The mean absolute deviation about the median for the data 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21 (rounded off to two decimal places) is ________________.
To find the Mean Absolute Deviation (MAD) about the median for the given data, follow these steps:
1. Order the Data
First, arrange the dataset in ascending order:
Dataset: 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21
Ordered Dataset: 3, 3, 4, 5, 7, 9, 10, 12, 18, 19, 21
There are $ N = 11 $ observations.
2. Calculate the Median
Since there are 11 data points (an odd number), the median is the middle value. The position of the median is $ \frac{N+1}{2} $.
Median Position = $ \frac{11+1}{2} = \frac{12}{2} = 6 $.
The 6th value in the ordered dataset is 9.
Therefore, the Median = 9.
3. Calculate Absolute Deviations from Median
Calculate the absolute difference between each data point ($ x_i $) and the median (9).
| Data Point ($ x_i $) | Absolute Deviation ($ |x_i - 9| $) |
|---|---|
| 3 | $ |3 - 9| = 6 $ |
| 3 | $ |3 - 9| = 6 $ |
| 4 | $ |4 - 9| = 5 $ |
| 5 | $ |5 - 9| = 4 $ |
| 7 | $ |7 - 9| = 2 $ |
| 9 | $ |9 - 9| = 0 $ |
| 10 | $ |10 - 9| = 1 $ |
| 12 | $ |12 - 9| = 3 $ |
| 18 | $ |18 - 9| = 9 $ |
| 19 | $ |19 - 9| = 10 $ |
| 21 | $ |21 - 9| = 12 $ |
The absolute deviations are: 6, 6, 5, 4, 2, 0, 1, 3, 9, 10, 12.
4. Calculate Mean Absolute Deviation (MAD)
Sum the absolute deviations:
$ \text{Sum} = 6 + 6 + 5 + 4 + 2 + 0 + 1 + 3 + 9 + 10 + 12 = 58 $
Calculate the mean by dividing the sum by the number of observations ($ N $):
$ \text{MAD} = \frac{\sum |x_i - \text{Median}|}{N} = \frac{58}{11} $
$ \text{MAD} \approx 5.2727... $
Rounding to two decimal places, the MAD is 5.27.
This value, 5.27, falls within the range of 5.26 to 5.28.
Important Questions from Mean Median Mode
- Five integers are picked from 0 to 20, with possible repetitions, such that their mean is 12, median is 18, and they have a single mode of 20.
Ignoring permutations, the number of ways to pick these five integers is _____ A data set is given to be $[1, 2, 0, -1, -3, 1, 2, 0, 1]$.
The median of the data set is ____. (rounded off to the nearest integer)
- The following sequence of numbers is arranged in increasing order: 1, x, x, x, y, y, 9,16,18. Given that the mean and median are equal, and are also equal to twice the mode, the value of y is
- The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ________.
- In a company with 100 employees, 45 earn Rs. 20,000 per month, 25 earn Rs. 30,000, 20 earn Rs. 40,000, 8 earn Rs. 60,000, and 2 earn Rs. 150,000. The median of the salaries is
