Example Question - calculating 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 Quartiles in a Sorted Dataset
To find the lower quartile, median, and upper quartile, we first need to understand that these are measures of central tendency and dispersion for a given set of data, specifically in a sorted list. The data provided is already sorted: 1, 2, 2, 2, 3, 3, 6, 6, 8, 8 The median (the middle value of the dataset) is found by taking the average of the two middle numbers if there is an even number of observations. Since we have 10 numbers here, the median will be the average of the 5th and 6th numbers: (2 + 3)/2 = 5/2 = 2.5 So the median is 2.5. The lower quartile (Q1), also known as the first quartile, is the median of the first half of the data. When determining quartiles, if the median is part of the data, for the lower quartile, we do not include the median as part of the lower half. So, for the lower quartile calculation, we only consider the first five numbers: 1, 2, 2, 2, 3. Since there is an odd number of numbers in this half, the lower quartile is the middle number, which is 2. The upper quartile (Q3), or the third quartile, is the median of the second half of the data. Similar to calculating Q1, we will disregard the median of the entire data set and use only the last five numbers: 3, 6, 6, 8, 8. The middle number here is 6, so the upper quartile is 6. Let's summarize: - Lower quartile = 2 - Median = 2.5 - Upper quartile = 6
Calculating Quartiles
To calculate the lower quartile, median, and upper quartile of the data set provided, here are the steps: 1. Arrange the data in ascending order: 74, 76, 76, 78, 78, 79, 79 2. Determine the median (the middle number): Since there are 7 numbers, the median is the fourth number: 78. 3. Determine the lower quartile (Q1), which is the median of the lower half of the data, not including the median of the dataset. The lower half of the data is 74, 76, 76. The median of this half is the second number: 76. 4. Determine the upper quartile (Q3), which is the median of the upper half of the data, not including the median of the dataset. The upper half of the data is 78, 79, 79. The median of this half is the second number: 79. Therefore, the lower quartile (Q1) is 76, the median is 78, and the upper quartile (Q3) is 79.
Calculating Quartiles
To solve for the lower quartile, median, and upper quartile, you first need to arrange the data in ascending order, which appears to be already done in the image: 74, 76, 76, 78, 78, 79, 79. The median (or second quartile) is the middle number of a data set that's been arranged in order, and since there are seven numbers, the middle one is the fourth one: 78. For odd-numbered data sets, the lower quartile (or first quartile, Q1) is the median of the first half of the data not including the overall median. The first half of the data is 74, 76, 76, so the lower quartile is the middle of these three numbers, which is 76. The upper quartile (or third quartile, Q3) is the median of the second half of the data not including the overall median. The second half of the data is 78, 79, 79, so the upper quartile is the middle of these three numbers, which is 79. Therefore, the lower quartile is 76, the median is 78, and the upper quartile is 79.
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 Quartiles from Sorted Data
To solve for the lower quartile, median, and upper quartile, we first need to arrange the data in ascending order, which is already done for us: 2, 6, 6, 7, 9. The median is the middle number when the data is ordered from least to greatest. In this case, since there are five numbers, the median will be the third number: Median = 6 The lower quartile (Q1) is the median of the lower half of the data. Since there are 5 data points, we will consider the lower half to be the first two numbers (2 and 6). The median of these two numbers is: Lower quartile = (2 + 6) / 2 = 8 / 2 = 4 The upper quartile (Q3) is the median of the upper half of the data. For the upper half, we take the last two numbers (7 and 9). The median of these is: Upper quartile = (7 + 9) / 2 = 16 / 2 = 8 Now we have our three values: - Lower quartile = 4 - Median = 6 - Upper quartile = 8
Calculating Quartiles from a Dataset
