Final answer:
Quartiles are used to divide ordered data into four equal parts, with each section containing an equal number of scores. Quartiles, such as the first (Q1) and third (Q3), represent the 25th and 75th percentiles, respectively. The interquartile range (IQR) is the spread of the middle 50% of the data.
Step-by-step explanation:
When dividing scores into quartiles, each section will have an equal number of scores. To understand this, first, recall that quartiles divide ordered data into four equal parts, each representing 25% of the dataset. This means that the first quartile (Q1) is at the 25th percentile, the second quartile (Q2 or median) at the 50th percentile, and the third quartile (Q3) at the 75th percentile. The interquartile range (IQR) is the range between the first and the third quartiles and represents the middle 50% of the data. By its very definition, quartiles do not necessarily have an equal range of scores, equal sum of scores, or equal mean of scores, as these can vary depending on the distribution of the data.
As an example, consider a dataset: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5. To find the quartiles, you first find the median, which in this case is 7. The first quartile (Q1) is then the median of the lower half, which is 2, and the third quartile (Q3) is the median of the upper half, which is 9. It is important to note that while each quartile represents an equal portion of the number of data points, the actual values within each quartile may vary greatly.