Let's calculate the range, sample standard deviation, and inter-quartile range (IQR) for the given data set:

Given Data Set:

21, 35, 38, 40, 40, 42, 45, 47, 61, 85

1. Range

The range is the difference between the maximum and minimum values of the data set.

• Maximum value: 85
• Minimum value: 21

$\text{Range} = 85 - 21 = 64$

2. Inter-Quartile Range (IQR)

The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). To calculate the quartiles:

• Sort the data (already sorted):
  21, 35, 38, 40, 40, 42, 45, 47, 61, 85

The data set has 10 numbers, so:

• $Q1$ is the median of the first half of the data (excluding the overall median).
  First half: 21, 35, 38, 40, 40
  Median of this set is $Q1 = 38$.

• $Q3$ is the median of the second half of the data (excluding the overall median).
  Second half: 42, 45, 47, 61, 85
  Median of this set is $Q3 = 47$.

$\text{IQR} = Q3 - Q1 = 47 - 38 = 9$

3. Sample Standard Deviation

The sample standard deviation is calculated as:

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2}$

Where:

• $n$ = 10 (number of data points)
• $\overline{x}$ = mean of the data
• $x_i$ = each data point

Step 1: Find the Mean ($\overline{x}$)

$\overline{x} = \frac{21 + 35 + 38 + 40 + 40 + 42 + 45 + 47 + 61 + 85}{10} = \frac{454}{10} = 45.4$

Step 2: Calculate $(x_i - \overline{x})^2$ for each data point:

$(21 - 45.4)^2 = 600.25, \quad (35 - 45.4)^2 = 108.16, \quad \dots$

The sum of squared differences is:
$600.25 + 108.16 + 55.76 + 28.96 + 28.96 + 11.56 + 0.16 + 2.56 + 244.36 + 1576.96 = 2657.2$

Step 3: Calculate the Standard Deviation:

$s = \sqrt{\frac{2657.2}{10 - 1}} = \sqrt{\frac{2657.2}{9}} = \sqrt{295.24} \approx 17.18$

Summary:

• Range: 64
• Inter-Quartile Range (IQR): 9
• Sample Standard Deviation: 17.18

Would you like more details or explanations on any of these steps?

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Here are 5 related questions:

1. How does the IQR help in identifying outliers?
2. What is the difference between sample and population standard deviation?
3. How would the standard deviation change if you added a constant value to each data point?
4. Can you explain why the range is not always a reliable measure of spread?
5. How does the shape of the data distribution affect the quartiles?

Tip: The IQR is often a better measure of spread than the range in skewed distributions, as it focuses on the middle 50% of the data.