Q16. - Algebra 1

Question

The table shows the number of servings of fruits and vegetables that Cole eats in one week. Find the range, median, lower quartile, and upper quartile.

Step-by-Step Solution

Answer

Range is 5

Median is 5

Lower quartile is $Q_1=3$

Lower quartile is $Q_3=7$

Step-1. Apply the concept of range, median, lower quartile, and upper quartile.

• The range is the difference between the largest observation and the smallest observation of a data set.
• The median is the middle value when a data set is ordered from least to greatest. It is the second quartile (Q2) and divides the data into two-half.

• The lower quartile Q1 is the median of the lower half of the data. 
• The upper quartile Q3 is the median of the upper half of the data.

• If the size of the data set is odd, include the median when finding the first and third quartiles.
• If the size of the data set is even, the median splits the data set into lower and upper halves.

Step-2. Calculation of range.

Given data for 7 days:

$\left\{5,7,5,4,3,3,8\right\}$

Arrange the data set in ascending order:

3,3,4,5,5,7,8

$\begin{array}{l}\mathrm{Range}=\mathrm{Largest}-\mathrm{Smallest}\\ \Rightarrow \mathrm{Range}=8-3\\ \Rightarrow \mathrm{Range}=5\end{array}$

Step-3. Calculation of median.

Given data for 7 days:

$\left\{5,7,5,4,3,3,8\right\}$

Arrange the data set in ascending order:

3,3,4,5,5,7,8

So, the median is average of ${\left(\frac{7+1}{2}\right)}^{th}=4^{th}$ and $5^{th}$ observation

Therefore, the median is $Q_2=5$

Step-4. Calculation of lower quartile.

Given data for 7 days:

$\left\{5,7,5,4,3,3,8\right\}$

Arrange the data set in ascending order:

3,3,4,5,5,7,8

From step 3, the median of the data set is 5

Hence, the lower half of the data set is:

3,3,4

The lower quartile is the median of lower half of the data set:

$Q_1=3$

Step-5. Calculation of upper quartile.

Given data for 7 days:

$\left\{5,7,5,4,3,3,8\right\}$

Arrange the data set in ascending order:

3,3,4,5,5,7,8

From step 3, the median of the data set is 5 

Hence, the upper half of the data set is:

5,7,8

The upper quartile is the median of upper half of the data set:

$Q_3=7$