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Sagot :
Let's address each part of the question step-by-step.
### 3.1
#### a) [tex]$\{10, 7, 9, 5, 8, 7, 13, 3, 2\}$[/tex]
I. Range
The range of a dataset is the difference between the maximum and minimum values.
Given data: [tex]$\{10, 7, 9, 5, 8, 7, 13, 3, 2\}$[/tex]
Maximum value = 13
Minimum value = 2
Range = Maximum value - Minimum value = 13 - 2 = 11
II. Median
To find the median, arrange the values in ascending order and find the middle value.
Sorted data: [tex]$\{2, 3, 5, 7, 7, 8, 9, 10, 13\}$[/tex]
Number of data points = 9 (which is odd). The median is the middle value (5th value in this sorted list).
Median = 7
III. Lower Quartile (Q1) and V. Upper Quartile (Q3)
The lower quartile (Q1) is the 25th percentile and the upper quartile (Q3) is the 75th percentile.
The sorted data: [tex]$\{2, 3, 5, 7, 7, 8, 9, 10, 13\}$[/tex]
Use the positions to calculate Q1 and Q3:
For Q1 (25th percentile), find the value at 25% of 9 (total data points):
Position = 0.25 (9 + 1) = 2.5
Average of the 2nd and 3rd values in sorted data: Q1 = (3 + 5) / 2 = 5.0
For Q3 (75th percentile), find the value at 75% of 9 (total data points):
Position = 0.75 (9 + 1) = 7.5
Average of the 7th and 8th values in sorted data: Q3 = (9 + 10) / 2 = 9.0
### Summary of 3.1:
- Range: 11
- Median: 7
- Lower Quartile (Q1): 5.0
- Upper Quartile (Q3): 9.0
### 3.2
#### Data: [tex]$\{43, 48, 62, 52, 46, 90, 58, 37, 48, 73, 84, 68, 54, 34, 78\}$[/tex]
Median
To find the median, arrange the values in ascending order and find the middle value.
Sorted data: [tex]$\{34, 37, 43, 46, 48, 48, 52, 54, 58, 62, 68, 73, 78, 84, 90\}$[/tex]
Number of data points = 15 (which is odd). The median is the middle value (8th value in this sorted list).
Median = 54
Range
The range of a dataset is the difference between the maximum and minimum values.
Maximum value = 90
Minimum value = 34
Range = Maximum value - Minimum value = 90 - 34 = 56
Interquartile Range (IQR)
For Q1 (25th percentile) and Q3 (75th percentile), find the respective positions and values:
For Q1:
Position = 0.25 (15 + 1) = 4
Q1 (4th value in sorted data) = 46
For Q3:
Position = 0.75 (15 + 1) = 12
Q3 (12th value in sorted data) = 73.5
IQR = Q3 - Q1 = 73.5 - 46 = 23.5
Box and Whisker Diagram
To draw a box and whisker diagram, you need the five-number summary: minimum, Q1, median, Q3, and maximum.
Five-number summary for the data:
- Minimum: 34
- Q1: 46
- Median: 54
- Q3: 73.5
- Maximum: 90
The box extends from Q1 to Q3, with a line at the median. The "whiskers" extend from the minimum to Q1 and from Q3 to the maximum.
Box and Whisker Diagram:
```
34 46 54 73.5 90
|--------|========|========|--------|
```
This visual representation helps to understand the spread and distribution of the data.
### Summary of 3.2:
- Median: 54
- Range: 56
- Interquartile Range (IQR): 23.5
This completes the detailed step-by-step solution for the given question.
### 3.1
#### a) [tex]$\{10, 7, 9, 5, 8, 7, 13, 3, 2\}$[/tex]
I. Range
The range of a dataset is the difference between the maximum and minimum values.
Given data: [tex]$\{10, 7, 9, 5, 8, 7, 13, 3, 2\}$[/tex]
Maximum value = 13
Minimum value = 2
Range = Maximum value - Minimum value = 13 - 2 = 11
II. Median
To find the median, arrange the values in ascending order and find the middle value.
Sorted data: [tex]$\{2, 3, 5, 7, 7, 8, 9, 10, 13\}$[/tex]
Number of data points = 9 (which is odd). The median is the middle value (5th value in this sorted list).
Median = 7
III. Lower Quartile (Q1) and V. Upper Quartile (Q3)
The lower quartile (Q1) is the 25th percentile and the upper quartile (Q3) is the 75th percentile.
The sorted data: [tex]$\{2, 3, 5, 7, 7, 8, 9, 10, 13\}$[/tex]
Use the positions to calculate Q1 and Q3:
For Q1 (25th percentile), find the value at 25% of 9 (total data points):
Position = 0.25 (9 + 1) = 2.5
Average of the 2nd and 3rd values in sorted data: Q1 = (3 + 5) / 2 = 5.0
For Q3 (75th percentile), find the value at 75% of 9 (total data points):
Position = 0.75 (9 + 1) = 7.5
Average of the 7th and 8th values in sorted data: Q3 = (9 + 10) / 2 = 9.0
### Summary of 3.1:
- Range: 11
- Median: 7
- Lower Quartile (Q1): 5.0
- Upper Quartile (Q3): 9.0
### 3.2
#### Data: [tex]$\{43, 48, 62, 52, 46, 90, 58, 37, 48, 73, 84, 68, 54, 34, 78\}$[/tex]
Median
To find the median, arrange the values in ascending order and find the middle value.
Sorted data: [tex]$\{34, 37, 43, 46, 48, 48, 52, 54, 58, 62, 68, 73, 78, 84, 90\}$[/tex]
Number of data points = 15 (which is odd). The median is the middle value (8th value in this sorted list).
Median = 54
Range
The range of a dataset is the difference between the maximum and minimum values.
Maximum value = 90
Minimum value = 34
Range = Maximum value - Minimum value = 90 - 34 = 56
Interquartile Range (IQR)
For Q1 (25th percentile) and Q3 (75th percentile), find the respective positions and values:
For Q1:
Position = 0.25 (15 + 1) = 4
Q1 (4th value in sorted data) = 46
For Q3:
Position = 0.75 (15 + 1) = 12
Q3 (12th value in sorted data) = 73.5
IQR = Q3 - Q1 = 73.5 - 46 = 23.5
Box and Whisker Diagram
To draw a box and whisker diagram, you need the five-number summary: minimum, Q1, median, Q3, and maximum.
Five-number summary for the data:
- Minimum: 34
- Q1: 46
- Median: 54
- Q3: 73.5
- Maximum: 90
The box extends from Q1 to Q3, with a line at the median. The "whiskers" extend from the minimum to Q1 and from Q3 to the maximum.
Box and Whisker Diagram:
```
34 46 54 73.5 90
|--------|========|========|--------|
```
This visual representation helps to understand the spread and distribution of the data.
### Summary of 3.2:
- Median: 54
- Range: 56
- Interquartile Range (IQR): 23.5
This completes the detailed step-by-step solution for the given question.
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