Concept explainers
Life Expectancy (years) | Infant Mortality Rate* | Fertility Rate# | ||||
Nation | 2010 | 2020 | 2010 | 2020 | 2010 | 2020 |
Canada | 81 | 82 | 5.0 | 4.4 | 1.6 | 1.6 |
China | 75 | 76 | 16.5 | 12.6 | 1.5 | 1.5 |
Egypt | 72 | 75 | 26.2 | 17.9 | 3.0 | 2.7 |
Germany | 79 | 81 | 4.0 | 3.6 | 1.4 | 1.5 |
Japan | 82 | 83 | 2.8 | 2.7 | 1.2 | 1.3 |
Mali | 52 | 57 | 113.7 | 91.9 | 6.5 | 5.5 |
Mexico | 76 | 78 | 17.8 | 13.2 | 2.3 | 2.1 |
Peru | 71 | 74 | 27.7 | 20.2 | 2.3 | 2.0 |
Ukraine | 69 | 70 | 8.7 | 7.3 | 1.3 | 1.4 |
U.S | 78 | 80 | 6.1 | 5.4 | 2.1 | 2.1 |
Zambia | 52 | 54 | 68.4 | 50.6 | 6.0 | 5.3 |
Source: U.S. Bureau of the Census, 2012. Statistical Abstract of the United States: 2012. p. 842.
Notes:
*Number of deaths of children under one year of age per 1000 live births.
#Average number of children per female.
To find:
The range, mean and standard deviation of male and female of the given variable
Answer to Problem 4.8P
Solution:
The range, mean and standard deviation of the given variable is given in the table below,
Life Expectancy | Infant Mortality Rate | Fertility Rate | ||||
2010 | 2020 | 2010 | 2020 | 2010 | 2020 | |
Range | 30 | 29 | 110.9 | 89.2 | 5.3 | 4.2 |
Mean | 71.54 | 73.63 | 26.99 | 20.89 | 2.65 | 2.45 |
Standard Deviation | 9.99 | 9.31 | 32.74 | 25.91 | 1.77 | 1.44 |
Explanation of Solution
Given:
The following table shows the data of 3 different variables of 11 nations.
Life Expectancy (years) | Infant Mortality Rate | Fertility Rate | ||||
Nation | 2010 | 2020 | 2010 | 2020 | 2010 | 2020 |
Canada | 81 | 82 | 5.0 | 4.4 | 1.6 | 1.6 |
China | 75 | 76 | 16.5 | 12.6 | 1.5 | 1.5 |
Egypt | 72 | 75 | 26.2 | 17.9 | 3.0 | 2.7 |
Germany | 79 | 81 | 4.0 | 3.6 | 1.4 | 1.5 |
Japan | 82 | 83 | 2.8 | 2.7 | 1.2 | 1.3 |
Mali | 52 | 57 | 113.7 | 91.9 | 6.5 | 5.5 |
Mexico | 76 | 78 | 17.8 | 13.2 | 2.3 | 2.1 |
Peru | 71 | 74 | 27.7 | 20.2 | 2.3 | 2.0 |
Ukraine | 69 | 70 | 8.7 | 7.3 | 1.3 | 1.4 |
U.S | 78 | 80 | 6.1 | 5.4 | 2.1 | 2.1 |
Zambia | 52 | 54 | 68.4 | 50.6 | 6.0 | 5.3 |
Formula used:
Let the data values be
The formula to calculate range is given by,
The formula to calculate mean is given by,
The formula to calculate standard deviation is given by,
Where,
Calculation:
Consider the data of life expectancy in the year 2010.
Arrange the data in the increasing order.
The data in increasing order is given by,
S. No | Life Expectancy |
1 | 52 |
2 | 52 |
3 | 69 |
4 | 71 |
5 | 72 |
6 | 75 |
7 | 76 |
8 | 78 |
9 | 79 |
10 | 81 |
11 | 82 |
The highest value is 82 and the lowest value is 52.
The range is given by,
Substitute 82 for highest value and 52 for lowest value in the above mentioned formula,
The size of the population is 11.
