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It looks like you need to calculate the **mean**, **median**, **mode**, **range**, and **standard deviation** for the given population of numbers:  
\[ 5, 9, 23, 20, 20, 11, 20, 20 \]

Let's go step by step and correct or verify your answers:

1. **Mean (Average)**:
   \[
   \text{Mean} = \frac{\text{Sum of all numbers}}{\text{Number of numbers}} = \frac{5 + 9 + 23 + 20 + 20 + 11 + 20 + 20}{8}
   \]
   Let me calculate that for you.

2. **Median**:
   - First, order the numbers:  
     \[ 5, 9, 11, 20, 20, 20, 20, 23 \]
   - Since there are 8 numbers, the median is the average of the 4th and 5th values:
     \[
     \text{Median} = \frac{20 + 20}{2} = 20
     \]
   - Your median of 20 is correct.

3. **Mode**:
   - The mode is the number that appears most frequently.
   - In this case, 20 appears **4 times**, so the mode is 20.
   - Your mode of 20 is also correct.

4. **Range**:
   - The range is the difference between the largest and smallest values:
     \[
     \text{Range} = 23 - 5 = 18
     \]
   - Your range of 18 is correct.

5. **Standard Deviation**:
   I'll calculate the standard deviation using the formula for population standard deviation:
   \[
   \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}
   \]
   where \( \mu \) is the mean and \( N \) is the number of values.

Let me complete these calculations.Here are the correct calculations for the given set of numbers:

1. **Mean**: 16.00  
2. **Median**: 20  
3. **Mode**: 20  
4. **Range**: 18  
5. **Standard Deviation**: 6.20 (rounded to two decimals)

Your entries for the median, mode, and range are correct, but the mean should be **16.00**, and the standard deviation should be **6.20**.

Would you like more details on any of these calculations or further clarification?

Here are some related questions to think about:
1. How does the presence of outliers affect the mean and standard deviation?
2. Why does the standard deviation measure variability in a dataset?
3. What would happen to the median if one of the values changed drastically?
4. How do the mode and median help in understanding the shape of a dataset?
5. What is the difference between population standard deviation and sample standard deviation?

**Tip**: When calculating standard deviation, remember that small datasets are sensitive to changes in values, so always double-check your inputs!

Find the mode, median, mean, range, and standard deviation for the population of numbers: 5, 9, 23, 20, 20, 11, 20, 20.

Learn how to find the mean, median, mode, range, and standard deviation of a set of numbers. This example covers descriptive statistics, suitable for college-level statistics courses.

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