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For the data set

83
36
23
43
18
5
37
68
13
39
28
find the first and third quartiles

7 answers

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
incorrect

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!

That is correct. Now find the IQR

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.

(c) Find the upper and lower outlier boundaries.

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).