The mean is given by,
Substitute 11 for
Consider the following table of sum of squares,
81 | 9.46 | 89.4916 |
75 | 3.46 | 11.9716 |
72 | 0.46 | 0.2116 |
79 | 7.46 | 55.6516 |
82 | 10.46 | 109.4116 |
52 | 381.8116 | |
76 | 4.46 | 19.8916 |
71 | 0.2916 | |
69 | 6.4516 | |
78 | 6.46 | 41.7316 |
52 | 381.8116 | |
From equation
Square the both sides of the equation.
Proceed in the same manner to calculate
The standard deviation is given by,
From equation
Thus, standard deviation of life expectancy in the year 2010 is 9.99.
Consider the data of life expectancy in the year 2020.
Arrange the data in the increasing order.
The data in increasing order is given by,
S. No | Life Expectancy |
1 | 54 |
2 | 57 |
3 | 70 |
4 | 74 |
5 | 75 |
6 | 76 |
7 | 78 |
8 | 80 |
9 | 81 |
10 | 82 |
11 | 83 |
The highest value is 83 and the lowest value is 54.
The range is given by,
Substitute 83 for highest value and 54 for lowest value in the above mentioned formula,
The size of the population is 11.
The mean is given by,
Substitute 11 for
Consider the following table of sum of squares,
82 | 8.37 | 70.0569 |
76 | 2.37 | 5.6169 |
75 | 1.37 | 1.8769 |
81 | 7.37 | 54.3169 |
83 | 9.37 | 87.7969 |
57 | 276.5569 | |
78 | 4.37 | 19.0969 |
74 | 0.37 | 0.1369 |
70 | 13.1769 | |
80 | 6.37 | 40.5769 |
54 | 385.3369 | |
From equation
Square the both sides of the equation.
Proceed in the same manner to calculate
The standard deviation is given by,
From equation
Thus, standard deviation of life expectancy in the year 2020 is 9.31.
Consider the data of infant mortality rate in the year 2010.
Arrange the data in the increasing order.
The data in increasing order is given by,
S. No | Infant Mortality Rate |
1 | 2.8 |
2 | 4 |
3 | 5 |
4 | 6.1 |
5 | 8.7 |
6 | 16.5 |
7 | 17.8 |
8 | 26.2 |
9 | 27.7 |
10 | 68.4 |
11 | 113.7 |
The highest value is 113.7 and the lowest value is 2.8.
The range is given by,
Substitute 113.7 for highest value and 2.8 for lowest value in the above mentioned formula,
The size of the population is 11.
The mean is given by,
Substitute 11 for
Consider the following table of sum of squares,
5 | 483.5601 | |
16.5 | 110.0401 | |
26.2 | 0.6241 | |
4 | 528.5401 | |
2.8 | 585.1561 | |
113.7 | 86.71 | 7518.624 |
17.8 | 84.4561 | |
27.7 | 0.71 | 0.5041 |
8.7 | 334.5241 | |
6.1 | 436.3921 | |
68.4 | 41.41 | 1714.788 |
WFrom equation
Square the both sides of the equation.
Proceed in the same manner to calculate
The standard deviation is given by,
From equation
Thus, standard deviation of infant mortality rate in the year 2010 is 32.74.
Consider the data of infant mortality rate in the year 2020.
Arrange the data in the increasing order.
The data in increasing order is given by,
S. No | Infant Mortality Rate |
1 | 2.7 |
2 | 3.6 |
3 | 4.4 |
4 | 5.4 |
5 | 7.3 |
6 | 12.6 |
7 | 13.2 |
8 | 17.9 |
9 | 20.2 |
10 | 50.6 |
11 | 91.9 |
The highest value is 91.9 and the lowest value is 2.7.
The range is given by,
Substitute 91.9 for highest value and 2.7 for lowest value in the above mentioned formula,
The size of the population is 11.
