Question

select the correct answer.

which of the following box plots is symmetric?

w.
x.
y.
z.

w
x
y
z

Explanation:

⚡ Using what you learned: displaying data: histograms & box plots

Step 1: Define symmetry in a box plot

A box plot is perfectly symmetric if the median line is exactly in the middle of the box, and the whiskers on both sides are of equal length.

Mathematically, this means:

• The distance from the minimum to the median equals the distance from the median to the maximum:

$$ \text{Median} - \text{Minimum} = \text{Maximum} - \text{Median} $$

• The distance from the first quartile (\(Q_1\)) to the median equals the distance from the median to the third quartile (\(Q_3\)):

$$ \text{Median} - Q_1 = Q_3 - \text{Median} $$

• The left whisker length equals the right whisker length:

$$ Q_1 - \text{Minimum} = \text{Maximum} - Q_3 $$

Step 2: Analyze the given box plots

Let's evaluate each option using its five-number summary:

• Box Plot W:
• \(\text{Minimum} = 2\)
• \(Q_1 = 8\)
• \(\text{Median} = 10\)
• \(Q_3 = 14\)
• \(\text{Maximum} = 18\)

Checking symmetry:

• Left whisker: \(Q_1 - \text{Minimum} = 8 - 2 = 6\)
• Right whisker: \(\text{Maximum} - Q_3 = 18 - 14 = 4\)
• Left box half: \(\text{Median} - Q_1 = 10 - 8 = 2\)
• Right box half: \(Q_3 - \text{Median} = 14 - 10 = 4\)

Since the left and right parts are not equal, W is not symmetric.

• Box Plot X:
• \(\text{Minimum} = 2\)
• \(Q_1 = 6\)
• \(\text{Median} = 10\)
• \(Q_3 = 12\)
• \(\text{Maximum} = 18\)

Checking symmetry:

• Left whisker: \(6 - 2 = 4\)
• Right whisker: \(18 - 12 = 6\)

Since the whiskers are unequal, X is not symmetric.

• Box Plot Y:
• \(\text{Minimum} = 2\)
• \(Q_1 = 4\)
• \(\text{Median} = 6\)
• \(Q_3 = 8\)
• \(\text{Maximum} = 18\)

Checking symmetry:

• Left whisker: \(4 - 2 = 2\)
• Right whisker: \(18 - 8 = 10\)

Since the whiskers are highly unequal, Y is not symmetric.

• Box Plot Z:
• \(\text{Minimum} = 2\)
• \(Q_1 = 6\)
• \(\text{Median} = 10\)
• \(Q_3 = 14\)
• \(\text{Maximum} = 18\)

Checking symmetry:

• Left whisker: \(Q_1 - \text{Minimum} = 6 - 2 = 4\)
• Right whisker: \(\text{Maximum} - Q_3 = 18 - 14 = 4\)
• Left box half: \(\text{Median} - Q_1 = 10 - 6 = 4\)
• Right box half: \(Q_3 - \text{Median} = 14 - 10 = 4\)

All corresponding left and right segments are perfectly equal in length (\(4\) units each). Therefore, Z is symmetric.

Answer:

Z