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Statement-1 : The median of a set of 9 d...

Statement-1 : The median of a set of 9 distinct observations is 20.5. if each of the largest 4 observations of the set Is increased by 2 then median of new set remains the same as that of the original.
Statement-2 : If the variable of a series are arranged in ascending or descending order, then the value of the middle variable is defined as median .

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To solve the given problem, we need to analyze both statements provided and determine their validity. ### Step-by-Step Solution: **Step 1: Understanding the Median in a Set of Observations** - The median of a set of observations is the middle value when the observations are arranged in ascending or descending order. For a set of 9 distinct observations, the median is the 5th observation when arranged in order. **Step 2: Identifying the Given Median** - We are given that the median of the 9 distinct observations is 20.5. This means that when arranged, the 5th observation (x5) is equal to 20.5. **Step 3: Increasing the Largest 4 Observations** - The problem states that if we increase each of the largest 4 observations by 2, we need to check if the median remains the same. The largest 4 observations are x6, x7, x8, and x9. **Step 4: Analyzing the Impact on the Median** - After increasing x6, x7, x8, and x9 by 2, the new observations will be: - New x6 = x6 + 2 - New x7 = x7 + 2 - New x8 = x8 + 2 - New x9 = x9 + 2 - Since x1, x2, x3, x4, and x5 remain unchanged, the new set of observations will still have x5 as the 5th observation when arranged in order. **Step 5: Conclusion on the Median** - The median of the new set will still be the 5th observation, which is x5 = 20.5. Therefore, the median remains the same after the increase. **Step 6: Evaluating Statement 1 and Statement 2** - **Statement 1** is true because the median remains unchanged after the increase. - **Statement 2** is false because it does not specify that the number of observations is odd, which is necessary for defining the median as the middle observation. ### Final Answer: - Statement 1: True - Statement 2: False
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Median of a distribution is the value of the variable which divides it into equal parts . In case of individual observations x_1,x_2 .. x_n , if the number of observations is odd, then median is the value of ((n+1)/2) th observation when the observations have been arranged in ascending or descending order of magnitude. In case of even number of observations median is the A.M. of the values of (n/2) th and (n/2+1) th observations, arranged in ascending or descending order of magnitude . The mode of distribution is that value of the variable for which the frequency is maximum . The mode of following series is {:("value (x)",40,44,48,52,56,62,64,72,76),("frequency (f)",10,12,14,20,15,19,18,8,4):}