Range and Interquartile Range

Show your work for each problem.

1.
What is the range of this data set: $8, 12, 15, 19, 23$ ?
1. a) $15$
2. b) $11$
3. c) $23$
4. d) $8$
Answer: $15$
Range $= 23 - 8 = 15$ . The maximum is $23$ and the minimum is $8$ .
2.
Find the range: $35, 42, 18, 56, 29$
Answer: 38
Maximum $= 56$ , Minimum $= 18$ . Range $= 56 - 18 = 38$ .
3.
In the data set $5, 9, 12, 15, 18, 22, 25$ , what is the median (Q2)?
1. a) $12$
2. b) $15$
3. c) $14$
4. d) $18$
Answer: $15$
With 7 values in order, the median is the 4th value: $15$ .
4.
For the data $3, 7, 9, 12, 15$ , what is Q1 (the first quartile)?
Answer: 5
Lower half: $3, 7$ . Q1 $= (3 + 7) \div 2 = 5$ .
5.
Find the IQR of: $4, 8, 10, 14, 18, 22, 26$
Answer: 14
1. What is the median of the full data set? 14
2. What is Q1 (median of 4, 8, 10)? 8
3. What is Q3 (median of 18, 22, 26)? 22
4. Calculate IQR = Q3 - Q1 14
6.
Which measure of spread is least affected by outliers?
1. a)Both Range and IQR
2. b)Interquartile Range (IQR)
3. c)Maximum
4. d)Range
Answer: Interquartile Range (IQR)
The IQR only uses Q1 and Q3, the edges of the middle half of the data, so one extreme value barely changes it. The range uses the maximum and minimum, which can be outliers.
7.
Find the IQR: $2, 5, 8, 11, 14, 17, 20, 23$
Answer: 12
1. What is the median of this even-sized data set? 12.5
2. What is Q1 (median of 2, 5, 8, 11)? 6.5
3. What is Q3 (median of 14, 17, 20, 23)? 18.5
4. Calculate IQR = Q3 - Q1 12
8.
The test scores are: $65, 72, 78, 82, 85, 88, 92, 98$ . What is Q3?
Answer: 90
Upper half: $85, 88, 92, 98$ . Q3 $= (88 + 92) \div 2 = 90$ .
9.
Data set A: Range = 50, IQR = 12. Data set B: Range = 50, IQR = 40. Which statement is true?
1. a)Both sets have the same distribution
2. b)Set A has data more clustered in the middle
3. c)We cannot compare them
4. d)Set B has data more clustered in the middle
Answer: Set A has data more clustered in the middle
Both have the same range (50), but Set A has a much smaller IQR (12 vs 40). This means Set A's middle 50% of data is more tightly clustered.
10.
Data: $15, 18, 22, 25, 28, 32, 35, 38, 95$ . Find the range and the IQR. Which one better describes the typical spread?
Answer: Range $= 80$ , IQR $= 16.5$
1. What is the range? 80
2. What is the median (5th value)? 28
3. What is Q1 (median of 15, 18, 22, 25)? 20
4. What is Q3 (median of 32, 35, 38, 95)? 36.5
5. What is the IQR? 16.5
11.
A data set has $Q_1 = 24$ and an IQR of $28$ . What is $Q_3$ ?
Answer: 52
Since IQR $= Q_3 - Q_1$ , we get $Q_3 = Q_1 + \text{IQR} = 24 + 28 = 52$ . Check: $52 - 24 = 28$ .
12.
A data set has Range = 100 and IQR = 8. What can you conclude?
1. a)The median equals the mean
2. b)The data likely has outliers
3. c)All values are close together
4. d)The data is evenly spread
Answer: The data likely has outliers
A range of 100 with an IQR of only 8 means most data is tightly clustered, but there are extreme values (outliers) stretching the range.
13.
Heights (cm): $155, 158, 162, 165, 168, 172, 175, 178, 182, 185$ . Find the five-number summary and IQR.
Answer: 16
1. What is the minimum? 155
2. What is Q1 (median of first 5 values)? 162
3. What is the median (average of 5th and 6th values)? 170
4. What is Q3 (median of last 5 values)? 178
5. What is the maximum? 185
6. What is the IQR? 16
14.
Monthly sales (thousands): $45, 52, 48, 61, 55, 58, 150, 62, 57, 54, 59, 51$ . Find both the range and IQR.
Answer: Range $= 105$ , IQR $= 8.5$
1. First, order the data. What is the minimum? 45
2. What is the maximum? 150
3. What is the range? 105
4. Ordered data: 45, 48, 51, 52, 54, 55, 57, 58, 59, 61, 62, 150. What is Q1? 51.5
5. What is Q3? 60
6. What is the IQR? 8.5
15.
Class A has IQR = 15. Class B has IQR = 5. Which class has more consistent test scores? Type A or B.
Answer: B
Class B has more consistent scores. A smaller IQR (5 vs 15) means the middle 50% of students scored within a narrower range, showing less variability.