To find the mean deviation from the median for the given data set, we will follow these steps:
### Step 1: Arrange the data in ascending order
The given data is:
36, 72, 46, 42, 60, 45, 53, 46, 51, 49.
Arranging this data in ascending order, we get:
36, 42, 45, 46, 46, 49, 51, 53, 60, 72.
### Step 2: Find the number of observations (n)
Count the number of observations in the data set:
n = 10 (there are 10 numbers).
### Step 3: Calculate the median
Since n is even, we use the formula for the median:
\[
\text{Median} (M) = \frac{\text{n/2 th observation} + \text{(n/2 + 1) th observation}}{2}
\]
Here, n/2 = 10/2 = 5, so we need the 5th and 6th observations.
From the ordered data:
- 5th observation = 46
- 6th observation = 49
Now, substituting these values into the median formula:
\[
M = \frac{46 + 49}{2} = \frac{95}{2} = 47.5
\]
### Step 4: Calculate the mean deviation from the median
The formula for mean deviation (MD) from the median is:
\[
MD = \frac{\sum |x_i - M|}{n}
\]
Where \(x_i\) are the data points and M is the median.
Now we will calculate \(|x_i - M|\) for each observation:
- For 36: \(|36 - 47.5| = 11.5\)
- For 42: \(|42 - 47.5| = 5.5\)
- For 45: \(|45 - 47.5| = 2.5\)
- For 46: \(|46 - 47.5| = 1.5\)
- For 46: \(|46 - 47.5| = 1.5\)
- For 49: \(|49 - 47.5| = 1.5\)
- For 51: \(|51 - 47.5| = 3.5\)
- For 53: \(|53 - 47.5| = 5.5\)
- For 60: \(|60 - 47.5| = 12.5\)
- For 72: \(|72 - 47.5| = 24.5\)
Now, summing these absolute deviations:
\[
\sum |x_i - M| = 11.5 + 5.5 + 2.5 + 1.5 + 1.5 + 1.5 + 3.5 + 5.5 + 12.5 + 24.5 = 70
\]
### Step 5: Calculate the mean deviation
Now, we divide the total absolute deviation by the number of observations:
\[
MD = \frac{70}{10} = 7
\]
### Final Answer
The mean deviation from the median is **7**.
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