Find the range, sample standard deviation and inter-quartile range (IQR) of the following data set.


21	35	38	40	40	42	45	47	61	85


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?

---

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.

Find the range, sample standard deviation and inter-quartile range (IQR) of the following data set: 21, 35, 38, 40, 40, 42, 45, 47, 61, 85

Learn how to calculate the range, sample standard deviation, and inter-quartile range (IQR) for the data set: 21, 35, 38, 40, 40, 42, 45, 47, 61, 85. This problem involves key concepts of descriptive statistics, perfect for students in Grades 9-12.

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