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N T E R D I S C I P L I N A R Y
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I V E L Y
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P P L I C A T I O N S
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R O J E C T
Survivalof
EarlyAmericans
Interdisciplinary Lively Application Project
Title: Survival of Early Americans
Authors: Lisa Pike
Bill Fox
Department of Biology and Department of Mathematics, Francis
Marion University, Florence, SC 29501
Editor: Richard D. West
Mathematics Classification: College Algebra and Precalculus
Disciplinary Classification: Demography and Human
Survivorship Curves
Prerequisite Skills:
1. Elementary Functions
2. Exponential Functions
3. Graphs
Physical Concepts Examined: Population Dynamics
Materials:
1. Instructor Notes
2. Problem Statement
3. Sample Solution
Computing Requirements: Graphing Calculator or Computer
Project Intermath 1-16. ©Copyright 2002 by COMAP, Inc. All rights reserved. Permission to make
digital or hard copies of part or all of this work for personal or classroom use is granted without fee
provided that copies are not made or distributed for profit or commercial advantage and that copies bear
this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by
others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to
redistribute to lists requires prior permission from COMAP.
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Project Intermath
Background Information
Demography is the study of population dynamics -- how populations grow and
decline.
The worldwide human population is currently experiencing a
population growth phase, and presently is increasing at an exponential rate.
Today, human population is approximately 6.1 billion people, and it is expected
to double in about 35 years (much faster than 600 or 1000 years ago). We are
unsure of the carrying capacity of the earth; some scientists fear we have already
reached it, meaning we may not have the resources (space, clean water, food etc)
for the additional people entering our global population every day.
Populations grow as more members are added, through births and immigration.
Populations decline as members are deleted, through deaths and emigration.
Stable populations have a balance between growth rates and decay rates (birth &
immigration rates and death & emigration rates). So a stable population has a
zero growth rate. One way to learn about populations of living organisms is to
examine age-specific rates of mortality and reproduction. Because accumulated
age-specific information is often presented in life tables, life table analysis is a
common component of courses in ecology and population biology. In this
project, we perfom life table analysis.
Results can be presented in the form of a survivorship curve, which traces the
decline in number, over time, of a group of individuals born at the same time (a
cohort). It can be thought of as the probability of an individual surviving to
various ages -- the Life Expectancy vs. Maximum Life Span. For example, the
American robin Turdus migratorius can live to be 7 years old but the probability
of a newly hatched robin doing so is less than 1%. Many robins live only a year
or two. Their life expectancy is 1-2 years, but their maximum life span is 7 years.
In order to determine age specific mortality and survivorship curves for a
specific population, ecologists follow a cohort, a group of individuals born
within the same time interval. For example, the ecologist might follow a cohort
for a year, 5 years, or a month, depending on the species they are looking at. The
cohort is followed until all members of the cohort are dead. The gender and age
at death are recorded for each individual. The ecologists find that each species
has a characteristic life span, with few reaching the maximum age.
The survival rate or life expectancy of human populations has increased
significantly in the past 100 - 300 years due to improved nutrition, preventative
Survival of Early Americans 3
medicine, life-style changes, improved sewage control and hygiene and new
technologies. In the early days of the Roman Empire, life expectancy was only
22 years. In America in 1900, the life expectancy was about 48 years; in 1998 it
was 76 years (mostly due to a decrease in infant mortality). In 1998, a man in
Italy lived to a ripe old age of 126 (the maximum human life span). Survival
rates are up and mortality rates, especially infant mortality rates, are down: this
increases the population growth.
There are three typical population survivorship curves: Type I, Type II and
Type III. Humans have a Type I growth rate, with low infant mortality and a
high probability of living until you are old (at which time the probability of
death increases).
100
Number
50
of
Individuals
0
Time
Type I
(Humans, large mammals)
Time
Type II
(songbirds)
Time
Type III
(marine invertebrates)
Situation 1. Survival of Early Americans
In this project we will make survivorship curves for humans living in the 1800’s
and humans living in the 1900’s. We will construct a life table from these data
using the attached data sheet. Then we will determine values for the number of
individuals who would have been alive in each interval of ten years. Also, we
will determine the number of individuals who died during each interval. This
curve is the opposite of a survivorship curve and is called a mortality curve. A
survivorship curve is prepared by plotting the logarithm of the number of
survivors against age.
