To determine the upper and lower bounds for outliers, we need to compute the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1). Here's the step-by-step process:

Step 1: Organize the Data

The given data is already ordered: $18, 44, 47, 55, 61, 62, 78, 79, 83, 145$

Step 2: Calculate Q1 and Q3

• Q1 (First Quartile): This is the median of the first half of the data set (not including the overall median). $Q1 = \frac{44 + 47}{2} = 45.5$

• Q3 (Third Quartile): This is the median of the second half of the data set (not including the overall median). $Q3 = \frac{78 + 79}{2} = 78.5$

Step 3: Calculate the Interquartile Range (IQR)

$IQR = Q3 - Q1 = 78.5 - 45.5 = 33$

Step 4: Calculate the Outlier Bounds

The formulas for detecting outliers are:

• Lower Bound: $Q1 - 1.5 \times IQR$ $\text{Lower Bound} = 45.5 - 1.5 \times 33 = 45.5 - 49.5 = -4$

• Upper Bound: $Q3 + 1.5 \times IQR$ $\text{Upper Bound} = 78.5 + 1.5 \times 33 = 78.5 + 49.5 = 128$

Conclusion:

• Lower bound: $-4$
• Upper bound: $128$

Thus, any data point below $-4$ or above $128$ would be considered an outlier. In the given data set, the value $145$ is an outlier because it exceeds the upper bound.

Would you like further details or have any questions?

Related questions:

1. How do you calculate the median if the number of data points is odd?
2. What happens if the data contains multiple outliers?
3. Can outliers affect the mean of a data set?
4. What is the purpose of detecting outliers in a data set?
5. How does changing the value of 1.5 affect the range for outliers?

Tip:

Detecting outliers helps identify unusual values that might be caused by errors or significant variations in your data.