The scores of a batsman in ten innings are:
48, 80, 58, 44, 52, 65, 73, 56, 64, 54. Find the mean deviation from the median.

Answer

To find the mean deviation from the median for the given scores of a batsman in ten innings, we will follow these steps:

### Step 1: Arrange the Scores in Ascending Order
First, we need to arrange the scores in ascending order.

**Scores:** 48, 80, 58, 44, 52, 65, 73, 56, 64, 54
**Ascending Order:** 44, 48, 52, 54, 56, 58, 64, 65, 73, 80

### Step 2: Find the Median
Since there are 10 scores (an even number), the median will be the average of the 5th and 6th scores in the ordered list.

**5th Score:** 56
**6th Score:** 58
**Median (M):** (56 + 58) / 2 = 114 / 2 = 57

### Step 3: Calculate the Mean Deviation from the Median
The mean deviation from the median is calculated using the formula:

\[ \text{Mean Deviation} = \frac{1}{n} \sum_{i=1}^{n} |x_i - M| \]

Where:
- \( n \) = number of observations (10)
- \( x_i \) = each score
- \( M \) = median (57)

### Step 4: Calculate the Absolute Deviations
Now, we will calculate the absolute deviations from the median for each score:

1. |44 - 57| = 13
2. |48 - 57| = 9
3. |52 - 57| = 5
4. |54 - 57| = 3
5. |56 - 57| = 1
6. |58 - 57| = 1
7. |64 - 57| = 7
8. |65 - 57| = 8
9. |73 - 57| = 16
10. |80 - 57| = 23

### Step 5: Sum the Absolute Deviations
Now we will sum all the absolute deviations calculated above:

\[ \text{Total Absolute Deviations} = 13 + 9 + 5 + 3 + 1 + 1 + 7 + 8 + 16 + 23 = 86 \]

### Step 6: Calculate the Mean Deviation
Now we can calculate the mean deviation:

\[ \text{Mean Deviation} = \frac{86}{10} = 8.6 \]

### Final Answer
The mean deviation from the median is **8.6**.

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