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Project Intermath
Data for requirements 1-5
Data Table (born 1810-1820): MALES
Age at Death
(yrs) as an
interval
0
0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110 +
Total People
# who died
within
interval
Survivorship (#
left alive within
interval)
% Survivorship
% Mortality
Log 10 of %
Survivorship
% Mortality
Log 10 of %
Survivorship
9
3
7
12
17
26
65
112
67
11
4
0
333
Data Table (born 1810-1820): FEMALES
Age at Death
(yrs) as an
interval
0
0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110 +
Total People
# who died
within
interval
4
4
8
10
13
28
41
90
65
11
1
0
275
Survivorship (#
left alive within
interval)
% Survivorship
Survival of Early Americans 5
Data Table (born 1910-1920): MALES
Age at Death
(yrs) as an
interval
0
0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110 +
Total People
# who died
within
interval
Survivorship (#
left alive within
interval)
% Survivorship
% Mortality
Log 10 of %
Survivorship
% Mortality
Log 10 of %
Survivorship
0
6
2
5
13
19
40
66
144
28
3
0
326
Data Table (born 1910-1920): FEMALES
Age at Death
(yrs) as an
interval
0
0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110 +
Total People
# who died
within
interval
1
3
1
4
11
17
36
78
98
24
5
1
279
Survivorship (#
left alive within
interval)
% Survivorship
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Requirements:
Part 1.
1. Complete the tables for males & females 1810-1820 and 1910-1920.
2. Compute from each table the
a. AVG Life Expectancy for men
b. AVG Life Expectancy for women
c. MAX Life Span for men
d. MAX Life Span for women
Part 2.
3. Graph the log (% survivorship) over time.
4. Compare your graph to the survivorship curves. What kind of
survivorship curve did you find for your data (Type I, II, or III)? Why?
5. What does this reflect about males and females in the early 1800ʹs and
the early 1900ʹs?
Part 3.
6. What was the average life expectancy for people living in the 1800’s?
for people living in the 1900’s?
7. What was the maximum life span for people living in the 1800’s? for
people living in the 1900’s?
Part 4.
8. Did your data show a difference in age at death between males and
females? For which cohort? Why do you think this happened?
Part 5.
9. If a third of the world population is now below the age 15, what effect
will this age distribution have on the growth rate of the human
population?
10. What sort of humane recommendations would you make to
encourage this age group to limit the number of children they plan to
have?
Survival of Early Americans 7
Situation 2. Age Structure and Survivorship
Populations, whether animal or plant, vary in their proportions of young and old
individuals. Time units are used to describe ages but could be in hours, days,
weeks, months or years. Sometimes they can be grouped into classes: nestling,
juvenile, subadult, and adult. The proportions collected into the categories are
referred to as age structure or age distribution data. We desire some statistical
measures for these ages. In this example, there are 13 categories for age.
Requirements:
Part 1.
1. Using our data from the tables produced in situation 1, compute the
following descriptive statistics for ages of males and females in the early
1800ʹs and 1900ʹs. (Use midpoints of the 13 categories for the value of the
counts in each class).
a. mean
b. median
c. mode
d. standard deviation
e. variance
f. range
g. coefficient of skewness, SK
(SK = 3 (mean - median) / Standard Deviation)
2. Interpret the meaning of each of these descriptive statistics.
Part 2.
3. Construct a histogram for each data set. What information is being
displayed?
4. Construct boxplots for males and females. Explain the interpretation of
the results of the boxplots.
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Project Intermath
Sample Solution: Survival of Early Americans
Situation 1. Survival of Early Americans
Part 1.
