Question

find the mean.
μ = 4.55
find the variance.
σ² = 1.4124
find the standard deviation. round to four decimal places, if necessary.
σ = 1.1884

Explanation:

1. Assume we have a data - set \(x_1,x_2,\cdots,x_n\):

• Step 1: Recall the formula for the mean \(\mu\)
• The mean \(\mu=\frac{\sum_{i = 1}^{n}x_i}{n}\). Since we are not given the data - set, we can't calculate it from scratch. But if we assume we have all the values \(x_i\), we first sum them up (\(\sum_{i=1}^{n}x_i\)) and then divide by the number of data points \(n\).
• Step 2: Recall the formula for the variance \(\sigma^{2}\)
• The variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\mu)^2}{n}\). After finding the mean \(\mu\), we subtract the mean from each data point \(x_i\), square the result \((x_i - \mu)^2\), sum up all these squared differences \(\sum_{i=1}^{n}(x_i - \mu)^2\), and then divide by \(n\).
• Step 3: Recall the formula for the standard deviation \(\sigma\)
• The standard deviation \(\sigma=\sqrt{\sigma^{2}}\). Once we have the variance \(\sigma^{2}\), we take the square - root of it to get the standard deviation.

1. Since no data - set is provided, we can't calculate the exact values:

• But if we assume we need to correct the given wrong values based on the formulas:
• Let's assume we have a data - set \(x_1,x_2,\cdots,x_n\). First, calculate the mean \(\mu=\frac{\sum_{i = 1}^{n}x_i}{n}\).
• Then, calculate the variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i - \mu)^2}{n}\).
• And the standard deviation \(\sigma=\sqrt{\sigma^{2}}\).
• Without the data - set, we can't give the correct numerical answers. But the general steps are as above.

Since we don't have the data - set, we can't provide the correct numerical answers. If we had the data - set values \(x_1,x_2,\cdots,x_n\):

Step1: Calculate the sum of data points

Let \(S=\sum_{i = 1}^{n}x_i\). Then the mean \(\mu=\frac{S}{n}\).

Step2: Calculate squared differences

For each \(i\) from \(1\) to \(n\), calculate \(d_i=(x_i - \mu)^2\). Then the variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}d_i}{n}\).

Step3: Calculate standard deviation

The standard deviation \(\sigma=\sqrt{\sigma^{2}}\).

Answer:

Since no data - set is given, we can't provide the correct values for \(\mu\), \(\sigma^{2}\), and \(\sigma\).