Q.31 - Elementary Statistics

Question

Life Expectancy. From The World Bank, in the document Life Expectancy af Birth, we obtained data on the expectation of life (in years) at birth for people in various countries. Those data are presented on the Weiss Stats site.

a. Obtain the mean, median, and mode(s) of the data. Determine which of these measures of center is best, and explain your answer:

b. Determine the range and sample standard deviation of the data.

c. Find the five-number summary and interquartile range of the data.

d. Identify potential outliers. if any:

e. Obtain and interpret a boxplot.

Step-by-Step Solution

Answer

(a) Mean & median$69.9684\&73.3$

     mode$=54.1,55.4,68.9,73.1,73.4,73.6,73.9,75.8,76.1,76.7,76.9,79.3,80.5,80.7, 80.9,81,81.8$

(b) Range$=35.6$, sample standard deviation$=9.8391$

(c) $47.8,64.4,73.3,76.7,83.4 and (I Q R)=12.3$

(d) There are no outliers.

(e) 

Part (a) Step 1: Given Information

Given in the question , from The World Bank, in the document Life Expectancy af Birth, we obtained data on the expectation of life (in years) at birth for people in various countries. Those data are presented on the Weiss Stats site are shown below

$$\begin{gathered} 47.8,48,48.1,48.3,48.4,48.7,48.7,49,49.5 \\ ,50.2,50.3,51.1,51.1,51.2,51.2, 51.4,51.6,51.9 52.6 \\ ,52.3,54.1,54.1,54.1,54.7,55.4,55.4,55.4,56,56.7,57,57.1 \\ ,57.4,57.9, 58.2,58.5,58.5 ,59.2,59.3,61,61.4,61.4,62.1,62.3 \\ ,62.5,62.7,62.8,63,64.2,64.6,65,65.2, 65.4,65.5,65.5, 66.6, \\ 66.7,67.3,67.4,67.5,67.9,68.3,68.5,68.7,68.7,68.8,68.9,68.9 \\ ,68.9,69, 69,69.2,69.3, 69.3,69.6,69.9,70,70.1,70.6,70.7,70.7 \\ ,70.8,71.1,71.1,71.9,72.1,72.3,72.3, 72.5,72.5,72.6, 72.8,73,73.1 \\ ,73.1,73.1,73.2,73.3,73.3,73.4,73.4,73.4,73.5,73.5,73.6,73.6 \\ , 73.6,73.9,73.9, 73.9,74,74,74.1,74.1,74.2,74.3,74.3,74.5,74.5,74.6,74.6,74.7,74.8,74.8, 74.9,74.9,75,75.1, 75.1,75.2,75.3,75.5,75.6,75.6,75.8,75.8,75.8,76,76.1,76.1,76.1,76.2,76.4, 76.6,76.7 \\ ,76.7,76.7, 82.3,82.4,82.6,82.7,83.3,83.4 \end{gathered}$$We have to  obtain the mean, median, and mode(s) of the data. Determine which of these measures of center is best,

Part (a) Step 2: Explanation

The mean can be calculated as follows

$$\begin{gathered} \overline{x}=\frac{47.8+48+48.1+\dots ..+82.6+82.7+83.3+83.4}{193} \\ =\frac{13503.9}{193} \\ \approx 69.9684 \end{gathered}$$

The median can be calculated as follows 

$\left(M\right)Q_2=73.3$

The mode can be calculated as follows 

$=54.1,55.4,68.9,73.1,73.4,73.6,73.9,75.8,76.1,76.7,76.9,79.3,80.5,80.7, 80.9,81,81.8$

Part (b) Step 1: Given Information

Given in the question , from The World Bank, in the document Life Expectancy af Birth, we obtained data on the expectation of life (in years) at birth for people in various countries. Those data are presented on the Weiss Stats site we have to determine the range and sample standard deviation of the data. 

Part (b) Step 2: Explanation

The formula of range is as follows:

$$\begin{gathered} Range=max-min \\ =83.4-47.8 \\ =35.6 \end{gathered}$$

The square root  of variance is called standard deviation.so,

$$\begin{gathered} s=\sqrt{\frac{(47.8-69.9684{)}^2+\dots \dots \dots +(83.4-69.9684{)}^2}{50-1}} \\ \approx 9.8391 \end{gathered}$$

Part (c) Step 1: Given Information

Given in the question , from The World Bank, in the document Life Expectancy af Birth, we obtained data on the expectation of life (in years) at birth for people in various countries. Those data are presented on the Weiss Stats. site we have to find the five-number summary and interquartile range of the data. 

Part (c) Step 2: Explanation

The median is the sorted data set's middle value. Because the number of data values is even, the median is the average of the sorted data set's two middle values:

$M=Q_2=73.3$

The median of the data values below the median (or at $25\%$ of the data) is the first quartile:

$Q_1=64.4$

The median of the data values above the median (or at $75\%$ of the data) is the third quartile:

$Q_3=76.7$

The minimum, first quartile, median, third quartile, and maximum are included in the five-number summary.

The five-number summary is as follows:

Minimum:$47.8$

$$\begin{gathered} Q_1=64.4 \\ M=73.3 \\ {Q}_3=76.7 \end{gathered}$$

Maximum$83.4$:

$IQR=Q_3-Q_1=76.7-64.4=12.3$

Part (d) Step 1: Given Information

Given in the question , from The World Bank, in the document Life Expectancy af Birth, we obtained data on the expectation of life (in years) at birth for people in various countries. Those data are presented on the Weiss Stats site. we have to identify potential outliers.  

Part (d) Step 2: Explanation

The observations that are more than $1.5$ times the IQR above $Q_3$ or below $Q_1$ are considered outliers.

$Q_3+1.5IQR=76.7+1.5(12.3)=95.15$

$Q_1-1.5IQR=64.4-1.5(12.3)=45.95$

 There are no outliers because all of the data values are between $45.95$ and $95.15$.

As a result, there are no potential outliers.

Part (e) Step 1: Given Information

Given in the question , from The World Bank, in the document Life Expectancy af Birth, we obtained data on the expectation of life (in years) at birth for people in various countries. Those data are presented on the Weiss Stats site  we have to obtain and interpret a boxplot.

Part (e) Step 2: Explanation