Question
For the data set
83
36
23
43
18
5
37
68
13
39
28
find the first and third quartiles
83
36
23
43
18
5
37
68
13
39
28
find the first and third quartiles
Answers
Answer
incorrect
Answer
That is correct. Now find the IQR
Answer
(c) Find the upper and lower outlier boundaries.
Answered by
GPT-4o mini
To find the first (Q1) and third (Q3) quartiles of the given data set, we first need to sort the data in ascending order:
**Data Set (Sorted):**
5, 13, 18, 23, 28, 36, 37, 39, 43, 68, 83
**Step 1: Determine Q1 (First Quartile)**
Q1 is the median of the first half of the data (the lower 50%).
- The first half of the data (lower 6 values) is:
5, 13, 18, 23, 28, 36
Since there are 6 data points in the first half, the median is the average of the 3rd and 4th values:
(18 + 23) / 2 = 20.5
**Thus, Q1 = 20.5**
**Step 2: Determine Q3 (Third Quartile)**
Q3 is the median of the second half of the data (the upper 50%).
- The second half of the data (upper 6 values) is:
37, 39, 43, 68, 83
Since there are 5 data points in the second half, the median is the middle value:
Q3 = 43
**Final Results:**
- Q1 = 20.5
- Q3 = 43
**Data Set (Sorted):**
5, 13, 18, 23, 28, 36, 37, 39, 43, 68, 83
**Step 1: Determine Q1 (First Quartile)**
Q1 is the median of the first half of the data (the lower 50%).
- The first half of the data (lower 6 values) is:
5, 13, 18, 23, 28, 36
Since there are 6 data points in the first half, the median is the average of the 3rd and 4th values:
(18 + 23) / 2 = 20.5
**Thus, Q1 = 20.5**
**Step 2: Determine Q3 (Third Quartile)**
Q3 is the median of the second half of the data (the upper 50%).
- The second half of the data (upper 6 values) is:
37, 39, 43, 68, 83
Since there are 5 data points in the second half, the median is the middle value:
Q3 = 43
**Final Results:**
- Q1 = 20.5
- Q3 = 43
Answered by
GPT-4o mini
Apologies for any confusion. Let's work through the steps again to find the first quartile (Q1) and the third quartile (Q3) accurately.
### Given Data Set:
83, 36, 23, 43, 18, 5, 37, 68, 13, 39, 28
### Step 1: Sort the Data
First, we need to sort the data in ascending order:
Sorted Data:
5, 13, 18, 23, 28, 36, 37, 39, 43, 68, 83
### Step 2: Finding Q1 and Q3
**Number of Data Points (n)**:
There are 11 data points.
#### For Q1 (First Quartile):
Q1 is located at the position \(\frac{(n + 1)}{4}\).
\[
Q1 \text{ position} = \frac{(11 + 1)}{4} = \frac{12}{4} = 3
\]
So, Q1 is the 3rd value in the sorted data.
**Values**:
1. 5
2. 13
3. 18 (this is Q1)
**Thus, Q1 = 18**.
#### For Q3 (Third Quartile):
Q3 is located at the position \(\frac{3(n + 1)}{4}\).
\[
Q3 \text{ position} = \frac{3(11 + 1)}{4} = \frac{3 \times 12}{4} = 9
\]
So, Q3 is the 9th value in the sorted data.
**Values**:
1. 5
2. 13
3. 18
4. 23
5. 28
6. 36
7. 37
8. 39
9. 43 (this is Q3)
**Thus, Q3 = 43**.
### Final Results:
- **Q1 = 18**
- **Q3 = 43**
Thank you for your patience, and I appreciate your understanding!
### Given Data Set:
83, 36, 23, 43, 18, 5, 37, 68, 13, 39, 28
### Step 1: Sort the Data
First, we need to sort the data in ascending order:
Sorted Data:
5, 13, 18, 23, 28, 36, 37, 39, 43, 68, 83
### Step 2: Finding Q1 and Q3
**Number of Data Points (n)**:
There are 11 data points.
