Find the mean deviation about the median for the following data:
xi: 3, 6, 9, 12, 13, 15, 21, 22
fi: 3, 4, 5, 2, 4, 5, 4, 3.
Get the complete, step-by-step math solution for: "Find the mean deviation about the median for the following data: x_{i}: 3, 6, 9, 12, 13, 15, 21, 22 f_{i}: 3, 4, 5, 2, 4, 5, 4, 3.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate the cumulative frequency and total number of observations
First, we need to organize the given data in a frequency distribution table and calculate the cumulative frequency (cf). The cumulative frequency for each class is the sum of its frequency and the frequencies of all preceding classes. We also find the total number of observations, N, by summing all frequencies.
Step 2: Determine the median
For discrete data, the median is the value of the (2N)th observation if N is even. Here, N=30, so we look for the 15th observation. By checking the cumulative frequency column, we find that the 15th observation falls in the class where xi=13.
Step 3: Calculate absolute deviations from the median
Next, we calculate the absolute deviation of each observation from the median, ∣xi−M∣, where M=13. Then, we multiply each absolute deviation by its corresponding frequency, fi∣xi−M∣, and sum these products to get ∑fi∣xi−M∣.
Step 4: Calculate the mean deviation about the median
Finally, we calculate the mean deviation about the median using the formula: N∑fi∣xi−M∣. We substitute the sum of the products of frequencies and absolute deviations, which is 149, and the total number of observations, which is 30.
Step 5: Simplify the result
Dividing 149 by 30 gives us the mean deviation about the median.