1. Complete the tables for males and females 1810-1820 and 1910-1920.
1810-1820
Age
Intervals
0
0-9
10 to 19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110+
TOTAL
Males
died
0
9
3
7
12
17
26
65
112
67
11
4
0
333
Cumulative
Survive
0
9
12
19
31
48
74
139
251
318
329
333
333
333
324
321
314
302
285
259
194
82
15
4
0
0
1810-1820
Age
Intervals
0
0-9
10 to 19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110+
TOTAL
Females
died
0
4
4
8
10
13
28
41
90
65
11
1
0
275
Cumulative
Survive
0
4
8
16
26
39
67
108
198
263
274
275
275
275
271
267
259
249
236
208
167
77
12
1
0
0
%
Survive
100
97.2973
96.3964
94.29429
90.69069
85.58559
77.77778
58.25826
24.62462
4.504505
1.201201
0
0
%
Mortality
0
2.702703
3.603604
5.705706
9.309309
14.41441
22.22222
41.74174
75.37538
95.4955
98.7988
100
100
Log10
Survival
2
1.988101
1.984061
1.974485
1.957563
1.932401
1.890856
1.765357
1.39137
0.653647
0.079616
0
0
%
Survive
100
98.54545
97.09091
94.18182
90.54545
85.81818
75.63636
60.72727
28
4.363636
0.363636
0
0
%
Mortality
0
1.454545
2.909091
5.818182
9.454545
14.18182
24.36364
39.27273
72
95.63636
99.63636
100
100
Log10
Survival
2
1.993637
1.987179
1.973967
1.956867
1.933579
1.878731
1.783384
1.447158
0.639849
-0.43933
0
0
Survival of Early Americans 9
1910-1920
Age
Intervals
0
0-9
10 to 19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110+
TOTAL
Males
died
0
0
6
2
5
13
19
40
66
144
28
3
0
326
Cumulative
Survive
0
0
6
8
13
26
45
85
151
295
323
326
326
326
326
320
318
313
300
281
241
175
31
3
0
0
1910-1920
Age
Intervals
0
0-9
10 to 19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
100-109
110+
TOTAL
Females
died
0
1
3
1
4
11
17
36
78
98
24
5
1
279
Cumulative
Survive
0
1
4
5
9
20
37
73
151
249
273
278
279
279
278
275
274
270
259
242
206
128
30
6
1
0
%
Survive
100
100
98.15951
97.54601
96.01227
92.02454
86.19632
73.92638
53.68098
9.509202
0.920245
0
0
%
Mortality
0
0
1.840491
2.453988
3.98773
7.97546
13.80368
26.07362
46.31902
90.4908
99.07975
100
100
Log10
Survival
2
2
1.991932
1.98921
1.982327
1.963904
1.935489
1.868799
1.72982
0.978144
-0.0361
0
0
%
Survive
100
99.64158
98.56631
98.20789
96.77419
92.83154
86.73835
73.83513
45.87814
10.75269
2.150538
0.358423
0
%
Mortality
0
0.358423
1.433692
1.792115
3.225806
7.168459
13.26165
26.16487
54.12186
89.24731
97.84946
99.64158
100
Log10
Survival
2
1.998441
1.993728
1.992146
1.98576
1.967696
1.938211
1.868263
1.661606
1.031517
0.332547
0
0
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Project Intermath
2. Compute from each table the
(a) AVG Life Expectancy for men : 22665/333= 68.06 years in the 1800ʹs
and 24710/326 = 75.79 years in the 1900ʹs
(b) AVG Life Expectancy for women : 18845/275 = 68.52 years in the 1800ʹs
and 21085/279 = 75.57 years in the 1900ʹs
(c) MAX Life Span for men 100-109 in both
(d) MAX Life Span for women 100-109 in the 1800ʹs and 110+ in the 1900ʹs
Part 2.
3. Graph the log (% survivorship) over time.
Log % Male
Survivors
Log 10 Survivors-Male
3
2
Series1
1
0
1
3
5
7
9
11
13
Age Intervals
3
2
1
Series1
13
11
9
7
5
-1
3
0
1
Log % female
Survivors
Log 10 Females
Time
3
2
Series1
1
Time
11
9
7
5
3
0
1
Log % females
survivors
1910-1920 Female Survivor Curve
Survival of Early Americans 11
1910-1920 Male Survivorship
2
1
Series1
11
9
7
5
-1
3
0
1
Log Male
Survivors
3
Time
3
2
Series1
1
13
11
9
7
5
3
0
1
Log % Male
Survivors
Log 10 Survivors-Male
Age Intervals
4. Compare your graph to the survivorship curves. What kind of survivorship
curve did you find for your data? (Type I, II, or III) Why ? It is type I. This
makes sense because the data is human.
5. What does this reflect about males and females in the early 1800ʹs and the
early 1900ʹs? By looking at the data and the percents in each intervals, it is clear
that males and females both increased their life span from the early 1800ʹs to the
early 1900ʹs.
