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Question 1
A dataset is given as $S = {3, 7, 11, 15, 19, 23, 27, 31, 35, 95}$ . Using the IB definition, which of the following values is an outlier?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $95$ only
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Find Q₁ and Q₃
Ordering the data: $3, 7, 11, 15, 19, 23, 27, 31, 35, 95$ . With 10 values, $Q_{1}$ is the median of the lower half ${3, 7, 11, 15, 19}$ , giving $Q_{1} = 11$ , and $Q_{3}$ is the median of the upper half ${23, 27, 31, 35, 95}$ , giving $Q_{3} = 31$ .
Step 2: Calculate the IQR
$IQR = Q_{3} - Q_{1} = 31 - 11 = 20$
Step 3: Calculate the fences
Lower fence: $Q_{1} - 1.5 \times IQR = 11 - 30 = - 19$ . Upper fence: $Q_{3} + 1.5 \times IQR = 31 + 30 = 61$ .
Step 4: Identify outliers
Any value below $- 19$ or above $61$ is an outlier. The value $3 > - 19$ , so it is not an outlier. The value $95 > 61$ , so 95 is an outlier. The correct answer is $95$ only.
Method #2Approach 2
Step 1: Identify what is being asked
We need to determine which data value(s) qualify as outliers using the IB rule: a value is an outlier if it lies more than $1.5 \times IQR$ beyond the nearest quartile.
Step 2: Eliminate 'Both 3 and 95'
The lower fence is $- 19$ . Since $3 > - 19$ , the value $3$ is within the fences and is not an outlier. So the option 'Both $3$ and $95$ ' is incorrect.
Step 3: Eliminate '3 only'
As established, $3$ lies well above the lower fence of $- 19$ . It cannot be the only outlier. This option is eliminated.
Step 4: Eliminate 'There are no outliers'
The upper fence is $61$ . Since $95 > 61$ , there is at least one outlier. This option is incorrect.
Step 5: Select the correct answer
Only $95$ exceeds the upper fence of $61$ . The correct answer is $95$ only.
Question 2
The ages (in years) of volunteers at a community event are recorded as: $18, 21, 22, 24, 25, 27, 29, 31, 33, 68$ . Which statement about the value $68$ is correct?
No clue? Show me the answer
Correct answerCorrect!Incorrect
AIt is an outlier because it exceeds the upper fence of $Q_{3} + 1.5 \times IQR$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Find Q₁ and Q₃
The 10 ordered values are: $18, 21, 22, 24, 25, 27, 29, 31, 33, 68$ . The lower half is ${18, 21, 22, 24, 25}$ , so $Q_{1} = 22$ . The upper half is ${27, 29, 31, 33, 68}$ , so $Q_{3} = 31$ .
Step 2: Compute IQR and upper fence
$IQR = 31 - 22 = 9$ $Upper fence = Q_{3} + 1.5 \times 9 = 31 + 13.5 = 44.5$
Step 3: Compare 68 to the fence
Since $68 > 44.5$ , the value $68$ is an outlier by the IB definition. The correct answer confirms this using the proper fence formula.
Method #2Approach 2
Step 1: Identify what is being asked
We must decide whether $68$ is an outlier and which reasoning is correct.
Step 2: Eliminate the 'not an outlier' option
The IQR is only $9$ , so the upper fence is $31 + 13.5 = 44.5$ . The value $68$ is well above $44.5$ , so the claim that the IQR is large enough to include it is false.
Step 3: Eliminate 'more than twice the next value'
The IB definition of an outlier does not use a ratio comparison with adjacent values. This reasoning is mathematically incorrect and not part of the IB syllabus.
Step 4: Eliminate 'cannot be determined without the mean'
The IB outlier rule uses $Q_{1}$ , $Q_{3}$ , and IQR — the mean is not needed. This option is incorrect.
Step 5: Select the correct answer
The only correct statement is that $68$ is an outlier because it exceeds the upper fence $Q_{3} + 1.5 \times IQR = 44.5$ .

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13 more questions in this topic

