Imagine tracking the number of hours spent studying per week by students in a class: 5, 7, 10, 8, 6, 9, 4, 12, 8, and 6. Now identify and compute the appropriate measure of dispersion using the step-by-step process 

Step 1. Understand your data: 
Step 2. Consider the scale of measurement: 
Step 3. Assess data distribution:
Step 4: determine the presence of outliers: 
Step 5. Select the measure:
   1. Put the numbers in ascending order:
   2. Divide your data into a lower half and an upper half. Don't include the median if you have an odd number of data point:
    Lower half:
    Upper half:
  3. Find the middle of the lower half: 
  4. Find the middle of the upper half:
  5. Compute the IQR:
Step 6. Interpret result:

Let's go through the steps to compute the appropriate measure of dispersion, in this case, the **Interquartile Range (IQR)**, using the dataLet's go through the steps to compute the appropriate measure of dispersion, in this case, the **Interquartile Range (IQR)**, using the  5, 7, 10, 8, 6, 9, 4, 12, 8, and 6.

### Step 1: Understand your data
We are working with a set of **10 data points**, representing the number of hours students spent studying per week. This data set contains numerical, interval-level data since the values represent a measurable quantity (hours) where both the order and differences between values matter.

### Step 2: Consider the scale of measurement
The scale of measurement here is **interval**, meaning the data is numerical and the differences between values are meaningful.

### Step 3: Assess data distribution
Before calculating the measure of dispersion, it's helpful to know how the data is distributed. The data points vary from a low of 4 hours to a high of 12 hours, with no extreme gaps between most values, suggesting a somewhat even distribution.

### Step 4: Determine the presence of outliers
The IQR will help us assess outliers later. First, let's complete the next few steps and then revisit this point.

### Step 5: Select the measure
We will use the **Interquartile Range (IQR)** as the measure of dispersion.

#### 1. Put the numbers in ascending order:
4, 5, 6, 6, 7, 8, 8, 9, 10, 12

#### 2. Divide your data into a lower half and an upper half.
Since we have **10 data points**, the median will be the average of the 5th and 6th values. Here, the two middle numbers are 7 and 8, so:
\[
\text{Median} = \frac{7 + 8}{2} = 7.5
\]
Now, we divide the data into two halves, excluding the median:

- **Lower half:** 4, 5, 6, 6, 7
- **Upper half:** 8, 8, 9, 10, 12

#### 3. Find the middle of the lower half (Q1):
The middle of the lower half (the 1st quartile, \(Q_1\)) is the median of the lower half of the data. In this case:
\[
Q_1 = 6
\]

#### 4. Find the middle of the upper half (Q3):
The middle of the upper half (the 3rd quartile, \(Q_3\)) is the median of the upper half of the data. In this case:
\[
Q_3 = 9
\]

#### 5. Compute the IQR:
The **Interquartile Range (IQR)** is calculated by subtracting \(Q_1\) from \(Q_3\):
\[
\text{IQR} = Q_3 - Q_1 = 9 - 6 = 3
\]

### Step 6: Interpret result
The IQR is **3 hours**, meaning that the middle 50% of students spent between 6 and 9 hours studying per week. The IQR tells us how spread out the central portion of the data is and helps in identifying outliers. Outliers are typically considered data points below \(Q_1 - 1.5 \times \text{IQR}\) or above \(Q_3 + 1.5 \times \text{IQR}\).

In this case:
- Lower bound = \( Q_1 - 1.5 \times IQR = 6 - 1.5 \times 3 = 1.5 \)
- Upper bound = \( Q_3 + 1.5 \times IQR = 9 + 1.5 \times 3 = 13.5 \)

Since all data points fall between 1.5 and 13.5, **there are no outliers** in this dataset.

---

Do you want details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the IQR change if there was an outlier in the data?
2. How does the IQR compare to the standard deviation for measuring spread?
3. Can we use the IQR for skewed data, and why?
4. What other methods can be used to detect outliers?
5. How can we visualize the IQR in a box plot?

**Tip:** The IQR is especially useful when dealing with skewed data, as it isn't affected by extreme values like the mean or standard deviation.

Imagine tracking the number of hours spent studying per week by students in a class: 5, 7, 10, 8, 6, 9, 4, 12, 8, and 6. Now identify and compute the appropriate measure of dispersion using the step-by-step process.

Learn how to compute the Interquartile Range (IQR) for a dataset representing hours spent studying by students per week. This step-by-step guide covers key statistical concepts such as measures of dispersion, median, and identifying outliers. Suitable for students in Grades 8-10.

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