To solve for the median and quantiles, we first need to organize the data and then calculate these measures step by step.

Step 1: Organize the Data (Ascending Order)

Arranging the scores:

12, 17, 19, 19, 20, 20, 20, 21, 21, 21, 23, 23, 23, 24, 24, 26, 27, 27, 28, 29, 29, 29, 29, 29, 32, 32, 34, 37

Step 2: Median Calculation

The median is the middle value in a sorted dataset. For a dataset with $n = 28$, the median is the average of the 14th and 15th values:

• 14th value = 24
• 15th value = 24
  Median = $\frac{24 + 24}{2} = 24$

Step 3: Calculation of Quartiles (Quantiles)

First Quartile (Q1): Lower 25%

For Q1, find the median of the lower half (first 14 values):

12, 17, 19, 19, 20, 20, 20, 21, 21, 21, 23, 23, 23, 24

• Median of these values = $\frac{20 + 20}{2} = 20$
  Q1 = 20

Third Quartile (Q3): Upper 25%

For Q3, find the median of the upper half (last 14 values):

24, 26, 27, 27, 28, 29, 29, 29, 29, 29, 32, 32, 34, 37

• Median of these values = $\frac{29 + 29}{2} = 29$
  Q3 = 29

Interquartile Range (IQR)

The IQR is the difference between Q3 and Q1:

IQR = Q3 - Q1 = 29 - 20 = 9

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Step 4: Interpretation and Analysis

1. Median (24): This indicates that half the students scored less than or equal to 24, and the other half scored higher than or equal to 24.
2. Q1 (20): A quarter (25%) of the students scored 20 or below, showing the lower performance boundary.
3. Q3 (29): Three-quarters (75%) of the students scored 29 or below, indicating the typical upper range of scores.
4. IQR (9): This range represents the spread of the middle 50% of the scores, suggesting moderate variability.
5. Data Range: The scores range from 12 to 37, showing significant variation among students' performance.

Would you like detailed visualization or any further breakdown of the data?

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5 Related Questions to Explore

1. How does the interquartile range relate to overall variability in this dataset?
2. What percentage of students scored above the median?
3. How would the addition of a new score, such as 40, affect the median and quartiles?
4. Can this data be used to identify outliers using the IQR method?
5. How do the quartiles compare if the data distribution was normal versus skewed?

Tip: When analyzing scores, the interquartile range is a robust measure of variability since it is not influenced by extreme values or outliers.