Example Question - interquartile range
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Calculating Mean, Median, and Interquartile Range
In order to solve this question, we need to find the mean, median, and interquartile range of the provided data set: 7, 22, 3, 18, 15, 7, 12, 2, 11, 13, 15, 7, 18 First, let's sort the numbers in ascending order: 2, 3, 7, 7, 7, 11, 12, 13, 15, 15, 18, 18, 22 Now we can calculate each measure: Mean: The mean is calculated by adding all the numbers together and dividing by the count of numbers. Mean = (2 + 3 + 7 + 7 + 7 + 11 + 12 + 13 + 15 + 15 + 18 + 18 + 22) / 13 Mean = 130 / 13 Mean ≈ 10 Median: The median is the middle number of the ordered data set. Since there are 13 numbers, the median will be the 7th number. Median = 12 Interquartile Range (IQR): IQR is the difference between the third quartile (Q3) and the first quartile (Q1). Since we have 13 numbers, the first quartile Q1 will be the median of the first half: Q1 = median of 2, 3, 7, 7, 7, 11 (which is the average of the 3rd and 4th values): Q1 = (7 + 7) / 2 Q1 = 7 The third quartile Q3 will be the median of the second half: Q3 = median of 13, 15, 15, 18, 18, 22 (which is the average of the 3rd and 4th values): Q3 = (15 + 15) / 2 Q3 = 15 IQR = Q3 - Q1 IQR = 15 - 7 IQR = 8 Summary: Mean = ≈10 Median = 12 Interquartile Range = 8
Calculating Interquartile Range
The interquartile range (IQR) is a measure of statistical dispersion and is calculated as the difference between the 75th percentile (Q3) and the 25th percentile (Q1) in a data set. To find the IQR, follow these steps: 1. Order the data set from smallest to largest (which is already done in this case). 2. Divide the data set into quartiles. Q1 is the median of the first half of the data, and Q3 is the median of the second half. 3. Calculate Q1 and Q3, and then find the difference between them. Let's go through the steps with the given data set: 10, 25, 31, 40, 45, 51, 59, 67, 96, 98 There are 10 data points, so we can exactly divide this set into two halves for finding Q1 and Q3: First half (for Q1): 10, 25, 31, 40, 45 Second half (for Q3): 51, 59, 67, 96, 98 Now let's find Q1, the median of the first half: Q1 = 31 (since 31 is the middle value in the first half) And Q3, the median of the second half: Q3 = 67 (since 67 is the middle value in the second half) Finally, calculate the IQR: IQR = Q3 - Q1 IQR = 67 - 31 IQR = 36 So, the interquartile range of the given data set is 36.
Calculating Interquartile Range of a Dataset
To find the interquartile range (IQR) of a dataset, you need to find the difference between the third quartile (Q3) and the first quartile (Q1). First, let's arrange the given data in ascending order if it is not already sorted: 1, 2, 4, 4, 4, 4, 4, 8, 8. The dataset is already sorted. The next step is to find Q1 and Q3: - The first quartile (Q1) is the median of the first half of the data. Since there are 9 data points, the first half will consist of the first 4 data points. Q1 is the median of 1, 2, 4, 4, which is the average of the 2nd and 3rd points, (2+4)/2 = 3. - The third quartile (Q3) is the median of the second half of the data. The second half of the dataset after excluding the median (4, in this case) is 4, 4, 8, 8. The median of these four numbers is the average of the 2nd and 3rd points, (4+4)/2 = 4. Now, we find the IQR: IQR = Q3 - Q1 = 4 - 3 = 1. So, the interquartile range of the given dataset is 1.
Calculating Interquartile Range from Data Set
To find the interquartile range of the data set, you must first find the first quartile (Q1), which is the median of the lower half of the data, and the third quartile (Q3), which is the median of the upper half of the data. The interquartile range is then calculated by subtracting Q1 from Q3. The data set provided is: 36, 44, 44, 68, 71, 75 Step 1: Find the median to separate the lower and upper halves of the data. Since there's an even number of data points, the median is the average of the middle two values. The middle values are the third and fourth values: Median = (44 + 68) / 2 = 112 / 2 = 56 Step 2: Find Q1, the median of the lower half of the data (not including the median of the dataset if the number of observations is odd). The lower half of the data is 36, 44, and 44. The median of these values is directly the second number (since there are an odd number of data points in the lower half): Q1 = 44 Step 3: Find Q3, the median of the upper half of the data (not including the median of the dataset if the number of observations is odd). The upper half of the data is 68, 71, and 75. The median of these values is directly the second number: Q3 = 71 Step 4: Calculate the interquartile range (IQR), which is Q3 - Q1. IQR = Q3 - Q1 IQR = 71 - 44 IQR = 27 The interquartile range for the provided data set is 27.
Understanding Interquartile Range in a Dataset
The interquartile range (IQR) is the difference between the first quartile (Q1) and the third quartile (Q3) in a dataset. It measures the middle spread of the data and represents the range where the central 50% of the data points lie. The dataset given is: 36, 44, 44, 68, 71, 75. To find the quartiles, we first need to find the median of the dataset, which is the middle value when the numbers are arranged in order. If there is an even number of data points, as is the case here, the median is the average of the two middle numbers. The median is the average of the third and fourth data points: (44 + 68) / 2 = 112 / 2 = 56. Now, Q1 (the first quartile) is the median of the first half of the dataset, and Q3 (the third quartile) is the median of the second half. For Q1, considering the first half of the dataset: 36, 44, 44. Since there's an odd number of data points, the median is the middle number, which is 44. For Q3, considering the second half of the dataset: 68, 71, 75. Again, the median is the middle number, which is 71. So the interquartile range (IQR) is Q3 - Q1: 71 - 44 = 27. The interquartile range of the dataset is 27.
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.
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