Families at a school in Namibia were surveyed, and the number of children in each family was recorded. The following data shows the results of the survey.

\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline 1 & 2 & 3 & 2 & 3 & 1 & 4 & 2 & 2 & 3 \\
\hline 2 & 4 & 2 & 2 & 1 & 2 & 2 & 3 & 3 & 2 \\
\hline 3 & 3 & 2 & 1 & 2 & 3 & 4 & 2 & 4 & 1 \\
\hline 2 & 2 & 4 & 2 & 3 & 1 & 3 & 2 & 2 & 4 \\
\hline 3 & 3 & 2 & 3 & 2 & 3 & 2 & 4 & 5 & 2 \\
\hline 2 & 2 & 4 & 2 & 3 & 2 & 5 & 2 & 2 & \\
\hline
\end{tabular}

(a) Copy and complete the table:

\begin{tabular}{|c|l|l|l|}
\hline \begin{tabular}{c}
Number of \\
children
\end{tabular} & Tally & Frequency & \begin{tabular}{c}
Relative \\
Frequency
\end{tabular} \\
\hline 1 & & & \\
\hline 2 & & & \\
\hline 3 & & & \\
\hline 4 & & & \\
\hline 5 & & & \\
\hline 6 & & & \\
\hline
\end{tabular}
[tex]$[4]$[/tex]

(b) Calculate the:
(i) mean
[tex]$[3]$[/tex]

(ii) mode
[tex]$[1]$[/tex]

(iii) median
[tex]$[2]$[/tex]

(c) The average Namibian family has 2.2 children. How does this compare to the national average?
[tex]$[2]$[/tex]

(d) The data set is skewed. Is the skewness positive or negative?
[tex]$[1]$[/tex]

(e) How has the skewness of the data affected the measures of the center of the data set?
[tex]$[2]$[/tex]



Answer :

(a) Let's fill in the table given the data provided.

\begin{tabular}{|c|l|l|l|}
\hline
\begin{tabular}{c}
Number of \\
children
\end{tabular} & Tally & Frequency & \begin{tabular}{c}
Relative \\
Frequency
\end{tabular} \\

\hline 1 & [tex]\(\| \| \| \| \| \|\)[/tex] & 6 & 0.1017 \\
\hline 2 & [tex]\(\| \| \| \| \| \|\ \| \| \| \| \| \| \| \| \|\)[/tex] & 28 & 0.4746 \\
\hline 3 & [tex]\(\| \| \| \| \| \| \|\ \| \| \| \| \| \| \| \|\)[/tex] & 15 & 0.2542 \\
\hline 4 & [tex]\(\| \| \| \| \|\ \| \| \|\)[/tex] & 8 & 0.1356 \\
\hline 5 & [tex]\(\| \|\| \)[/tex] & 2 & 0.0339 \\
\hline 6 & & 0 & 0.0000 \\
\hline
\end{tabular}

[tex]$[4]$[/tex]

(b) Now, let's perform the calculations.

(i) Mean: The mean is calculated by summing all the values and dividing by the number of values.
[tex]\[ \text{Mean} = 2.5254 \][/tex]
[tex]$[3]$[/tex]

(ii) Mode: The mode is the value that appears most frequently in the data set.
[tex]\[ \text{Mode} = 2 \][/tex]
[tex]$[1]$[/tex]

(iii) Median: The median is the middle value when the data is ordered.
[tex]\[ \text{Median} = 2 \][/tex]
[tex]$[2]$[/tex]

(c) Comparison to National Average:

The average number of children in the surveyed families is 2.52, which is higher than the national average of 2.2 children. This likely indicates that the families surveyed at the school have, on average, more children than the typical Namibian family.
[tex]$[2]$[/tex]

(d) Skewness Direction:

The data is positively skewed since the mean (2.52) is greater than the median (2).
[tex]\[ \text{Skewness is positive.} \][/tex]
[tex]$[1]$[/tex]

(e) Effect of Skewness on Measures of Centre:

Positive skewness typically pulls the mean to the right of the median. In this case, the mean is somewhat higher than the median due to the presence of families with a larger number of children, which affects the average.
[tex]\[ \text{Positive skewness typically pulls the mean to the right of the median.} \][/tex]
[tex]$[2]$[/tex]