Chapter 3:
Frequency Distributions
March 16
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
• A stemplot is a graphical technique that
organizes data into a histogram-like display
You can observe a lot by looking –
Yogi Berra
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
Stemplot illustration (cont.)
• Draw an axis for the stem-values:
0|
1|
2| 1
3|
4|
5|
×10  axis multiplier (important!)
• Place leaves next to their stem value
• 21 plotted (animation)
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
2
5 1 1 0 2 0
-----------0 1 2 3 4 5
-----------Rotated stemplot
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)
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
Shape: Idealized Density Curve
A large dataset is introduced
An density curve is superimposed to better discuss shape
Symmetrical Shapes
Asymmetrical shapes
Modality (no. of peaks)
Kurtosis (steepness)
 fat tails
Mesokurtic (medium)
Platykurtic (flat)
 skinny tails
Leptokurtic (steep)
Kurtosis is not be easily judged by eye
Location: Mean
“Eye-ball method”  visualize where plot would balance
Arithmetic method = sum values and divide by n
8
7
4
2
5 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
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
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”
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
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
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
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
Fourth Example: Body weights (n = 53)
Data range from 100 to 260 lbs:
192
152
135
110
128
180
260
170
165
150
110
120
185
165
212
119
165
210
186
100
195
170
120
185
175
203
185
123
139
106
180
130
155
220
140
157
150
172
175
133
170
130
101
180
187
148
106
180
127
124
215
125
194
Data range from 100 to 260 lbs:
×100 axis multiplier  only two stemvalues (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
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
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:
*
t
f
s
.
for
for
for
for
for
leaves
leaves
leaves
leaves
leaves
0 and 1
two and three
four and five
six and seven
eight and nine
For example, this is 120:
1t|2
(x100)
SPSS Stemplot
SPSS provides frequency counts w/ its
stemplots:
Frequency
Stem &
2.00
3
9.00
4
28.00
5
37.00
6
54.00
7
85.00
8
94.00
9
81.00
10
90.00
11
57.00
12
43.00
13
25.00
14
19.00
15
13.00
16
8.00
17
9.00 Extremes
Stem width:
Each leaf:
Leaf
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 . 0 means 3.0 years
0
0000
00000000000000
000000000000000000
000000000000000000000000000
000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
0000000000000000000000000000
000000000000000000000
000000000000
000000000
000000
0000
(>=18)
1
2 case(s)
Because of large n, each leaf
represents 2 observations
Frequency Table
AGE
• Frequency = count
• Relative frequency
= proportion or %
• Cumulative
frequency  % less
than or equal to
level
|
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 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
Class Intervals
Uniform class intervals table (width 10) for data:
05 11 21 24 27 28 30 42 50 52
Class
0 – 9.99
10 – 19
20 – 29
30 – 39
40 – 44
50 – 59
Total
Freq
Relative
Freq. (%)
Cumulative
Freq (%)
1
1
4
1
1
2
10
10
10
40
10
10
20
100
10
20
60
70
80
100
--
Histogram
A histogram is a frequency chart for a quantitative
measurement. Notice how the bars touch.
5
4
3
2
1
Age Class
50
-5
9
40
-4
9
30
-3
9
20
-2
9
09
10
_1
9
0
Bar Chart
A bar chart with non-touching bars is reserved for
categorical measurements and non-uniform class
intervals
500
450
400
350
300
250
200
150
100
50
0
Pre-
Elem.
Middle
School-level
High