Identifying Outliers

Show your work for each problem.

1.
What is an outlier?
1. a)The difference between Q3 and Q1
2. b)The middle value of the ordered data
3. c)A value unusually far from the rest of the data
4. d)The value that appears most often
Answer: A value unusually far from the rest of the data
An outlier is a value that lies unusually far from the other values, much smaller or much larger. The 1.5 × IQR rule decides exactly how far is "unusually far".
2.
A data set has $Q_1 = 10$ and $Q_3 = 18$ . What is $1.5 \times \text{IQR}$ ?
Answer: 12
$\text{IQR} = 18 - 10 = 8$ , so $1.5 \times 8 = 12$ .
3.
The lower fence of a data set is $12$ and the upper fence is $36$ . Which value is an outlier?
1. a)35
2. b)14
3. c)40
4. d)12
Answer: 40
$40 > 36$ , so $40$ is an outlier. $14$ and $35$ lie between the fences, and $12$ is exactly on the lower fence, so it is not an outlier.
4.
A data set has $Q_1 = 40$ and $Q_3 = 50$ . What is the upper fence?
Answer: 65
$\text{IQR} = 10$ , $1.5 \times 10 = 15$ , upper fence $= 50 + 15 = 65$ .
5.
The data set $10, 11, 12, 13, 14$ gets one new value: $90$ . Which measure changes the least?
1. a)The median
2. b)The mean
3. c)The range
4. d)The maximum
Answer: The median
Median: $12 \to 12.5$ . Mean: $12 \to 25$ . Range: $4 \to 80$ . Maximum: $14 \to 90$ . The median is resistant to the outlier.
6.
A data set has $Q_1 = 24$ and $Q_3 = 36$ . What is the lower fence?
Answer: 6
$\text{IQR} = 12$ , $1.5 \times 12 = 18$ , lower fence $= 24 - 18 = 6$ .
7.
Find the outlier in $10, 12, 13, 15, 16, 18, 19, 20, 35$ .
Answer: 35
1. The median is $16$ . What is $Q_1$ (median of $10, 12, 13, 15$ )? 12.5
2. What is $Q_3$ (median of $18, 19, 20, 35$ )? 19.5
3. What is the IQR? 7
4. What is the upper fence? 30
5. Which value is the outlier? 35
8.
Find the mean of $5, 6, 7, 8, 30$ . (The median is $7$ .)
Answer: 11.2
$5 + 6 + 7 + 8 + 30 = 56$ and $56 \div 5 = 11.2$ . The outlier $30$ pulls the mean far above the median $7$ , and above four of the five values.
9.
Check the data set $2, 14, 15, 16, 17, 18, 19, 20$ for a low outlier.
Answer: 2
1. With 8 values the halves are $2, 14, 15, 16$ and $17, 18, 19, 20$ . What is $Q_1$ ? 14.5
2. What is $Q_3$ ? 18.5
3. What is the IQR? 4
4. What is the lower fence? 8.5
5. Which value is below the lower fence? 2
10.
In a box plot, the whiskers go from $15$ to $42$ and there is a separate dot at $70$ . What does the dot mean?
1. a)70 is where the upper whisker should end
2. b)70 is the median
3. c)70 is an outlier: it lies above the upper fence
4. d)70 is the third quartile Q3
Answer: 70 is an outlier: it lies above the upper fence
Outliers are drawn as separate dots. The whisker stops at $42$ , the largest value that is not an outlier, so $70$ lies above the upper fence.
11.
The data set $4, 6, 7, 8, 10$ gets one more value: $49$ . How do the mean and the median change?
Answer: 14
1. What is the mean of $4, 6, 7, 8, 10$ ? 7
2. What is the mean after adding $49$ ? 14
3. What is the median after adding $49$ ? 7.5
4. By how much did the mean increase? 7
12.
For $3, 5, 6, 7, 8, 9, 31$ : show that $31$ is an outlier and compare the range with and without it.
Answer: 28
1. The median is $7$ . What is $Q_1$ (median of $3, 5, 6$ )? 5
2. What is $Q_3$ (median of $8, 9, 31$ )? 9
3. What is the upper fence? 15
4. What is the range with $31$ ? 28
5. What is the range without $31$ ? 6
13.
Test scores: $60, 65, 70, 72, 75, 78, 80, 95$ . Is the top score $95$ an outlier?
Answer: 0
1. What is $Q_1$ (median of $60, 65, 70, 72$ )? 67.5
2. What is $Q_3$ (median of $75, 78, 80, 95$ )? 79
3. What is the IQR? 11.5
4. What is the upper fence? 96.25
5. Is $95$ an outlier? Type 1 for yes or 0 for no. 0
14.
Find the lower fence of $1, 8, 9, 10, 11, 12, 13$ .
Answer: 2
$Q_1 = 8$ , $Q_3 = 12$ , $\text{IQR} = 4$ , $1.5 \times 4 = 6$ , lower fence $= 8 - 6 = 2$ . Since $1 < 2$ , the value $1$ is an outlier.
15.
How many outliers does $1, 20, 22, 23, 24, 25, 26, 27, 28, 50$ have?
Answer: 2
1. 10 values: the lower half is $1, 20, 22, 23, 24$ . What is $Q_1$ ? 22
2. The upper half is $25, 26, 27, 28, 50$ . What is $Q_3$ ? 27
3. What is the IQR? 5
4. What is the lower fence? 14.5
5. What is the upper fence? 34.5
6. How many values lie outside the fences? 2
16.
A runner's last 7 times for 5 km (minutes): $24, 25, 25, 26, 26, 27, 52$ . On the 52-minute day she sprained her ankle and walked home. The upper fence is $30$ . What is the best way to describe her usual time?
1. a)Use the mean of all 7 times
2. b)Use the range of all 7 times
3. c)Use the median (26 minutes), or leave out 52 and say why
4. d)Change 52 to 25 so it fits the others
Answer: Use the median (26 minutes), or leave out 52 and say why
The 52 is a real time, but it does not show her usual running. The median ( $26$ ) is not affected by it, or you can leave it out and say why. The mean (about $29.3$ ) is pulled up, the range is stretched, and inventing a value like 25 is never allowed.
17.
Heights in cm: $150, 155, 158, 160, 162, 1600$ . The $1600$ is a typo for $160$ . Compare the mean and median before and after correcting it.
Answer: 157.5
1. What is the sum of the five correct values $150, 155, 158, 160, 162$ ? 785
2. What is the mean with the typo $1600$ ? 397.5
3. What is the mean after correcting $1600$ to $160$ ? 157.5
4. What is the median after the correction? 159
18.
Draw the box plot of $3, 10, 11, 12, 13, 14, 15, 16, 30$ : find the fences and where the upper whisker ends.
Answer: 16
1. The median is $13$ . What is $Q_1$ (median of $3, 10, 11, 12$ )? 10.5
2. What is $Q_3$ (median of $14, 15, 16, 30$ )? 15.5
3. What is the lower fence? 3
4. What is the upper fence? 23
5. Is $3$ an outlier? Type 1 for yes or 0 for no. 0
6. Where does the upper whisker end? 16