1. Chapter 3: Frequency Distributions
4/17/2017
Chapter 3: Frequency Distributions
April 17
Basic Biostatistics

2. In Chapter 3: 3.1 Stemplots 3.2 Frequency Tables
4/17/2017
In Chapter 3:
3.1 Stemplots
3.2 Frequency Tables
3.3 Additional Frequency Charts
Basic Biostatistics

3. You can observe a lot by looking – Yogi Berra
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

4. Stemplot: Illustration
10 ages (data sequenced as an ordered array)
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:
0|5 1|1 2| |0 4|2 5|02 ×10
Here is the plot horizontally
Rotated stemplot

6. Interpreting Distributions
Shape
Central location
Spread

7. Shape “Shape” refers to the distributional pattern
Here’s the silhouette of our data X X X X X X X X X X
Mound-shaped, symmetrical, no outliers
Do not “over-interpret” plots when n is small

8. Shape (cont.) Consider this large data set of IQ scores
An density curve is superimposed on the graph

9. Examples of Symmetrical Shapes

10. Examples of Asymmetrical shapes
Chapter 3
4/17/2017
Examples of Asymmetrical shapes
Basic Biostatistics

11. Modality (no. of peaks)

12. Kurtosis (steepness) Kurtosis is not be easily judged by eye
 fat tails
Mesokurtic (medium)
Platykurtic (flat)
 skinny tails
Leptokurtic (steep)
Kurtosis is not be easily judged by eye

13. 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
^ Grav.Center

14. Central location: Median
Ordered array:
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

15. 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)

16. 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

17. 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| |3| ×10
Too squished to see shape

18. 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| |3|5678 ×10
Negative skew now evident)

19. 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

20. Fourth Example: n = 53 body weights
Data range from 100 to 260 lbs:

21. 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

22. Fourth Stemplot Example (n = 53)
Chapter 3
4/17/2017
Fourth Stemplot Example (n = 53)
10|0166
11|009
12|
13|00359
14|08
15|00257
16|555
17|000255
18|
19|245
20|3
21|025
22|0
23|
24|
25|
26|0
(×10)
Shape: Positive skew high outlier (260)
Central Location: L(M) = (53 + 1) / 2 = 27 Median = 165 (underlined)
Spread: from 100 to 260
The student should construct a stem & leaf plot here using the first two digits as the stem and the last digit as the leaf. The shape of the stem & leaf plot should look similar to the bar graph shown on an upcoming slide.
Basic Biostatistics

23. Quintuple-Split Stem Values
Chapter 3
4/17/2017
Quintuple-Split Stem Values
1*|
1t|
1f|
1s|
1.|
2*|0111
2t|2
2f|
2s|6
(×100)
Codes for stem values:
* for leaves 0 and 1 t for leaves two and three f for leaves four and five s for leaves six and seven . for leaves eight and nine
For example, 120 is: 1t|2 (x100)
The student should construct a stem & leaf plot here using the first two digits as the stem and the last digit as the leaf. The shape of the stem & leaf plot should look similar to the bar graph shown on an upcoming slide.
Basic Biostatistics

24. SPSS Stemplot, n = 654 Frequency counts 3 . 0 means 3.0 years
Frequency Stem & Leaf
Extremes (>=18)
Stem width: 1 Each leaf: case(s)
means 3.0 years
Because n large, each leaf represents 2 observations

25. Frequency Table Frequency ≡ count Relative frequency ≡ proportion
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
Cumulative [relative] frequency ≡ proportion less than or equal to current value

26. 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)

27. Class Intervals Freq Table
Data:
Class
Freq
Relative Freq. (%)
Cumulative Freq (%)
0 – 9 1
10%
10 – 19 10 20
20 – 29 4 40 60
30 – 39 70
40 – 44 80
50 – 59 2
100%
Total --

28. For a quantitative measurement only.
Histogram
For a quantitative measurement only.
Bars touch.

29. Bar Chart
For categorical and ordinal measurements and continuous data in non-uniform class intervals  bars do not touch.