To solve the problem, we need to find the mean, median, and mode of the given data set: 25, 15, 20, 20, 9, 20, 25, 15, 7, 13, 20, 12, 10, 15, 8.
### Step 1: Organize the Data
First, let's arrange the data in ascending order:
- Data: 7, 8, 9, 10, 12, 13, 15, 15, 15, 20, 20, 20, 20, 25, 25
### Step 2: Calculate the Mean
The mean is calculated by taking the sum of all observations and dividing it by the number of observations.
1. **Sum of observations**:
\[
7 + 8 + 9 + 10 + 12 + 13 + 15 + 15 + 15 + 20 + 20 + 20 + 20 + 25 + 25
\]
- Calculating step by step:
- \(7 + 8 = 15\)
- \(15 + 9 = 24\)
- \(24 + 10 = 34\)
- \(34 + 12 = 46\)
- \(46 + 13 = 59\)
- \(59 + 15 = 74\)
- \(74 + 15 = 89\)
- \(89 + 20 = 109\)
- \(109 + 20 = 129\)
- \(129 + 20 = 149\)
- \(149 + 25 = 174\)
- \(174 + 25 = 199\)
- **Total Sum = 199**
2. **Number of observations (n)**: There are 15 observations.
3. **Mean Calculation**:
\[
\text{Mean} = \frac{\text{Sum of observations}}{n} = \frac{199}{15} \approx 13.27
\]
### Step 3: Calculate the Median
The median is the middle value when the data is arranged in ascending order. Since there are 15 observations (an odd number), the median will be the value at position \(\frac{n + 1}{2}\).
1. **Position of Median**:
\[
\text{Position} = \frac{15 + 1}{2} = 8
\]
- The 8th value in the ordered list is 15.
2. **Median**:
\[
\text{Median} = 15
\]
### Step 4: Calculate the Mode
The mode is the value that appears most frequently in the data set.
1. **Frequency of each value**:
- 7: 1 time
- 8: 1 time
- 9: 1 time
- 10: 1 time
- 12: 1 time
- 13: 1 time
- 15: 3 times
- 20: 4 times
- 25: 2 times
2. **Maximum frequency**: The value 20 appears 4 times, which is more than any other value.
3. **Mode**:
\[
\text{Mode} = 20
\]
### Final Results
- **Mean**: approximately 13.27
- **Median**: 15
- **Mode**: 20
