To solve the problem, we need to analyze how the reported weights of the horses change when one horse's weight is corrected. We will determine which of the four statistical values (mean, median, range, and standard deviation) remains unchanged after this correction.
### Step-by-Step Solution:
1. **Identify the Original Weights**:
Let's denote the weights of the 15 horses as \( x_1, x_2, x_3, \ldots, x_{15} \), where \( x_1 \) is the lowest weight and \( x_{15} \) is the highest weight.
2. **Correct the Weight of the Lowest Horse**:
The horse with the lowest weight \( x_1 \) is found to weigh 10 pounds less than reported. Therefore, the corrected weight for this horse will be \( x_1 - 10 \). The weights now become \( x_1 - 10, x_2, x_3, \ldots, x_{15} \).
3. **Calculate the Mean**:
The mean of the original data is given by:
\[
\text{Mean}_{\text{original}} = \frac{x_1 + x_2 + x_3 + \ldots + x_{15}}{15}
\]
The mean of the new data becomes:
\[
\text{Mean}_{\text{new}} = \frac{(x_1 - 10) + x_2 + x_3 + \ldots + x_{15}}{15} = \frac{x_1 + x_2 + x_3 + \ldots + x_{15} - 10}{15} = \text{Mean}_{\text{original}} - \frac{10}{15}
\]
Thus, the mean changes.
4. **Determine the Median**:
The median of the original data is the middle value when the weights are arranged in order. Since there are 15 weights, the median is \( x_8 \) (the 8th value).
In the new data set, the order remains the same, and since \( x_1 - 10 \) is still the lowest weight, the 8th value remains \( x_8 \). Therefore, the median does not change.
5. **Calculate the Range**:
The range is defined as the difference between the highest and lowest weights:
\[
\text{Range}_{\text{original}} = x_{15} - x_1
\]
The new range becomes:
\[
\text{Range}_{\text{new}} = x_{15} - (x_1 - 10) = x_{15} - x_1 + 10
\]
Thus, the range changes.
6. **Evaluate the Standard Deviation**:
The standard deviation measures the spread of the data points from the mean. Since the mean has changed and one of the data points has also changed, the standard deviation will also change.
### Conclusion:
After analyzing the changes in the four statistical values, we find that:
- **Mean**: Changes
- **Median**: Remains unchanged
- **Range**: Changes
- **Standard Deviation**: Changes
Thus, the value that remains unchanged is the **median**.