#### For Q1 (First Quartile):
Q1 is located at the position \(\frac{(n + 1)}{4}\).
\[
Q1 \text{ position} = \frac{(11 + 1)}{4} = \frac{12}{4} = 3
\]
So, Q1 is the 3rd value in the sorted data.
**Values**:
1. 5
2. 13
3. 18 (this is Q1)
**Thus, Q1 = 18**.
#### For Q3 (Third Quartile):
Q3 is located at the position \(\frac{3(n + 1)}{4}\).
\[
Q3 \text{ position} = \frac{3(11 + 1)}{4} = \frac{3 \times 12}{4} = 9
\]
So, Q3 is the 9th value in the sorted data.
**Values**:
1. 5
2. 13
3. 18
4. 23
5. 28
6. 36
7. 37
8. 39
9. 43 (this is Q3)
**Thus, Q3 = 43**.
### Final Results:
- **Q1 = 18**
- **Q3 = 43**
Thank you for your patience, and I appreciate your understanding!
Answered by
GPT-4o mini
The Interquartile Range (IQR) is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
**Formula:**
\[
\text{IQR} = Q3 - Q1
\]
Given:
- \( Q1 = 18 \)
- \( Q3 = 43 \)
We can substitute these values into the formula:
\[
\text{IQR} = 43 - 18 = 25
\]
### Final Result:
**The Interquartile Range (IQR) is 25.**
**Formula:**
\[
\text{IQR} = Q3 - Q1
\]
Given:
- \( Q1 = 18 \)
- \( Q3 = 43 \)
We can substitute these values into the formula:
\[
\text{IQR} = 43 - 18 = 25
\]
### Final Result:
**The Interquartile Range (IQR) is 25.**
Answered by
GPT-4o mini
To identify the boundaries for detecting outliers in a data set, we can use the following formulas based on the Interquartile Range (IQR):
1. **Lower Bound**:
\[
\text{Lower Bound} = Q1 - 1.5 \times \text{IQR}
\]
2. **Upper Bound**:
\[
\text{Upper Bound} = Q3 + 1.5 \times \text{IQR}
\]
### Previously calculated values:
- \( Q1 = 18 \)
- \( Q3 = 43 \)
- \( \text{IQR} = 25 \)
### Calculating the Lower Bound:
\[
\text{Lower Bound} = 18 - 1.5 \times 25 = 18 - 37.5 = -19.5
\]
### Calculating the Upper Bound:
\[
\text{Upper Bound} = 43 + 1.5 \times 25 = 43 + 37.5 = 80.5
\]
### Final Results:
- **Lower Bound**: -19.5
- **Upper Bound**: 80.5
These boundaries can be used to identify any outliers in the data set. Any data points below -19.5 or above 80.5 would be considered outliers. In this case, the only outlier in the provided data points seems to be 83 (as it is greater than 80.5).
1. **Lower Bound**:
\[
\text{Lower Bound} = Q1 - 1.5 \times \text{IQR}
\]
2. **Upper Bound**:
\[
\text{Upper Bound} = Q3 + 1.5 \times \text{IQR}
\]
### Previously calculated values:
- \( Q1 = 18 \)
- \( Q3 = 43 \)
- \( \text{IQR} = 25 \)
### Calculating the Lower Bound:
\[
\text{Lower Bound} = 18 - 1.5 \times 25 = 18 - 37.5 = -19.5
\]
### Calculating the Upper Bound:
\[
\text{Upper Bound} = 43 + 1.5 \times 25 = 43 + 37.5 = 80.5
\]
### Final Results:
- **Lower Bound**: -19.5
- **Upper Bound**: 80.5
These boundaries can be used to identify any outliers in the data set. Any data points below -19.5 or above 80.5 would be considered outliers. In this case, the only outlier in the provided data points seems to be 83 (as it is greater than 80.5).