Question

The estimated distribution (in millions) of the population by age in a certain country for the year 2015 is shown in the pie chart. Make a frequency distribution for the
data. Then use the table to estimate the sample mean and the sample standard deviation of the data set. Use 70 as the midpoint for "65 years and over."

Answer 1

A frequency distribution table displays the frequency of each data element in the dataset.

• The sample mean is 37.81
• The sample standard deviation is 22.02

The dataset is given as:

$\left[\begin{array}{cc}Years & Frequency &Under\ 4& 21.5 & 5-14 & 39.9 &15-19 & 20.3 & 20-24 & 22.3 & 25-34 & 48.4 &35-44 & 37.8 & 45-64 & 75.7 & 65 - over & 54.3\end{array}\right]$

(a) The frequency distribution table

The frequency distribution table should include, the years, the frequency, and the class midpoint (x)

The class midpoint is the average of each class

For instance;

The midpoint of under 4 is:

$x = \frac{0 + 4}{2} = 2$

The midpoint of 5-14 is:

$x = \frac{5 + 14}{2} = 9.5$

And so on.....

So, we have:

$\left[\begin{array}{ccc}Years & Frequency & x & Under\ 4& 21.5 & 2 & 5-14 & 39.9&9.5 &15-19 & 20.3&17 & 20-24 & 22.3 & 22 & 25-34 & 48.4& 29.5 &35-44 & 37.8& 39.5 & 45-64 & 75.7 & 54.5 & 65 - over & 54.3 & 70\end{array}\right]$

(b) Sample mean and Sample standard deviation

The sample mean is calculated using:

$\bar x = \frac{\sum fx}{\sum f}$

So, we have:

$\bar x = \frac{21.5 \times 2 + 39.9 \times 9.5 + 20.3 \times 17 + 22.3 \times 22 + 48.4 \times 29.5 + 37.8 \times 39.5 + 75.7 \times 54.5 + 54.3 \times 70}{21.5 + 39.9 + 20.3 + 22.3 + 48.4 + 37.8 + 75.7 + 54.3}$

$\bar x = \frac{12105.3}{320.2}$

$\bar x = 37.81$

The sample standard deviation is calculated using:

$\sigma_x = \sqrt\frac{\sum f(x - \bar x)^2}{\sum f - 1}$

So, we have:

$\sigma_x = \sqrt{\frac{21.5 \times (2 - 37.81)^2 + 39.9 \times (9.5 - 37.81)^2+..................+ 75.7 \times (54.5 - 37.81)^2+ 54.3 \times (70- 37.81)^2}{21.5 + 39.9 + 20.3 + 22.3 + 48.4 + 37.8 + 75.7 + 54.3 - 1}}$

$\sigma_x = \sqrt\frac{154716.23522}{319.2}$

$\sigma_x = \sqrt{484.699985025}$

$\sigma_x = 22.02$

Hence, the sample mean and the sample standard deviation are 37.81 and 22.02, respectively.

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