To find the lower quartile, median, and upper quartile from a set of numbers, you first ensure that the numbers are ordered from smallest to largest, which they already are in the image you provided: 14, 20, 20, 31, 38, 45, 54, 81, 81, 87, 92 There are 11 numbers in this dataset. The median (the middle number) is easy to find because there is an odd number of data points. The median will be the value in the middle of the dataset: Median: Since there are 11 numbers, the median will be the 6th number (as 5 numbers are on either side): 14, 20, 20, 31, 38, (45), 54, 81, 81, 87, 92 So, the median is 45. For the quartiles, we divide the dataset into four equal parts. The lower quartile (Q1) is the median of the lower half of the dataset (not including the median if there is an odd number of data points), and the upper quartile (Q3) is the median of the upper half of the dataset. Lower quartile: The lower half of the dataset (before the median) is: 14, 20, 20, 31, 38 Since there are 5 numbers in this lower half, the lower quartile is the middle value, which is the 3rd number: 14, 20, (20), 31, 38 So, the lower quartile is 20. Upper quartile: The upper half of the dataset (after the median) is: 54, 81, 81, 87, 92 Again, there are 5 numbers in this upper half, so the upper quartile will also be the middle value, which is the 3rd number: 54, 81, (81), 87, 92 So, the upper quartile is 81. In summary: Lower quartile (Q1) = 20 Median (Q2) = 45 Upper quartile (Q3) = 81
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 from a Set of Numbers
The image presents a set of numbers and asks to find the lower quartile, median, and upper quartile. The numbers in the set are: 13, 25, 34, 37, 45, 47, 52, 56, 58, 66, 81 The median is the middle number when the numbers are ordered from least to greatest. Since there are 11 numbers, the median is the 6th number in the list. Let's find it: Median = 47 Quartiles divide the data into four equal parts. The lower quartile (Q1) is the median of the lower half of the data, and the upper quartile (Q3) is the median of the upper half. For the lower quartile, we'll consider the numbers before the median: 13, 25, 34, 37, 45 Since there are 5 numbers, the lower quartile is the middle one, which is the 3rd number: Lower quartile (Q1) = 34 For the upper quartile, we take the numbers after the median: 52, 56, 58, 66, 81 Similarly, since there are 5 numbers, the upper quartile is the middle one, which is the 3rd number in this set: Upper quartile (Q3) = 58 To summarize: Lower quartile (Q1) = 34 Median = 47 Upper quartile (Q3) = 58
Calculating Quartiles from Sorted Data Set
The image shows a list of numbers, which we'll assume is a data set that is sorted in ascending order. The task is to find the lower quartile, median, and upper quartile of the data. First, let's count how many data points there are: 13, 25, 34, 37, 45, 47, 52, 56, 58, 66, 81 There are 11 numbers in this data set. The median is the middle number when the data is ordered from least to greatest, which is the (11 + 1) / 2 = 6th number. Therefore, the median is the value of the 6th data point, which is 47. To find the quartiles, we will look at the data below and above the median: Lower half (excluding the median): 13, 25, 34, 37, 45 Upper half (excluding the median): 52, 56, 58, 66, 81 Each half has 5 numbers. The lower quartile (Q1) is the median of the lower half, and the upper quartile (Q3) is the median of the upper half. For 5 numbers, the median will be the 3rd number in order (since it's the middle number): (5 + 1) / 2 = 3rd number. For the lower half: The lower quartile (Q1) is the 3rd data point of the lower half, which is 34. For the upper half: The upper quartile (Q3) is the 3rd data point of the upper half, which is 58. So the answers are: - Lower quartile = 34 - Median = 47 - Upper quartile = 58
Calculating Quartiles of Given Numbers
The image shows a set of numbers in ascending order, and the task is to find the lower quartile, median, and upper quartile of these numbers. Let's calculate each of them: The numbers given are: 13, 23, 34, 37, 45, 47, 52, 56, 58, 66, 81. First, let's find the median, which is the middle number of the data set when arranged in order. Since there are 11 numbers, the median will be the 6th number (as there are 5 numbers on either side of it). So the median is 47. Now let's calculate the lower quartile, which is the median of the lower half of the data set. Exclude the overall median (47) when considering the halves. The lower half of this data set is: 13, 23, 34, 37, 45. The median of this subset is the third number, which is 34, so the lower quartile is 34. Lastly, let's find the upper quartile, which is the median of the upper half of the data set. Again, exclude the overall median (47). The upper half of the data set is: 52, 56, 58, 66, 81. The median of this subset is the third number, which is 58, so the upper quartile is 58. In summary: - The lower quartile is 34. - The median is 47. - The upper quartile is 58.
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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