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### Given Data:
- Numbers: 2, 4, 5, 7, 8

#### a. Find the median of the distribution.
To find the **median**, arrange the numbers in order (already done) and pick the middle number. Since there are 5 numbers, the middle one is:
- Median = 5

#### b. Find the mean of the distribution.
The **mean** is calculated by adding all the numbers together and dividing by how many there are.

\[
\text{Mean} = \frac{2 + 4 + 5 + 7 + 8}{5} = \frac{26}{5} = 5.2
\]

#### c. Find the standard deviation of the distribution.
The **standard deviation** is calculated using the formula:

\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}
\]
Where \( \mu \) is the mean and \( n \) is the number of elements.

1. Subtract the mean from each number:
   - \( 2 - 5.2 = -3.2 \)
   - \( 4 - 5.2 = -1.2 \)
   - \( 5 - 5.2 = -0.2 \)
   - \( 7 - 5.2 = 1.8 \)
   - \( 8 - 5.2 = 2.8 \)

2. Square each result:
   - \( (-3.2)^2 = 10.24 \)
   - \( (-1.2)^2 = 1.44 \)
   - \( (-0.2)^2 = 0.04 \)
   - \( (1.8)^2 = 3.24 \)
   - \( (2.8)^2 = 7.84 \)

3. Sum these squares:
   - \( 10.24 + 1.44 + 0.04 + 3.24 + 7.84 = 22.8 \)

4. Divide by the number of elements (5):
   - \( \frac{22.8}{5} = 4.56 \)

5. Take the square root:
   - \( \sqrt{4.56} \approx 2.14 \)

So, the standard deviation is approximately **2.14**.

#### d. Make a new distribution by replacing 8 with 800. Find the mean, median, and standard deviation.
New distribution: 2, 4, 5, 7, 800.

1. **Mean**:
\[
\text{Mean} = \frac{2 + 4 + 5 + 7 + 800}{5} = \frac{818}{5} = 163.6
\]

2. **Median**:
   Since the middle number is still 5, the median remains **5**.

3. **Standard deviation**:
   Repeating the same process as above with the new numbers:
   
   - \( 2 - 163.6 = -161.6 \)
   - \( 4 - 163.6 = -159.6 \)
   - \( 5 - 163.6 = -158.6 \)
   - \( 7 - 163.6 = -156.6 \)
   - \( 800 - 163.6 = 636.4 \)

   Squaring these:
   - \( (-161.6)^2 = 26114.56 \)
   - \( (-159.6)^2 = 25459.36 \)
   - \( (-158.6)^2 = 25153.96 \)
   - \( (-156.6)^2 = 24513.96 \)
   - \( (636.4)^2 = 405010.56 \)

   Summing these squares:
   - \( 26114.56 + 25459.36 + 25153.96 + 24513.96 + 405010.56 = 506252.4 \)

   Divide by 5:
   - \( \frac{506252.4}{5} = 101250.48 \)

   Square root:
   - \( \sqrt{101250.48} \approx 318.1 \)

So, the new standard deviation is approximately **318.1**.

#### e. Which statistic is most changed by the outlier?
The **standard deviation** changes the most, jumping from approximately 2.14 to 318.1, while the mean also changes significantly (from 5.2 to 163.6). The **median**, however, stays the same at 5, indicating that the median is more resistant to outliers.

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Would you like further details or clarification on any part of this? Here are 5 related questions for deeper exploration:

1. How would the range of the distribution change with the outlier?
2. What does it mean for a measure (like the median) to be resistant to outliers?
3. How would adding another small number, like 1, affect the mean and standard deviation?
4. How do you interpret the effect of a large outlier on data analysis?
5. Can you think of a real-life example where the mean is misleading due to an outlier?

**Tip:** The median is often a better measure of central tendency than the mean when the data contains significant outliers.

Given numbers 2, 4, 5, 7, 8, calculate the median, mean, and standard deviation. Then, replace 8 with 800 and recalculate these values. Identify which statistic is most affected by the outlier.

Learn how to calculate the mean, median, and standard deviation for the dataset 2, 4, 5, 7, 8, and observe the effect of replacing 8 with 800. Understand how outliers significantly change certain statistics while leaving others resistant.

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