1Apr 18, 2023

Chapter 3: Chapter 3: Frequency DistributionsFrequency Distributions

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In Chapter 3:

3.1 Stemplots

3.2 Frequency Tables

3.3 Additional Frequency Charts

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Stemplots

• 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

You can observe a lot by looking – Yogi Berra

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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)

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Stemplot illustration continued …

• Plot all data points in rank order:

0|5 1|1 2|1478 3|0 4|2 5|02 ×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• Here’s 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)

Mesokurtic (medium) Platykurtic (flat)

Leptokurtic (steep)

skinny tails

fat tails

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 nsum = 290n = 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, median’s depth = (10+1) ÷ 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)

|1|4|2|03|3|4779|4|4(×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:|1|4789|2|223466789|3|000123445678×10

• 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|1|4|1|789|2|2234|2|66789|3|00012344|3|5678×10

• 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 (1×100 and 2×100) 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)

10|016611|00912|003457813|0035914|0815|0025716|55517|00025518|00005556719|24520|321|02522|023|24|25|26|0(×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

1*|00001111t|2222222333331f|44555551s|6667777771.|8888888889992*|01112t|22f|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, 120 is: 1t|2(x100)

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SPSS Stemplot, n = 654

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)

Because n large, each leaf represents 2 observations

3 . 0 means 3.0 years

Frequency counts

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Frequency Table

• Frequency ≡ count

• Relative frequency ≡ proportion

• Cumulative [relative] frequency ≡ proportion less than or equal to current value

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%

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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

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% --

Data: 05 11 21 24 27 28 30 42 50 52

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HistogramFor a quantitative measurement only.

Bars touch.

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Bar ChartFor categorical and ordinal measurements and continuous data in non-uniform class

intervals bars do not touch.