Charles wants to analyze his last 9 math test scores with a box plot to see how spread apart they are.
Arrange the scores in order, from lowest (on the left) to highest.
What is the range of Charles's scores?
What is the median of Charles's scores?
What is the IQR of Charles's scores?
Answer (2)
Calculate the range by subtracting the minimum score from the maximum score: R an g e = 100 − 80 = 20 .
Find the median, which is the middle value of the ordered data set: M e d ian = 90 .
Calculate the first quartile (Q1) and the third quartile (Q3): Q 1 = 85 , Q 3 = 93 .
Calculate the interquartile range (IQR) by subtracting Q1 from Q3: I QR = 93 − 85 = 8 . The final answers are: Range = 20 , Median = 90 , IQR = 8 .
Explanation
Understanding the Problem We are given Charles's math test scores: 80, 84, 86, 87, 90, 91, 92, 94, 100. The scores are already arranged in ascending order. We need to find the range, median, and IQR of the scores.
Calculating the Range The range is the difference between the highest and lowest scores. In this case, the highest score is 100 and the lowest score is 80. Therefore, the range is: R an g e = 100 − 80 = 20
Finding the Median The median is the middle value of the ordered data set. Since there are 9 scores, the median is the 5th score, which is 90.
Calculating the First Quartile (Q1) To find the IQR, we first need to find the first quartile (Q1) and the third quartile (Q3). Q1 is the median of the lower half of the data (80, 84, 86, 87). Since there are 4 scores in the lower half, Q1 is the average of the 2nd and 3rd lowest scores: Q 1 = ( 84 + 86 ) /2 = 170/2 = 85
Calculating the Third Quartile (Q3) Q3 is the median of the upper half of the data (91, 92, 94, 100). Since there are 4 scores in the upper half, Q3 is the average of the 7th and 8th lowest scores: Q 3 = ( 92 + 94 ) /2 = 186/2 = 93
Calculating the Interquartile Range (IQR) The interquartile range (IQR) is the difference between Q3 and Q1: I QR = Q 3 − Q 1 = 93 − 85 = 8
Final Answer Therefore, the range of Charles's scores is 20, the median is 90, and the IQR is 8.
Examples
Understanding the range, median, and IQR can help Charles analyze the distribution of his test scores. For example, if the IQR is small, it means that his scores are clustered closely around the median, indicating consistent performance. If the range is large, it means that there is a wide variation in his scores, suggesting that his performance is not consistent. This type of analysis can be used in many real-life situations, such as analyzing sales data, stock prices, or weather patterns. By understanding the distribution of data, we can make better decisions and predictions.
The range of Charles's test scores is 20, the median is 90, and the IQR is 8. These calculations help determine the distribution of his scores. Understanding these measures can provide insights into his performance consistency.
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