1.11 Presidential Ages
(a) To construct a histogram, first you must make a frequency table (on paper or in
your calculator). A tip for doing this is to enter the list in your calculator first of ages,
then “Sort” the list. You were told to use a class size of 5. Be sure that you include the
left-hand endpoint but not the right-hand end-point in your count for each class (for
example, 45 would fall in the class of 45-50, not 40-45).
Your histogram should include a title, clearly defined labels, a “shark bite” if needed to
show the gap between 0 and your first class value, and the frequency count on top of
each bar (to facilitate reading).
Ages of Presidents at Inauguration
13
12
7
6
3
2
0
age
Shape, Center & Spread
(b) You were asked to describe the shape, center and spread of the data. Even when
not asked, you should always look at your graph to analyze the overall pattern and any
striking deviations from that pattern. Describe the pattern in terms of Shape, Center
& Spread (see pp 52-53 in your text).
Shape: The distribution is approximately symmetric with a single peak, therefore it is
unimodal.
Center: The data is distributed around the center at approximately 55 years. (Looking
at the graph, we can see that the center is approximately 55. Note that since we have
the actual data, we can calculate the mean and median values to give us the center,
which also calculates to be approximately 55).
Spread: The youngest president was 42 and the oldest president was 69 at
inauguration. Therefore, the range is 69-42=27 years.
1.11 Continued…
(c) The youngest president was Teddy Roosevelt
who was 42 years old. The oldest president was
Ronald Reagan who was 69 years old.
(d) Bill Clinton was 46 years old at his inauguration.
Although he was one of the youngest presidents, he
was not unusually young. He is certainly not an
outlier. He is one of four presidents who was 46
years old or younger.
Ogive Chart (pronounced O-Jive)
Histograms are helpful to display the
distribution of values for quantitative variables.
However, they do not tell us much about an
individual observation.
To find out how an individual observation
relates to the rest of the data, we can construct
an ogive to represent the cumulative frequency.
Cumulative Frequency
1.
2.
Calculate the Cumulative Frequency by adding the frequency of each class to the
cumulative frequency of the class that proceeds it. What do you know about the
value of the cumulative frequency of the 1st class? The last class?
The relative cumulative frequency is the percentage of values that fall at or
below that class value. To calculate this value, we must divide the cumulative
frequency by the total number of individuals (in this case 43). What do you
notice about the value of the relative cumulative frequency of the 1st class? The
last class? Will this always be the case?
Relative Cumulative Frequency Table
Class
40-45
45-50
50-55
55-60
60-65
65-70
Total
Frequency
2
7
13
12
7
3
44
Relative
Frequency
4.5%
15.9%
29.5%
27.3%
15.9%
6.8%
* Note: Class includes left-end data
Cumulative
Frequency
2
9
22
34
41
44
Relative
Cumulative
Frequency
4.5%
20.5%
50.0%
77.3%
93.2%
100.0%
Ogive
3. Plot each cumulative frequency for each class using the LEFT endpoint of that class.
For example, for the 1st class of ages 40-45, we plot the cumulative frequency of 4.7%
at 45. This shows us that 4.7% of all US Presidents were inaugurated before they were
45 years old.
Relative Cumulative
Frequency (%)
Relative Cumulative Frequency Plot for Presidents at Inauguration
0
age
Using the Ogive (going backwards)
•
Now if we want to determine where the middle 50% of individuals fall, we can use
the ogive to “work backwards.” Find the 50% first, then find the x value that
corresponds to this point on the line.
What does this tell us?
That 50% of the presidents were inaugurated before the age of 55.
Relative Cumulative Frequency Plot for Presidents at Inauguration
50%
Relative Cumulative
Frequency (%)
•
0
age
Calculating a Percentile Value
• Determine the 70th percentile of Presidential
Inauguration Ages.
The 70th percentile falls at approximately the age of 58 years old.
Relative Cumulative
Frequency (%)
Relative Cumulative Frequency Plot for Presidents at Inauguration
0
age