Let's break down the required statistical measures one by one for the given data set: 17, 9, 12, 11, 30, 26, 28, and 22.

Step 1: Order the Data

The data is: 17, 9, 12, 11, 30, 26, 28, 22.

First, we will order the data from smallest to largest: 9, 11, 12, 17, 22, 26, 28, 30.

Step 2: Calculate the Median

The median is the middle number in a sorted, ordered list. If there is an even number of observations, the median is the average of the two middle numbers.

Here, we have 8 data points, so the median is the average of the 4th and 5th numbers.

• 4th number = 17
• 5th number = 22

So, the median is:

$\text{Median} = \frac{17 + 22}{2} = \frac{39}{2} = 19.5$

Step 3: Calculate the Mean

The mean is the sum of all the numbers divided by the total number of data points.

$\text{Mean} = \frac{9 + 11 + 12 + 17 + 22 + 26 + 28 + 30}{8} = \frac{155}{8} = 19.375$

Step 4: Calculate the Mode

The mode is the value that appears most frequently in the data set.

In this case, all the numbers appear only once, so there is no mode.

Step 5: Calculate the Range

The range is the difference between the largest and smallest numbers in the data set.

$\text{Range} = 30 - 9 = 21$

Step 6: Calculate the Sample Standard Deviation

To calculate the sample standard deviation, we use the following formula:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Where:

• $x_i$ represents each individual data point.
• $\bar{x}$ is the mean of the data set.
• $n$ is the number of data points (8 in this case).

Steps:

1. Subtract the mean from each data point:

   • $9 - 19.375 = -10.375$
   • $11 - 19.375 = -8.375$
   • $12 - 19.375 = -7.375$
   • $17 - 19.375 = -2.375$
   • $22 - 19.375 = 2.625$
   • $26 - 19.375 = 6.625$
   • $28 - 19.375 = 8.625$
   • $30 - 19.375 = 10.625$

2. Square each of these differences:

   • $(-10.375)^2 = 107.640625$
   • $(-8.375)^2 = 70.140625$
   • $(-7.375)^2 = 54.515625$
   • $(-2.375)^2 = 5.640625$
   • $2.625^2 = 6.890625$
   • $6.625^2 = 43.890625$
   • $8.625^2 = 74.390625$
   • $10.625^2 = 113.390625$

3. Sum the squared differences:

$107.640625 + 70.140625 + 54.515625 + 5.640625 + 6.890625 + 43.890625 + 74.390625 + 113.390625 = 476.000$

4. Divide by $n - 1 = 7$ (since it's a sample, not a population):

$\frac{476.000}{7} = 68$

5. Take the square root to find the sample standard deviation:

$s = \sqrt{68} \approx 8.246$

Summary of Results:

• Median: 19.5
• Mean: 19.375
• Mode: None
• Range: 21
• Sample Standard Deviation: 8.25 (rounded to the nearest hundredth)

Let me know if you need further clarification or details on any of these calculations!

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Related Questions:

1. How do you calculate the population standard deviation instead of the sample standard deviation?
2. What is the effect of removing outliers from the data set on the mean and standard deviation?
3. Can a data set have more than one mode? If so, what is that called?
4. How do you calculate the variance of a data set?
5. How does the range provide insights about the spread of the data?

Tip: When calculating the standard deviation, remember that the formula divides by $n-1$ for a sample to account for the fact that you're estimating the population standard deviation, which makes it more accurate for small samples.