Part 3.
6. What was the average life expectancy for people living in the 1800ʹs? for
people living in the 1900ʹs?
1800ʹs Æ 68.27 years
1900ʹsÆ 75.69 years
People live about 7.4 years longer in the 1900ʹs.
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Project Intermath
7. The maximum life span for people living in the 1800ʹs? for people living in
the 1900ʹs?
MAX Life Span for men 100-109 in both
MAX Life Span for women 100-109 in the 1800ʹs and 110+ in the 1900
Part 4.
8. Did your data show a difference in age at death between males and females?
For which cohort? Why do you think this happened?
Women live longer than men. The 1900 women live longer than both men and
the 1800 women. Many factors including technology, medicine, gender, and
adaptability.
Part 5.
9. If a third of the world population is now below the age 15, what effect will
this age distribution have on the growth rate of the human population?
Since the average life expectancy is now over 75 years then over time this one
third of the world population will continue to survive for many years
(approximately 60 more years). The population growth could cause problems in
terms of space, food, water, jobs, etc.
10. What sort of humane recommendations would you make to encourage this
age group to limit the number of children they plan to have?
Population growth will exceed capacity to produce food and other essentials. If
this occurs then the growth rate will slow and the death rate will climb. We will
experience a shorter life expectancy.
Survival of Early Americans 13
Situation 2. Age Structure and Survivorship
Part 1.
1. Compute the following descriptive statistics for each data set.
M_1810
F_1810
M_1910
F_1910
N
MEAN
MEDIAN
TRMEAN
STDEV
SEMEAN
333
275
326
279
68.06
68.53
75.798
79.09
75.00
75.00
85.000
75.00
69.62
70.02
77.245
80.58
19.21
18.84
16.266
18.50
1.05
1.14
0.901
1.11
M_1810
F_1810
M_1910
F_1910
Statistic
Mean
Median
Mode
Std Deviation
Variance
Range
SK
MIN
5.00
5.00
15.000
5.00
Male, 1800
68.06
75
75
19.21
369.02
0-105
-1.08
MAX
105.00
105.00
105.000
115.00
Q1
65.00
65.00
65.000
65.00
Female, 1800
68.53
75
75
18.84
354.95
0-105
-1.03
Q3
75.00
85.00
85.000
95.00
Male, 1900
75.798
85
85
16.26
264.39
0-115
-1.69
Female, 1900
79.09
75
85
18.5
342.25
0-105
-.958
2. Interpret the meaning of these descriptive statistics.
All have SKʹs that are negative. This implies that the distribution is skewed to the
left. The locations of mean, median, and mode support this result.
Part 2.
3. Construct a stem and leaf and histogram for each data set. What information is
being displayed?
Histograms plot the frequencies in each age class. They provide information
about the shape (symmetric or skewed) of the distribution. Boxplot plot the 5
number summary (median, quartiles, and range) to view symmetry, skewness,
and outliers. Data sets are more easily compared with boxplots.
Project Intermath
Frequency
100
50
0
0
50
100
M_1810
90
80
Frequency
70
60
50
40
30
20
10
0
0
50
100
F_1810
150
Frequency
14
100
50
0
0
50
M_1910
100
Survival of Early Americans 15
Frequency
100
50
0
0
50
100
F_1910
All plots are negatively skewed.
4. Construct boxplots for males and females. Explain the interpretation of the
results of the boxplots.
MTB > BoxPlot c1
----------I
+----*
*
-----+---------+---------+---------+---------+---------+-----O
M_1810
O
0
MTB > GPro.
MTB > GStd.
MTB > BoxPlot c2
20
*
0
MTB > GPro.
MTB > GStd.
MTB > BoxPlot c3
40
*
60
80
100
*
20
40
60
80
100
-------------------------I
+---------------------------+---------+---------+---------+---------+-------*
M_1910
*
-------------------------I
+
I-------------------+---------+---------+---------+---------+---------+-----*
F_1810
O
MTB > GPro.
MTB > GStd.
MTB > BoxPlot c4
20
*
40
60
80
100
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Project Intermath
-----------------------------------I
+
I--------------------------------+---------+---------+---------+---------+-------*
F_1910
*
20
40
60
80
100
These are close to same shape. The five number summaries for these four data
sets are extremely similar.