october 15. in chapter 3: 3.1 stemplot 3.2 frequency tables 3.3 additional frequency charts
TRANSCRIPT
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Apr 19, 2023
Chapter 3: Chapter 3: Frequency DistributionsFrequency Distributions
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In Chapter 3:
3.1 Stemplot
3.2 Frequency Tables
3.3 Additional Frequency Charts
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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
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Stemplot Illustrative Example
• Select an SRS of 10 ages
• List data as an ordered array05 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
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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
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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
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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)
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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
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Shape: Idealized Density Curve A large dataset is introduced
An density curve is superimposed to better discuss shape
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Symmetrical Shapes
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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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Location: Mean“Eye-ball method” visualize where plot would balance
Arithmetic method = sum values and divide by n
8 7 4 25 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
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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
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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”
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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)
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
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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(×1)
• Too squished to see shape
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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
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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
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Fourth Example: Body weights (n = 53)
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 broad
×100 axis-multiplier w/ split stem only 4 stem values might be OK(?)
×10 axis-multiplier see 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)
Looks good!
Shape: Positive skew, high outlier (260)
Location: median underlined (about 165)
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, this is 120: 1t|2(x100)
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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
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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%
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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
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Class Intervals
Class Freq Relative Freq. (%)
Cumulative Freq (%)
0 – 9.99 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 --
Uniform class intervals table (width 10) for data:05 11 21 24 27 28 30 42 50 52
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HistogramA histogram is a frequency chart for a
quantitative measurement. Notice how the bars touch.
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Bar ChartA bar chart with non-touching bars is
reserved for categorical measurements and non-uniform class intervals