3: Frequency Distributions

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Chapter 3 Frequency Distributions
2
In Chapter 3

• 3.1 Stemplots
• 3.2 Frequency Tables
• 3.3 Additional Frequency Charts

3
Stemplots
You can observe a lot by looking Yogi Berra

• Start by exploring the data with Exploratory Data
  Analysis (EDA)
• A popular univariate EDA technique is the
  stem-and-leaf plot
• The stem of the stemplot is an number-line (axis)
• Each leaf represents a data point

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Stemplot Illustration

• 10 ages (data sequenced as an ordered array)
• 05 11 21 24 27 28 30 42 50 52
• Draw the stem to cover the range 5 to 52
• 0 1 2 3 4 5 10 ? axis multiplier
• Divide each data point into a stem-value (in this
  example, the tens place) and leaf-value (the
  ones-place, in this example)
• Place leaves next to their stem value
• Example of a leaf 21 (plotted)

1
5
Stemplot illustration continued

• Plot all data points in rank order
• 05 11 21478 30 42 502 10
• Here is the plot horizontally

8 7 4 25 1 1 0 2
0------------0 1 2 3 4 5------------Rotated
stemplot
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Interpreting Distributions

• Shape
• Central location
• Spread

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Shape

• Shape refers to the distributional pattern
• Heres the silhouette of our data X
  X X X X X X X X X
  ----------- 0 1 2 3 4 5 -----------
• Mound-shaped, symmetrical, no outliers
• Do not over-interpret plots when n is small

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Shape (cont.)
Consider this large data set of IQ scores
An density curve is superimposed on the graph
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Examples of Symmetrical Shapes
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Examples of Asymmetrical shapes
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Modality (no. of peaks)
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Kurtosis (steepness)
? fat tails
Mesokurtic (medium)
Platykurtic (flat)
? skinny tails
Leptokurtic (steep)
Kurtosis is not be easily judged by eye
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Gravitational Center (Mean)

• Gravitational center arithmetic mean
• Eye-ball method ? visualize where plot would
  balance on see-saw
• around 30 (takes practice)
• Arithmetic method sum values and divide by n
• sum 290
• n 10
• mean 290 / 10 29

8 7 4 25 1 1 0 2
0------------0 1 2 3 4 5 ------------
Grav.Center
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Central location Median

• Ordered array
• 05 11 21 24 27 28 30 42 50 52
• The median has depth (n 1) 2
• n 10, medians depth (101) 2 5.5
• ? falls between 27 and 28
• When n is even, average adjacent values? Median
  27.5

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Spread Range

• For now, report the range (minimum and maximum
  values)
• Current data range is 5 to 52
• The range is the easiest but not the best way to
  describe spread (better methods described later)

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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)

142033477944(1)

• Center median between 3.4 3.7 (underlined)
• Spread 1.4 to 4.4
• Shape mound, no outliers

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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
• 147892223466789300012344567810
• Too squished to see shape

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Third Illustration Split Stem

• Split stem-values into two ranges, e.g., first
  1 holds leaves between 0 to 4, and second 1
  will holds leaves between 5 to 9
• Split-stem
• 1417892223426678930001234435678
  10
• Negative skew now evident)

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How many stem-values?

• Start with between 4 and 12 stem-values
• Then, use trial and error using different stem
  multipliers and splits ? use plot that shows
  shape most clearly

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Fourth Example n 53 body weights
Data range from 100 to 260 lbs
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Data range from 100 to 260 lbs

• 100 axis multiplier ? only two stem-values
  (1100 and 2100) ? too few
• 100 axis-multiplier w/ split stem ? 4 stem
  values ? might be OK(?)
• 10 axis-multiplier ? 16 stem values next slide

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Fourth Stemplot Example (n 53)
100166 11009 120034578 1300359 1408 1500257
16555 17000255 18000055567 19245 203 21025
220 23 24 25 260 (10)
Shape Positive skewhigh outlier (260) Central
Location L(M) (53 1) / 2 27 Median 165
(underlined) Spread from 100 to 260
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Quintuple-Split Stem Values
10000111 1t222222233333 1f4455555 1s66677777
7 1.888888888999 20111 2t2 2f 2s6 (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, 120 is 1t2(x100)
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SPSS Stemplot, n 654
Frequency counts
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 . 00000000000000000000000000000000
000000000000000 81.00 10 .
0000000000000000000000000000000000000000
90.00 11 . 00000000000000000000000000000000
0000000000000 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 (gt18) Stem width
1 Each leaf 2 case(s)
3 . 0 means 3.0 years
Because n large, each leaf represents 2
observations
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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    47.210        81   1
2.4    59.611        90   13.8    73.412   
     57    8.7    82.113        43    6.6   
 88.714        25    3.8    92.515        1
9    2.9    95.416        13    2.0    97.4
17         8    1.2    98.618         6    0.
9    99.519         3    0.5   100.0------
-----------------------Total    654  100.0

• Frequency count
• Relative frequency proportion
• Cumulative relative frequency proportion less
  than or equal to current value

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Class Intervals

• When data sparse, group data into class intervals
  
• Classes intervals can be uniform or non-uniform
• Use end-point convention, so data points fall
  into unique intervals include lower boundary,
  exclude upper boundary
• (next slide)

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Class Intervals Freq Table
Data 05 11 21 24 27 28 30 42 50 52
Class Freq Relative Freq. () Cumulative Freq ()
0 9 1 10 10
10 19 1 10 20
20 29 4 40 60
30 39 1 10 70
40 44 1 10 80
50 59 2 20 100
Total 10 100 --
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Histogram
For a quantitative measurement only. Bars touch.
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Bar Chart
For categorical and ordinal measurements and
continuous data in non-uniform class intervals ?
bars do not touch.