A group of \(\text{15}\) learners count the number of
sweets they each have. This is the data they collect:
\[\begin{array}{c c c c c}
4 & 11 & 14 & 7 & 14 \\
5 & 8 & 7 & 12 & 12 \\
5 & 13 & 10 & 6 & 7
\end{array}\]
Calculate the range of values in the data set.
We first need to order the data set:
\[\{4; 5; 5; 6; 7; 7; 7; 8; 10; 11; 12; 12; 13; 14; 14\}\]
Next we find the maximum value in the data set:
\[\text{maximum value } = \text{14}\]
Then we find the minimum value in the data set:
\[\text{minimum value } = \text{4}\]
Finally, we calculate the range of the data set:
\begin{align*}
\text{range } & = \text{(maximum value) } - \text{ (minimum value)}\\
& = (\text{14}) - (\text{4}) \\
& = \text{10}
\end{align*}
A group of \(\text{10}\) learners count the number of
playing cards they each have. This is the data they collect:
\[\begin{array}{c c c c c}
5 & 1 & 3 & 1 & 4 \\
10 & 1 & 3 & 3 & 4
\end{array}\]
Calculate the range of values in the data set.
We first need to order the data set:
\[\{1; 1; 1; 3; 3; 3; 4; 4; 5; 10\}\]
Next we find the maximum value in the data set:
\[\text{maximum value } = \text{10}\]
Then we find the minimum value in the data set:
\[\text{minimum value } = \text{1}\]
Finally, we calculate the range of the data set:
\begin{align*}
\text{range} & = \text{(maximum value) } - \text{ (minimum value)}\\
& = \text{10} - \text{1} \\
& = \text{9}
\end{align*}
Find the range of the data set
\[\left\{1; 2; 3; 4; 4; 4; 5; 6; 7; 8; 8; 9; 10; 10\right\}\]
The data set is already ordered.
Firstly, we find the maximum value in the data set:
\[\text{maximum value } = \text{10}\]
Secondly, we find the minimum value in the data set:
\[\text{minimum value } = \text{1}\]
Finally, we calculate the range of the data set:
\begin{align*}
\text{range} & = \text{(maximum value) } - \text{ (minimum value)}\\
& = \text{10} - \text{1} \\
& = \text{9}
\end{align*}
What are the quartiles of this data set?
\[\left\{3; 5; 1; 8; 9; 12; 25; 28; 24; 30; 41; 50\right\}\]
We first order the data set.
\[\left\{1; 3; 5; 8; 9; 12; 24; 25; 28; 30; 41; 50\right\}\]
Next we find the ranks of the quartiles. Using the percentile formula with
\(n = 12\), we can find the rank of the \(25^{\text{th}}\),
\(50^{\text{th}}\) and \(75^{\text{th}}\) percentiles:
\begin{align*}
{r}_{25} & = \frac{25}{100}\left(12 - 1\right) + 1 \\
& = \text{3,75} \\
{r}_{50} & = \frac{50}{100}\left(12 - 1\right) + 1 \\
& = \text{6,5} \\
{r}_{75} & = \frac{75}{100}\left(12 - 1\right) + 1 \\
& = \text{9,25}
\end{align*}
Find the values of the quartiles. Note that each of these ranks is a
fraction, meaning that the value for each percentile is somewhere in
between two values from the data set.
For the \(25^{\text{th}}\) percentile the rank is \(\text{3,75}\), which is
between the third and fourth values. Therefore the \(25^{\text{th}}\)
percentile is \(\frac{5 + 8}{2} = \text{6,5}\).
For the \(50^{\text{th}}\) percentile (the median) the rank is
\(\text{6,5}\), meaning halfway between the sixth and seventh values.
Therefore the median is \(\frac{12 + 24}{2} = \text{18}\). For the
\(75^{\text{th}}\) percentile the rank is \(\text{9,25}\), meaning
between the ninth and tenth values. Therefore the \(75^{\text{th}}\)
percentile is \(\frac{28 + 30}{2} = 29\).
Therefore we get the following values for the quartiles: \(Q_1 =
\text{6,5}\); \(Q_2 = 18\); \(Q_3 = 29\).
A class of \(\text{12}\) learners writes a test and the results are as
follows:
\[\left\{20; 39; 40; 43; 43; 46; 53; 58; 63; 70; 75; 91\right\}\]
Find the range, quartiles and the interquartile range.
The data set is ordered.
The range is:
\begin{align*}
\text{range} & = \text{(maximum value) } - \text{ (minimum value)}\\
& = (\text{91}) - (\text{20}) \\
& = \text{71}
\end{align*}
To find the quartiles we start by finding the ranks of the quartiles. Using
the percentile formula with \(n = 12\), we can find the rank of the
\(25^{\text{th}}\), \(50^{\text{th}}\) and \(75^{\text{th}}\)
percentiles:
\begin{align*}
{r}_{25} & = \frac{25}{100}\left(12 - 1\right) + 1 \\
& = \text{3,75} \\
{r}_{50} & = \frac{50}{100}\left(12 - 1\right) + 1 \\
& = \text{6,5} \\
{r}_{75} & = \frac{75}{100}\left(12 - 1\right) + 1 \\
& = \text{9,25}
\end{align*}
Find the values of the quartiles. Note that each of these ranks is a
fraction, meaning that the value for each percentile is somewhere in
between two values from the data set.
For the \(25^{\text{th}}\) percentile the rank is \(\text{3,75}\), which is
between the third and fourth values. Therefore the \(25^{\text{th}}\)
percentile is \(\frac{40 + 43}{2} = \text{41,5}\).
For the \(50^{\text{th}}\) percentile (the median) the rank is
\(\text{6,5}\), meaning halfway between the sixth and seventh values.
Therefore the median is \(\frac{46 + 53}{2} = \text{49,5}\). For the
\(75^{\text{th}}\) percentile the rank is \(\text{9,25}\), meaning
between the ninth and tenth values. Therefore the \(75^{\text{th}}\)
percentile is \(\frac{63 + 70}{2} = \text{66,5}\).
Therefore we get the following values for the quartiles: \(Q_1 =
\text{41,5}\); \(Q_2 = \text{49,5}\); \(Q_3 = \text{66,5}\).
Interquartile range:
\begin{align*}
\text{interquartile range } & = \text{quartile 3 } - \text{ quartile 1} \\
& = \text{66,5} - \text{41,5} \\
& = 25
\end{align*}
