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Following is a list of the number of hazardous waste sites in states in a recent year. The list has been sorted into numerical order. Construct a boxplot for these data. The median is 20, q1 is 13, and q3 is 34.

1
2
3
3
4
6
7
7
9
9
10
10
13
14
14
14
15
15
15
15
17
17
17
18
20
20
21
22
23
23
24
25
27
28
30
31
33
34
36
41
42
46
47
48
56
72
75
93
95
112
Construct a boxplot for the given data (including outliers if any).

1 answer

To construct a boxplot for the given data, we need the five-number summary, which consists of the minimum, Q1, median (Q2), Q3, and the maximum value. You've already provided Q1, Q2 (median), and Q3. The next steps involve identifying the minimum and maximum values and determining if there are any outliers.

Given Statistics:

  • Minimum (Min): 1
  • Q1: 13
  • Median (Q2): 20
  • Q3: 34
  • Maximum (Max): 112

Step 1: Calculate the Interquartile Range (IQR)

The IQR is calculated as follows: \[ \text{IQR} = Q3 - Q1 = 34 - 13 = 21 \]

Step 2: Calculate the lower and upper bounds for outliers

  • Lower Bound: \[ Q1 - 1.5 \times \text{IQR} = 13 - 1.5 \times 21 = 13 - 31.5 = -18.5 \]
  • Upper Bound: \[ Q3 + 1.5 \times \text{IQR} = 34 + 1.5 \times 21 = 34 + 31.5 = 65.5 \]

Step 3: Identify Outliers

Any data points below -18.5 or above 65.5 are considered outliers in this dataset.

From the sorted list of data:

  • Values above 65.5 are: 72, 75, 93, 95, 112
  • There are no values below -18.5.

Step 4: Construct the Boxplot

Now we can summarize our boxplot data:

  • Minimum: 1 (not an outlier)
  • Q1: 13
  • Median (Q2): 20
  • Q3: 34
  • Maximum (Max): 56 (not an outlier)
  • Outliers: 72, 75, 93, 95, 112

Boxplot Visualization Components:

  1. Box: The box is drawn between Q1 (13) and Q3 (34).
  2. Line inside the box: Represents the median (20).
  3. Whiskers: Extend from the minimum (1) to Q1 (13) and from Q3 (34) to the maximum (56).
  4. Outliers: Marked separately, which include 72, 75, 93, 95, and 112.

Boxplot Structure:

                                      Outliers
                                         |
            1 -----------|----13  |---------------- 34 --------- 56
                          |_______20_______|
             Minimum     Q1    Median   Q3    Maximum

Final Notes:

  • The data points 72, 75, 93, 95, and 112 will be marked as individual points beyond the whiskers.
  • This boxplot provides a visual representation of the distribution of hazardous waste sites across the given states, revealing both the central tendency and the variability in the data.