Answer :
Sure, I'll help you determine the five-number summary and the interquartile range (IQR) of the data, and guide you through drawing a box plot.
Given ages: 10, 12, 9, 7, 10, 12, 14, 14, 10, 16
### Step 1: Order the Data
First, list the ages in ascending order:
[tex]\[ 7, 9, 10, 10, 10, 12, 12, 14, 14, 16 \][/tex]
### Step 2: Calculate the Five-Number Summary
1. Minimum (Min):
The smallest value in the dataset.
[tex]\[ \text{Min} = 7 \][/tex]
2. First Quartile (Q1):
The median of the lower half of the dataset.
[tex]\[ Q1 = \text{median of } [7, 9, 10, 10, 10] \][/tex]
Since the number of values is odd: 9 is the middle one.
[tex]\[ Q1 = 9 \][/tex]
3. Median:
The middle value of the dataset.
[tex]\[ \text{Median} = \text{median of } [7, 9, 10, 10, 10, 12, 12, 14, 14, 16] \][/tex]
Since there is an even number of values, the median is the average of the 5th and 6th values.
[tex]\[ \text{Median} = (10 + 12) / 2 = 11 \][/tex]
4. Third Quartile (Q3):
The median of the upper half of the dataset.
[tex]\[ Q3 = \text{median of } [12, 12, 14, 14, 16] \][/tex]
Since the number of values is odd: 14 is middle one.
[tex]\[ Q3 = 14 \][/tex]
5. Maximum (Max):
The largest value in the dataset.
[tex]\[ \text{Max} = 16 \][/tex]
Thus, the five-number summary is:
[tex]\[ (7, 9, 11, 14, 16) \][/tex]
### Step 3: Calculate the Interquartile Range (IQR)
The IQR is the difference between the third quartile Q3 and the first quartile Q1.
[tex]\[ \text{IQR} = Q3 - Q1 = 14 - 9 = 5 \][/tex]
### Step 4: Draw a Box Plot
Here's how you would construct the box plot:
1. Horizontal Axis: This represents the ages.
2. Vertical Lines: Mark the minimum (7), Q1 (9), median (11), Q3 (14), and maximum (16) values.
3. Box: Draw a box from Q1 (9) to Q3 (14). The height of the box plot does not vary, it is usually constant.
4. Median Line: Divide the box with a line at the median value (11).
5. Whiskers: Draw lines (whiskers) from the edges of the box (Q1 and Q3) to the min (7) and max (16) values.
The box plot visually summarizes the distribution of the data, indicating the central tendency, spread, and potential outliers.
### Result:
- Five-number summary: [tex]\( (7, 9, 11, 14, 16) \)[/tex]
- Interquartile Range (IQR): [tex]\( 5 \)[/tex]
These calculations and the box plot representation give a clear picture of how the ages are distributed at the birthday party.
Given ages: 10, 12, 9, 7, 10, 12, 14, 14, 10, 16
### Step 1: Order the Data
First, list the ages in ascending order:
[tex]\[ 7, 9, 10, 10, 10, 12, 12, 14, 14, 16 \][/tex]
### Step 2: Calculate the Five-Number Summary
1. Minimum (Min):
The smallest value in the dataset.
[tex]\[ \text{Min} = 7 \][/tex]
2. First Quartile (Q1):
The median of the lower half of the dataset.
[tex]\[ Q1 = \text{median of } [7, 9, 10, 10, 10] \][/tex]
Since the number of values is odd: 9 is the middle one.
[tex]\[ Q1 = 9 \][/tex]
3. Median:
The middle value of the dataset.
[tex]\[ \text{Median} = \text{median of } [7, 9, 10, 10, 10, 12, 12, 14, 14, 16] \][/tex]
Since there is an even number of values, the median is the average of the 5th and 6th values.
[tex]\[ \text{Median} = (10 + 12) / 2 = 11 \][/tex]
4. Third Quartile (Q3):
The median of the upper half of the dataset.
[tex]\[ Q3 = \text{median of } [12, 12, 14, 14, 16] \][/tex]
Since the number of values is odd: 14 is middle one.
[tex]\[ Q3 = 14 \][/tex]
5. Maximum (Max):
The largest value in the dataset.
[tex]\[ \text{Max} = 16 \][/tex]
Thus, the five-number summary is:
[tex]\[ (7, 9, 11, 14, 16) \][/tex]
### Step 3: Calculate the Interquartile Range (IQR)
The IQR is the difference between the third quartile Q3 and the first quartile Q1.
[tex]\[ \text{IQR} = Q3 - Q1 = 14 - 9 = 5 \][/tex]
### Step 4: Draw a Box Plot
Here's how you would construct the box plot:
1. Horizontal Axis: This represents the ages.
2. Vertical Lines: Mark the minimum (7), Q1 (9), median (11), Q3 (14), and maximum (16) values.
3. Box: Draw a box from Q1 (9) to Q3 (14). The height of the box plot does not vary, it is usually constant.
4. Median Line: Divide the box with a line at the median value (11).
5. Whiskers: Draw lines (whiskers) from the edges of the box (Q1 and Q3) to the min (7) and max (16) values.
The box plot visually summarizes the distribution of the data, indicating the central tendency, spread, and potential outliers.
### Result:
- Five-number summary: [tex]\( (7, 9, 11, 14, 16) \)[/tex]
- Interquartile Range (IQR): [tex]\( 5 \)[/tex]
These calculations and the box plot representation give a clear picture of how the ages are distributed at the birthday party.
