Example Question - quartile calculation
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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 Quartiles from Data Set
In the image, you are given a set of data and asked to find the lower quartile, median, and upper quartile. To find these measures, you first need to make sure the data is sorted in ascending order, which it is. The dataset contains the following numbers: 13, 17, 20, 26, 37, 39, 40, 82, 85, 88, 88 To find the median (the middle value), since there are 11 data points, you take the middle value, which is the 6th number when the data is ordered. The 6th number in the sequence is 39, so the median is 39. The lower quartile (Q1) is the median of the first half of the data. Since there are 11 data points, we take the lower 5 data points (ignoring the median): 13, 17, 20, 26, 37 The middle of this set is the third value, which is 20, so the lower quartile is 20. The upper quartile (Q3) is the median of the second half of the data. Again, we ignore the median (39) and take the upper 5 data points: 40, 82, 85, 88, 88 The middle of this set is the third value, which is 85, so the upper quartile is 85. To summarize: Lower quartile (Q1) = 20 Median = 39 Upper quartile (Q3) = 85
Calculating Interquartile Range of a Data Set
To find the interquartile range (IQR) of a data set, we must first determine the first quartile (Q1) and the third quartile (Q3) and then subtract the first quartile from the third quartile: IQR = Q3 - Q1. The data set provided is: 36, 44, 44, 68, 71, 75 Since there are six data points, Q1 will be the value between the second and third data points, and Q3 will be the value between the fifth and sixth data points. To calculate Q1, we take the average of the second and third values (since they're both 44, the average is 44), so Q1 = 44. To calculate Q3, we take the average of the fifth and sixth values: (71 + 75)/2 = 146/2 = 73. Now, let's calculate the IQR: IQR = Q3 - Q1 IQR = 73 - 44 IQR = 29 Thus, the interquartile range of the data set is 29.
Statistical Analysis of Numerical Data
The image shows a set of numerical data: 31, 33, 35, 57, 80, 87. To find the lower quartile, median, and upper quartile of this data set, we need to follow these steps: 1. **Median**: Since there are 6 numbers, the median will be the average of the 3rd and 4th numbers. Therefore, the median is (35 + 57) / 2 = 92 / 2 = 46. 2. **Lower Quartile**: The lower quartile (Q1) is the median of the lower half of the data set (excluding the median if the number of observations is odd). With the data set split into two halves (31, 33, 35) and (57, 80, 87), the median of the lower half is 33, since it is the middle value of the first three numbers. 3. **Upper Quartile**: The upper quartile (Q3) is the median of the upper half of the data set (again excluding the median if the number of observations is odd). For the upper half (57, 80, 87), the median is 80, as it is the middle value of the last three numbers. Thus, the lower quartile is 33, the median is 46, and the upper quartile is 80.
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