Box Plots (Box and Whisker)

Show your work for each problem.

1.
What does the line inside a box plot represent?
1. a)The mean
2. b)The mode
3. c)The range
4. d)The median
Answer: The median
The line inside the box represents the median (Q2), which is the middle value of the data set. It shows where 50% of the data is above and 50% is below.
2.
About what percentage of the data lies inside the box of a box plot?
1. a)25%
2. b)75%
3. c)50%
4. d)100%
Answer: 50%
The box goes from Q1 (25th percentile) to Q3 (75th percentile), so it holds about 50% of the data, the middle half. The width of the box is the interquartile range (IQR).
3.
For the data set: 5, 8, 12, 15, 18, 22, 25, what is the median?
Answer: 15
There are 7 values, so the median is the 4th value. Counting: 5, 8, 12, $\mathbf{15}$ , 18, 22, 25. The median is 15.
4.
What is the minimum value in this data set: 23, 45, 12, 67, 34, 56, 8?
Answer: 8
When ordered: 8, 12, 23, 34, 45, 56, 67. The minimum (smallest value) is 8.
5.
If Q1 = 20 and Q3 = 45, what is the interquartile range (IQR)?
1. a)32.5
2. b)25
3. c)65
4. d)20
Answer: 25
IQR = Q3 - Q1 = 45 - 20 = 25. The interquartile range represents the spread of the middle 50% of the data.
6.
Find Q1 for the data set: 4, 7, 10, 12, 15, 18, 21, 24, 27
Answer: 8.5
1. How many values are in the data set? 9
2. What is the median of the entire data set? 15
3. What are the values in the lower half (before the median)? 4, 7, 10, 12
4. What is Q1 (median of the lower half)? 8.5
7.
A box plot shows Q1 = 30 and Q3 = 50. Calculate the upper fence for identifying outliers (use the formula: Q3 + 1.5 × IQR).
Answer: 80
IQR = 50 - 30 = 20. Upper fence = Q3 + 1.5 × IQR = 50 + 1.5 × 20 = 50 + 30 = 80. Any value above 80 would be an outlier.
8.
Find the five-number summary for: 3, 7, 8, 12, 14, 18, 21
Answer: 3, 7, 12, 18, 21
1. What is the minimum? 3
2. What is the maximum? 21
3. What is the median (4th value of 7)? 12
4. What is Q1 (median of 3, 7, 8)? 7
5. What is Q3 (median of 14, 18, 21)? 18
9.
Two box plots have the same median, but plot A has a smaller box than plot B. What can you conclude?
1. a)Data A has more values
2. b)Data B has more outliers
3. c)Data A is more consistent
4. d)Data B is more consistent
Answer: Data A is more consistent
A smaller box means a smaller IQR (less spread in the middle 50% of data). This indicates that Data A is more consistent - the values are closer together around the median.
10.
Is 95 an outlier for data with Q1 = 40, Q3 = 60?
Answer: Yes, 95 is an outlier
1. Calculate the IQR (Q3 - Q1) 20
2. Calculate 1.5 × IQR 30
3. Calculate the upper fence (Q3 + 1.5 × IQR) 90
4. Is 95 greater than the upper fence (90)? Yes
11.
A box plot shows the median line closer to Q1 than to Q3. This indicates the data is:
1. a)Perfectly symmetric
2. b)Bimodal
3. c)Skewed right (positively skewed)
4. d)Skewed left (negatively skewed)
Answer: Skewed right (positively skewed)
When the median is closer to Q1, it means the upper half of the data (between median and Q3) is more spread out, with potential extreme values on the right. This is called right-skewed or positively skewed.
12.
A data set has Min = 10, Q1 = 25, Median = 40, Q3 = 55, Max = 100. To check whether 100 is an outlier, calculate the upper fence.
Answer: 100
IQR = 55 - 25 = 30. Upper fence = 55 + 1.5 × 30 = 55 + 45 = 100. Only values greater than the fence are outliers, and 100 is equal to it, so 100 is not an outlier (it is right on the boundary).
13.
Compare two classes: Class A (Median = 78, IQR = 12) and Class B (Median = 75, IQR = 24). Which class performed better and which was more consistent?
Answer: Class A: higher median and more consistent
1. Which class has the higher median? Class A
2. Which class has the smaller IQR? Class A
3. A smaller IQR means more or less consistency? more
4. Which class performed better overall? Class A
14.
Find the complete five-number summary for: 2, 5, 7, 9, 11, 14, 16, 19, 22, 25, 28
Answer: 2, 7, 14, 22, 28
1. How many data values are there? 11
2. What is the minimum? 2
3. What is the maximum? 28
4. What is the median (6th value)? 14
5. What is Q1 (median of 2, 5, 7, 9, 11)? 7
6. What is Q3 (median of 16, 19, 22, 25, 28)? 22
15.
Calculate the lower fence for data with Q1 = 35 and Q3 = 65.
Answer: -10
IQR = 65 - 35 = 30. Lower fence = Q1 - 1.5 × IQR = 35 - 1.5(30) = 35 - 45 = -10. Any value below -10 would be considered an outlier.
16.
What do the whiskers on a box plot show?
1. a)The mean values
2. b)The range from minimum to Q1 and Q3 to maximum
3. c)Only the outliers
4. d)The standard deviation
Answer: The range from minimum to Q1 and Q3 to maximum
The whiskers extend from the box to show the range of data: one from Q1 down to the minimum (excluding outliers) and one from Q3 up to the maximum (excluding outliers).