbs: • ×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 threef for leaves four and fives for leaves six and seven. for leaves eight and nine For example, this is 120: 1t|2(x100)

  25. SPSS Stemplot SPSS provides frequency counts w/ its stemplots: 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

  26. Frequency Table 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 level

  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 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

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