Problem Analysis and Solution
The worksheet focuses on understanding statistical measures such as range, interquartile range (IQR), and box-and-whisker plots. Let's solve each part step by step.
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1. Would range or IQR be more appropriate in this data set:
Data set: \( 32, 35, 76, 29, 30, 32, 31, 39, 28, 34 \)
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Range: The difference between the maximum and minimum values.
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IQR (Interquartile Range): The difference between the third quartile (Q3) and the first quartile (Q1). It measures the spread of the middle 50% of the data.
To decide which is more appropriate:
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Range is sensitive to outliers. If there are extreme values, the range can be misleading.
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IQR is less sensitive to outliers because it only considers the middle 50% of the data.
Let's calculate both:
##### Step 1: Arrange the data in ascending order:
\[ 28, 29, 30, 31, 32, 32, 34, 35, 39, 76 \]
##### Step 2: Calculate the range:
\[ \text{Range} = \text{Max} - \text{Min} = 76 - 28 = 48 \]
##### Step 3: Calculate the IQR:
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Median (Q2): The middle value. Since there are 10 data points, the median is the average of the 5th and 6th values:
\[ Q2 = \frac{32 + 32}{2} = 32 \]
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Q1 (First Quartile): Median of the lower half (excluding the median if the dataset has an odd number of values):
Lower half: \( 28, 29, 30, 31, 32 \)
\[ Q1 = \text{Median of } \{28, 29, 30, 31, 32\} = 30 \]
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Q3 (Third Quartile): Median of the upper half:
Upper half: \( 32, 34, 35, 39, 76 \)
\[ Q3 = \text{Median of } \{32, 34, 35, 39, 76\} = 35 \]
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IQR:
\[ \text{IQR} = Q3 - Q1 = 35 - 30 = 5 \]
##### Conclusion:
The range is 48, which is heavily influenced by the outlier (76). The IQR is 5, which represents the spread of the middle 50% of the data. Since the data set contains an outlier,
IQR is more appropriate.
Answer: IQR
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2. Find the max, min, Q1, Q2, and Q3 of the data set:
Data set: \( 8, 2, 10, 9, 7, 5, 13, 4, 8, 12, 16 \)
##### Step 1: Arrange the data in ascending order:
\[ 2, 4, 5, 7, 8, 8, 9, 10, 12, 13, 16 \]
##### Step 2: Identify max and min:
\[ \text{Max} = 16 \]
\[ \text{Min} = 2 \]
##### Step 3: Calculate Q2 (Median):
Since there are 11 data points (odd number), the median is the 6th value:
\[ Q2 = 8 \]
##### Step 4: Calculate Q1 (First Quartile):
Lower half: \( 2, 4, 5, 7, 8 \)
\[ Q1 = \text{Median of } \{2, 4, 5, 7, 8\} = 5 \]
##### Step 5: Calculate Q3 (Third Quartile):
Upper half: \( 9, 10, 12, 13, 16 \)
\[ Q3 = \text{Median of } \{9, 10, 12, 13, 16\} = 12 \]
##### Final Answers:
\[ \text{Max} = 16, \quad \text{Min} = 2, \quad Q1 = 5, \quad Q2 = 8, \quad Q3 = 12 \]
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3. Label the Q1, Q2, Q3 on the box and whisker plot. Then, find the range and IQR.
Given box-and-whisker plot:
- Minimum: 0
- Q1: 10
- Median (Q2): 20
- Q3: 30
- Maximum: 40
##### Step 1: Label the plot:
- Left whisker ends at the minimum (0).
- Left edge of the box is Q1 (10).
- Line inside the box is Q2 (20).
- Right edge of the box is Q3 (30).
- Right whisker ends at the maximum (40).
##### Step 2: Calculate the range:
\[ \text{Range} = \text{Max} - \text{Min} = 40 - 0 = 40 \]
##### Step 3: Calculate the IQR:
\[ \text{IQR} = Q3 - Q1 = 30 - 10 = 20 \]
##### Final Answers:
\[ \text{Range} = 40, \quad \text{IQR} = 20 \]
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4. Draw a box and whisker plot using the following data:
Data set: \( 62, 73, 65, 70, 72, 95, 109, 106, 99, 73, 85, 89, 91 \)
##### Step 1: Arrange the data in ascending order:
\[ 62, 65, 70, 72, 73, 73, 85, 89, 91, 95, 99, 106, 109 \]
##### Step 2: Identify max and min:
\[ \text{Max} = 109 \]
\[ \text{Min} = 62 \]
##### Step 3: Calculate Q2 (Median):
There are 13 data points (odd number), so the median is the 7th value:
\[ Q2 = 85 \]
##### Step 4: Calculate Q1 (First Quartile):
Lower half: \( 62, 65, 70, 72, 73 \)
\[ Q1 = \text{Median of } \{62, 65, 70, 72, 73\} = 70 \]
##### Step 5: Calculate Q3 (Third Quartile):
Upper half: \( 89, 91, 95, 99, 106, 109 \)
\[ Q3 = \text{Median of } \{89, 91, 95, 99, 106, 109\} = \frac{95 + 99}{2} = 97 \]
##### Step 6: Draw the box-and-whisker plot:
- Minimum: 62
- Q1: 70
- Median (Q2): 85
- Q3: 97
- Maximum: 109
##### Final Plot:
```
|----| |-----| |------|
62 70 85 97 109
```
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5-10. Use the box and whisker plot to answer the questions:
Box-and-whisker plot for daily temperatures in January:
- Minimum: 32
- Q1: 36
- Median (Q2): 38
- Q3: 40
- Maximum: 44
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5. What was the lowest temperature recorded?
\[ \text{Lowest temperature} = \text{Minimum} = 32 \]
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6. What percentage of the temperatures were above 37°?
- Temperatures above 37° include the upper half of the data (above Q2).
- Since Q2 is 38, all temperatures above 37° are in the upper half.
- The upper half represents 50% of the data.
\[ \text{Percentage above 37°} = 50\% \]
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7. What percentage of the temperatures were above 40°?
- Temperatures above 40° are in the region between Q3 (40) and the maximum (44).
- This region represents the top 25% of the data (since Q3 marks the 75th percentile).
\[ \text{Percentage above 40°} = 25\% \]
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8. What is the Q1 and Q3 for this data set?
\[ Q1 = 36, \quad Q3 = 40 \]
#####
9. What percentage of the temperatures were below 37°?
- Temperatures below 37° include the lower half of the data (below Q2).
- Since Q2 is 38, all temperatures below 37° are in the lower half.
- The lower half represents 50% of the data.
\[ \text{Percentage below 37°} = 50\% \]
#####
10. What was the median temperature in January?
\[ \text{Median temperature} = Q2 = 38 \]
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Final Answers:
1.
IQR
2. Max: 16, Min: 2, Q1: 5, Q2: 8, Q3: 12
3. Range: 40, IQR: 20
4. Box-and-whisker plot drawn with: Min = 62, Q1 = 70, Q2 = 85, Q3 = 97, Max = 109
5. 32
6. 50%
7. 25%
8. Q1 = 36, Q3 = 40
9. 50%
10. 38
Boxed Final Answer:
\[
\boxed{38}
\]
Parent Tip: Review the logic above to help your child master the concept of box and whisker plot worksheet with answers.
