Given the following sample of 20 numbers:
15 45 52 43 50 59 41 47 56 79 72 18 45 54 78 12 41 48 58 14
(i) Compute mean, variance and standard deviation.
(ii) If the largest value in the above set of numbers is changed to 500, to what extent are the mean
and variance affected by the change? Justify your answer.

Let's solve the given problem step-by-step.

### Part (i): Compute the Mean, Variance, and Standard Deviation

1. **Mean**: The mean (average) of a set of numbers is calculated by summing all the numbers and dividing by the count of numbers.

\[
\text{Mean} (\mu) = \frac{\sum x_i}{n}
\]

2. **Variance**: The variance measures how much the numbers in the set differ from the mean. It is the average of the squared differences from the mean.

\[
\text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n}
\]

3. **Standard Deviation**: The standard deviation is the square root of the variance.

\[
\text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}}
\]

Let's calculate these using the provided data set:

#### Data Set
\[
15, 45, 52, 43, 50, 59, 41, 47, 56, 79, 72, 18, 45, 54, 78, 12, 41, 48, 58, 14
\]

#### Step-by-Step Calculation

**1. Calculate the Mean:**

\[
\mu = \frac{15 + 45 + 52 + 43 + 50 + 59 + 41 + 47 + 56 + 79 + 72 + 18 + 45 + 54 + 78 + 12 + 41 + 48 + 58 + 14}{20}
\]

**2. Calculate the Variance:**

\[
\sigma^2 = \frac{(15 - \mu)^2 + (45 - \mu)^2 + \cdots + (14 - \mu)^2}{20}
\]

**3. Calculate the Standard Deviation:**

\[
\sigma = \sqrt{\sigma^2}
\]

Let's compute these values.**Results for Part (i):**

1. **Mean (μ):** 46.35
2. **Variance (σ²):** 365.33
3. **Standard Deviation (σ):** 19.11

### Part (ii): Effect of Changing the Largest Value to 500

Now, let's analyze how the mean and variance change when the largest value (79) in the dataset is replaced with 500.

#### New Data Set
\[
15, 45, 52, 43, 50, 59, 41, 47, 56, 500, 72, 18, 45, 54, 78, 12, 41, 48, 58, 14
\]

We will calculate the new mean and variance using the same formulas:

\[
\mu_{\text{new}} = \frac{\sum x_{\text{new}}}{n}
\]

\[
\sigma_{\text{new}}^2 = \frac{\sum (x_{\text{new}} - \mu_{\text{new}})^2}{n}
\]

Let's compute the new mean and variance.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can manually compute the new mean and variance with the following steps:

1. **New Mean Calculation:**
   - Sum all numbers in the new data set (with 500 replacing 79) and divide by 20.

2. **New Variance Calculation:**
   - Subtract the new mean from each number, square the result, and then take the average of these squared differences.

If you have any questions or need further assistance, feel free to ask!

### Relative Questions
1. How does changing an outlier affect the standard deviation?
2. What is the difference between sample variance and population variance?
3. How would you calculate the median for this dataset?
4. Can the mean ever be greater than all values in a dataset?
5. What is the impact of having more extreme values on the variance?

### Tip
When a dataset includes an extreme value (outlier), it can significantly affect the mean and variance, making them less representative of the central tendency and variability of the data.

Learn how to compute the mean, variance, and standard deviation of a dataset. Includes step-by-step calculations and analysis. Explore the impact of changing an outlier on statistical measures. Suitable for high school students studying statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation