Example Question - data set quartiles
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Calculating Quartiles for a Data Set
To find the lower quartile, median, and upper quartile for the given data set, we follow these steps: 1. Median: This is the middle value when the data set is ordered from smallest to largest. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. The given data set is already ordered: 1, 2, 2, 2, 3, 3, 6, 8, 8 There are 9 values, so the middle one will be the 5th value (since (9 + 1) / 2 = 5): 1, 2, 2, 2, 3, 3, 6, 8, 8 Hence, the median is 3. 2. Lower Quartile (Q1): This is the median of the lower half of the data (excluding the median if the number of values is odd). For our data set, the lower half (before the median) is: 1, 2, 2, 2 In this set of 4 values, the lower quartile is the average of the 2nd and 3rd value (because (4 + 1) / 2 = 2.5, which means we take the average of 2nd and 3rd): (2 + 2) / 2 = 4 / 2 = 2 Hence, the lower quartile is 2. 3. Upper Quartile (Q3): This is the median of the upper half of the data (excluding the median if the number of values is odd). For our data set, the upper half (after the median) is: 3, 6, 8, 8 In this set of 4 values, the upper quartile is the average of the 6 and 8: (6 + 8) / 2 = 14 / 2 = 7 Hence, the upper quartile is 7. In summary: - Lower Quartile (Q1) = 2 - Median = 3 - Upper Quartile (Q3) = 7
Calculating Interquartile Range (IQR) for a Data Set
To find the interquartile range (IQR) for a given data set, you need to subtract the first quartile (Q1) from the third quartile (Q3). The first quartile (Q1) is the median of the lower half of the data set, and the third quartile (Q3) is the median of the upper half. The given data set: 1, 5, 5, 8, 8, 9, 9 Step 1: Find the median, which is the middle value when the data set is ordered. In this case, the median is 8 (there are 3 numbers to the left and 3 numbers to the right). Step 2: Find Q1 and Q3. - For Q1, you take the lower half of the data set, which is 1, 5, 5. The median of this half is 5. - For Q3, you take the upper half of the data set, which is 8, 9, 9. The median of this half is 9. Step 3: Subtract Q1 from Q3 to get the IQR. IQR = Q3 - Q1 = 9 - 5 = 4 So, the interquartile range of the given data set is 4.
Calculating Quartiles of a Data Set
To solve the given problem, we need to find the lower quartile (Q1), median (Q2), and upper quartile (Q3) of the provided data set. The data set is as follows: 15, 25, 42, 42, 42, 48, 74, 84, 90, 90, 91 First, let's arrange the data in ascending order, which is already done. There are 11 numbers in this data set. To find the median (Q2), we locate the middle number in the set. Since there are 11 numbers, the median will be the 6th number (as 11 + 1 divided by 2 gives us 6). Therefore, the median is 48. Next, we divide the data set into two halves to find the lower and upper quartiles. The lower half for the lower quartile will be the numbers before the median, and the upper half for the upper quartile will be the numbers after the median: Lower half (before median): 15, 25, 42, 42, 42 Upper half (after median): 74, 84, 90, 90, 91 Now, to find the lower quartile (Q1), we find the median of the lower half of the data set. Because there are 5 numbers in the lower half, the middle one is the 3rd, making Q1 equal to 42. To find the upper quartile (Q3), we do the same for the upper half of the data. There are 5 numbers in the upper half, so the middle one is also the 3rd number in the upper half, making Q3 equal to 90. Therefore, the lower quartile (Q1) is 42, the median (Q2) is 48, and the upper quartile (Q3) is 90.
Calculating Quartiles in a Data Set
To find the lower quartile, median, and upper quartile of a data set, we'll follow these steps: 1. **Median**: The median is the middle value of a data set when it's ordered from least to greatest. If the number of data points is odd, the median is the middle number. If it's even, the median is the average of the two middle numbers. For this set, there is an odd number of data points (7), so the median is the value in the middle - the fourth data point: \( \text{Median} = 35 \) 2. **Lower Quartile (Q1)**: The lower quartile is the median of the lower half of the data set (not including the median if the number of data points is odd). In this case, it's the median of the first three numbers: \( \text{Lower quartile} = 26 \) (as it is the middle of the first three numbers: 10, 26, 35) 3. **Upper Quartile (Q3)**: The upper quartile is the median of the upper half of the data set (again, not including the median if the number of data points is odd). The upper half of this data set is the last three numbers: \( \text{Upper quartile} = 55 \) (as it is the middle of the last three numbers: 43, 55, 73) To summarize: - **Lower Quartile**: 26 - **Median**: 35 - **Upper Quartile**: 55
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