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.

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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.