The mean is given by,
Substitute 11 for
Consider the following table of sum of squares,
4.4 | 271.9201 | |
12.6 | 68.7241 | |
17.9 | 8.9401 | |
3.6 | 298.9441 | |
2.7 | 330.8761 | |
91.9 | 71.01 | 5042.42 |
13.2 | 59.1361 | |
20.2 | 0.4761 | |
7.3 | 184.6881 | |
5.4 | 239.9401 | |
50.6 | 29.71 | 882.6841 |
From equation
Square the both sides of the equation.
Proceed in the same manner to calculate
The standard deviation is given by,
From equation
Thus, standard deviation of infant mortality in the year 2020 is 25.91.
Consider the data of fertility rate in the year 2010.
Arrange the data in the increasing order.
The data in increasing order is given by,
S. No | Fertility Rate |
1 | 1.2 |
2 | 1.3 |
3 | 1.4 |
4 | 1.5 |
5 | 1.6 |
6 | 2.1 |
7 | 2.3 |
8 | 2.3 |
9 | 3 |
10 | 6 |
11 | 6.5 |
The highest value is 6.5 and the lowest value is 1.2.
The range is given by,
Substitute 6.5 for highest value and 1.2 for lowest value in the above mentioned formula,
The size of the population is 11.
The mean is given by,
Substitute 11 for
Consider the following table of sum of squares,
1.6 | 1.1025 | |
1.5 | 1.3225 | |
3 | 0.35 | 0.1225 |
1.4 | 1.5625 | |
1.2 | 2.1025 | |
6.5 | 3.85 | 14.8225 |
2.3 | 0.1225 | |
2.3 | 0.1225 | |
1.3 | 1.8225 | |
2.1 | 0.3025 | |
6 | 3.35 | 11.2225 |
From equation
Square the both sides of the equation.
Proceed in the same manner to calculate
The standard deviation is given by,
From equation
Thus, standard deviation of fertility rate in the year 2010 is 1.77.
Consider the data of fertility rate in the year 2020.
Arrange the data in the increasing order.
The data in increasing order is given by,
S. No | Fertility Rate |
1 | 1.3 |
2 | 1.4 |
3 | 1.5 |
4 | 1.5 |
5 | 1.6 |
6 | 2 |
7 | 2.1 |
8 | 2.1 |
9 | 2.7 |
10 | 5.3 |
11 | 5.5 |
The highest value is 5.5 and the lowest value is 1.3.
The range is given by,
Substitute 5.5 for highest value and 1.3 for lowest value in the above mentioned formula,
The size of the population is 11.
The mean is given by,
Substitute 11 for
Consider the following table of sum of squares,
1.6 | 0.7225 | |
1.5 | 0.9025 | |
2.7 | 0.25 | 0.0625 |
1.5 | 0.9025 | |
1.3 | 1.3225 | |
5.5 | 3.05 | 9.3025 |
2.1 | 0.1225 | |
2 | 0.2025 | |
1.4 | 1.1025 | |
2.1 | 0.1225 | |
5.3 | 2.85 | 8.1225 |
From equation
Square the both sides of the equation.
Proceed in the same manner to calculate
The standard deviation is given by,
From equation
Thus, standard deviation of fertility rate in the year 2020 is 1.44.
The mean life expectancy in the year 2010 is 71.54 and in the year 2020 it is 73.63, on an average the life expectancy of the nations has increased over the ten year period. The standard deviation of life expectancy decreased in the ten year duration, there is less variability in 2020 as compared to year 2010. The mean infant mortality rate of year 2010 is more than the year 2020, the number of infant deaths has decreased during the time period. The mean fertility rate of year 2010 is 2.65 and in year 2020 it is 2.45, the number of children born has decreased between the years. The nations are becoming more diverse.
Conclusion:
Therefore, the range, mean and standard deviation of the given variable is given in the table below,
Life Expectancy | Infant Mortality Rate | Fertility Rate | ||||
2010 | 2020 | 2010 | 2020 | 2010 | 2020 | |
Range | 30 | 29 | 110.9 | 89.2 | 5.3 | 4.2 |
Mean | 71.54 | 73.63 | 26.99 | 20.89 | 2.65 | 2.45 |
Standard Deviation | 9.99 | 9.31 | 32.74 | 25.91 | 1.77 | 1.44 |
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