To solve the problem of finding the mean, median, mode, and mean deviation from the mean, median, and mode of the given marks obtained by 10 students, we will proceed step by step.
### Step 1: Organize the Data
First, we need to arrange the marks in ascending order.
**Given Marks:**
15, 10, 6, 15, 12, 9, 3, 5, 4, 2
**Sorted Marks:**
2, 3, 4, 5, 6, 9, 10, 12, 15, 15
### Step 2: Calculate the Mean
The mean is calculated using the formula:
\[
\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}
\]
**Sum of Observations:**
\[
2 + 3 + 4 + 5 + 6 + 9 + 10 + 12 + 15 + 15 = 81
\]
**Number of Observations:**
10
**Mean Calculation:**
\[
\text{Mean} = \frac{81}{10} = 8.1
\]
### Step 3: Calculate the Median
To find the median, we need to identify the middle value(s). Since there are 10 observations (an even number), the median will be the average of the 5th and 6th values.
**5th Value:** 6
**6th Value:** 9
**Median Calculation:**
\[
\text{Median} = \frac{6 + 9}{2} = \frac{15}{2} = 7.5
\]
### Step 4: Calculate the Mode
The mode is the value that appears most frequently in the data set.
In our sorted data:
2, 3, 4, 5, 6, 9, 10, 12, 15, 15
The value 15 appears twice, while all other values appear only once.
**Mode:**
15
### Step 5: Calculate Mean Deviation from Mean, Median, and Mode
Mean deviation is calculated using the formula:
\[
\text{Mean Deviation} = \frac{\sum |X - \text{Measure}|}{N}
\]
where \(X\) is each observation, "Measure" is the mean, median, or mode, and \(N\) is the number of observations.
#### Mean Deviation from Mean
1. Calculate \( |X - \text{Mean}| \):
- For 2: |2 - 8.1| = 6.1
- For 3: |3 - 8.1| = 5.1
- For 4: |4 - 8.1| = 4.1
- For 5: |5 - 8.1| = 3.1
- For 6: |6 - 8.1| = 2.1
- For 9: |9 - 8.1| = 0.9
- For 10: |10 - 8.1| = 1.9
- For 12: |12 - 8.1| = 3.9
- For 15: |15 - 8.1| = 6.9 (twice)
**Sum of Deviations from Mean:**
\[
6.1 + 5.1 + 4.1 + 3.1 + 2.1 + 0.9 + 1.9 + 3.9 + 6.9 + 6.9 = 41
\]
**Mean Deviation from Mean:**
\[
\text{Mean Deviation from Mean} = \frac{41}{10} = 4.1
\]
#### Mean Deviation from Median
1. Calculate \( |X - \text{Median}| \):
- For 2: |2 - 7.5| = 5.5
- For 3: |3 - 7.5| = 4.5
- For 4: |4 - 7.5| = 3.5
- For 5: |5 - 7.5| = 2.5
- For 6: |6 - 7.5| = 1.5
- For 9: |9 - 7.5| = 1.5
- For 10: |10 - 7.5| = 2.5
- For 12: |12 - 7.5| = 4.5
- For 15: |15 - 7.5| = 7.5 (twice)
**Sum of Deviations from Median:**
\[
5.5 + 4.5 + 3.5 + 2.5 + 1.5 + 1.5 + 2.5 + 4.5 + 7.5 + 7.5 = 42
\]
**Mean Deviation from Median:**
\[
\text{Mean Deviation from Median} = \frac{42}{10} = 4.2
\]
#### Mean Deviation from Mode
1. Calculate \( |X - \text{Mode}| \):
- For 2: |2 - 15| = 13
- For 3: |3 - 15| = 12
- For 4: |4 - 15| = 11
- For 5: |5 - 15| = 10
- For 6: |6 - 15| = 9
- For 9: |9 - 15| = 6
- For 10: |10 - 15| = 5
- For 12: |12 - 15| = 3
- For 15: |15 - 15| = 0 (twice)
**Sum of Deviations from Mode:**
\[
13 + 12 + 11 + 10 + 9 + 6 + 5 + 3 + 0 + 0 = 69
\]
**Mean Deviation from Mode:**
\[
\text{Mean Deviation from Mode} = \frac{69}{10} = 6.9
\]
### Final Results
- **Mean:** 8.1
- **Median:** 7.5
- **Mode:** 15
- **Mean Deviation from Mean:** 4.1
- **Mean Deviation from Median:** 4.2
- **Mean Deviation from Mode:** 6.9
