Step-by-Step Solution

Organizing the Data

The provided data is already arranged in ascending order: 1, 10, 20, 30, 50, 50, 50, 60, 75, 80, 80, 80, 83, 85, 90, 90, 90, 90, 90, 90, 95, 95, 97, 99, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100

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Part C: MEASURES OF RELATIVE POSITION

1. Quartile 1 (Q1)

Definition: The first quartile is the median of the lower half of the data (below the overall median).

• Steps:

  1. Identify the lower half of the data: 1, 10, 20, 30, 50, 50, 50, 60, 75, 80, 80, 80, 83, 85, 90
  2. Find the median of this subset:
     • Lower half size = 15 elements. Median is the 8th value (60).

• Q1: 60

2. Quartile 2 (Q2)

Definition: The second quartile is the overall median.

• Steps:

  1. Identify the full data set.
  2. Find the median of the 35 elements:
     • Median position: $\frac{35 + 1}{2} = 18^{\text{th}}$.
     • The 18th value is 90.

• Q2: 90

3. Quartile 3 (Q3)

Definition: The third quartile is the median of the upper half of the data (above the overall median).

• Steps:

  1. Identify the upper half of the data: 90, 90, 90, 90, 95, 95, 97, 99, 100, 100, 100, 100, 100, 100, 100
  2. Find the median of this subset:
     • Upper half size = 15 elements. Median is the 8th value (99).

• Q3: 99

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BOX-AND-WHISKER PLOT

To create the box-and-whisker plot:

• Minimum: 1
• Q1: 60
• Median (Q2): 90
• Q3: 99
• Maximum: 100

Visual Representation

(Approximation of Box Plot)

plaintextCopy code
|----|-----------------|---------|-----------|----|
  1   Q1=60          Q2=90     Q3=99      100

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INTERPRETATION OF THE BOX-AND-WHISKER PLOT

1. The dataset is heavily skewed toward the upper end since the median (Q2 = 90) is close to the third quartile (Q3 = 99).
2. There is a wide range between the minimum (1) and Q1 (60), showing that the lowest values are far from the central tendency.
3. The upper half of the data (90–100) contains frequent repetitions, showing a clustering of high values.

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Explanation of Steps

How I Found Each Answer:

1. Quartile 1 (Q1): Divided the dataset into lower half and computed its median.
2. Quartile 2 (Q2): Found the overall median of the dataset.
3. Quartile 3 (Q3): Divided the dataset into the upper half and computed its median.
4. Box-and-Whisker Plot: Identified the five-number summary: minimum, Q1, Q2, Q3, maximum.
5. Interpretation: Assessed data distribution based on the plot and quartiles.

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Follow-Up Questions

1. What is the interquartile range (IQR) of this dataset?
2. How would removing outliers affect the quartiles and box plot?
3. Can you describe the skewness of this dataset numerically?
4. How does the repeated frequency of "100" affect Q3?
5. What is the role of outliers in a box-and-whisker plot?

Tip: Always verify your median positions using $\frac{n+1}{2}$ for odd datasets and averaging the two middle values for even datasets.