Free preview 2/15

Question 1
A dataset is given as $S = {3, 7, 11, 15, 19, 23, 27, 31, 35, 95}$ . Using the IB definition, which of the following values is an outlier?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $95$ only
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Find Q₁ and Q₃
Ordering the data: $3, 7, 11, 15, 19, 23, 27, 31, 35, 95$ . With 10 values, $Q_{1}$ is the median of the lower half ${3, 7, 11, 15, 19}$ , giving $Q_{1} = 11$ , and $Q_{3}$ is the median of the upper half ${23, 27, 31, 35, 95}$ , giving $Q_{3} = 31$ .
Step 2: Calculate the IQR
$IQR = Q_{3} - Q_{1} = 31 - 11 = 20$
Step 3: Calculate the fences
Lower fence: $Q_{1} - 1.5 \times IQR = 11 - 30 = - 19$ . Upper fence: $Q_{3} + 1.5 \times IQR = 31 + 30 = 61$ .
Step 4: Identify outliers
Any value below $- 19$ or above $61$ is an outlier. The value $3 > - 19$ , so it is not an outlier. The value $95 > 61$ , so 95 is an outlier. The correct answer is $95$ only.
Method #2Approach 2
Step 1: Identify what is being asked
We need to determine which data value(s) qualify as outliers using the IB rule: a value is an outlier if it lies more than $1.5 \times IQR$ beyond the nearest quartile.
Step 2: Eliminate 'Both 3 and 95'
The lower fence is $- 19$ . Since $3 > - 19$ , the value $3$ is within the fences and is not an outlier. So the option 'Both $3$ and $95$ ' is incorrect.
Step 3: Eliminate '3 only'
As established, $3$ lies well above the lower fence of $- 19$ . It cannot be the only outlier. This option is eliminated.
Step 4: Eliminate 'There are no outliers'
The upper fence is $61$ . Since $95 > 61$ , there is at least one outlier. This option is incorrect.
Step 5: Select the correct answer
Only $95$ exceeds the upper fence of $61$ . The correct answer is $95$ only.
Question 2
The ages (in years) of volunteers at a community event are recorded as: $18, 21, 22, 24, 25, 27, 29, 31, 33, 68$ . Which statement about the value $68$ is correct?
No clue? Show me the answer
Correct answerCorrect!Incorrect
AIt is an outlier because it exceeds the upper fence of $Q_{3} + 1.5 \times IQR$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Find Q₁ and Q₃
The 10 ordered values are: $18, 21, 22, 24, 25, 27, 29, 31, 33, 68$ . The lower half is ${18, 21, 22, 24, 25}$ , so $Q_{1} = 22$ . The upper half is ${27, 29, 31, 33, 68}$ , so $Q_{3} = 31$ .
Step 2: Compute IQR and upper fence
$IQR = 31 - 22 = 9$ $Upper fence = Q_{3} + 1.5 \times 9 = 31 + 13.5 = 44.5$
Step 3: Compare 68 to the fence
Since $68 > 44.5$ , the value $68$ is an outlier by the IB definition. The correct answer confirms this using the proper fence formula.
Method #2Approach 2
Step 1: Identify what is being asked
We must decide whether $68$ is an outlier and which reasoning is correct.
Step 2: Eliminate the 'not an outlier' option
The IQR is only $9$ , so the upper fence is $31 + 13.5 = 44.5$ . The value $68$ is well above $44.5$ , so the claim that the IQR is large enough to include it is false.
Step 3: Eliminate 'more than twice the next value'
The IB definition of an outlier does not use a ratio comparison with adjacent values. This reasoning is mathematically incorrect and not part of the IB syllabus.
Step 4: Eliminate 'cannot be determined without the mean'
The IB outlier rule uses $Q_{1}$ , $Q_{3}$ , and IQR — the mean is not needed. This option is incorrect.
Step 5: Select the correct answer
The only correct statement is that $68$ is an outlier because it exceeds the upper fence $Q_{3} + 1.5 \times IQR = 44.5$ .

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Step-by-step walkthrough

Step 1: Find Q₁ and Q₃

Step 2: Calculate the IQR

Step 3: Calculate the fences

Step 4: Identify outliers

Step 1: Identify what is being asked

Step 2: Eliminate 'Both 3 and 95'

Step 3: Eliminate '3 only'

Step 4: Eliminate 'There are no outliers'

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Find Q₁ and Q₃

Step 2: Compute IQR and upper fence

Step 3: Compare 68 to the fence

Step 1: Identify what is being asked

Step 2: Eliminate the 'not an outlier' option

Step 3: Eliminate 'more than twice the next value'

Step 4: Eliminate 'cannot be determined without the mean'

Step 5: Select the correct answer

13 more questions in this topic

Step-by-step walkthrough

Step 1: Find Q₁ and Q₃

Step 2: Calculate the IQR

Step 3: Calculate the fences

Step 4: Identify outliers

Step 1: Identify what is being asked

Step 2: Eliminate 'Both 3 and 95'

Step 3: Eliminate '3 only'

Step 4: Eliminate 'There are no outliers'

Step 5: Select the correct answer

Step-by-step walkthrough

Step 1: Find Q₁ and Q₃

Step 2: Compute IQR and upper fence

Step 3: Compare 68 to the fence

Step 1: Identify what is being asked

Step 2: Eliminate the 'not an outlier' option

Step 3: Eliminate 'more than twice the next value'

Step 4: Eliminate 'cannot be determined without the mean'

Step 5: Select the correct answer

13 more questions in this topic