SOLUTION: Quartile decile percentile - Studypool - Free Printable

Problem Statement:

We are tasked with finding the 1st Quartile (Q₁), 3rd Quartile (Q₃), 7th Decile (D₇), and the 50th Percentile of the given ungrouped data set:
\[ 8, 10, 15, 20, 20, 22, 32, 33, 35 \]

Step-by-Step Solution:

#### Step 1: Organize the Data
The data is already sorted in ascending order:
\[ 8, 10, 15, 20, 20, 22, 32, 33, 35 \]

The total number of observations, \( N \), is:
\[ N = 9 \]

#### Step 2: Formula for Position Calculation
The general formula to find the position of a quartile, decile, or percentile in an ungrouped data set is:
\[
P_k = L + \left( \frac{kN}{100} - c \right) \cdot \frac{d}{f}
\]
where:
- \( P_k \) is the value at the \( k \)-th percentile.
- \( L \) is the lower limit of the class containing \( P_k \).
- \( c \) is the cumulative frequency up to the class before the class containing \( P_k \).
- \( f \) is the frequency of the class containing \( P_k \).
- \( d \) is the class width.

For ungrouped data, the formula simplifies to:
\[
P_k = \text{Value at the position } \left( \frac{kN}{100} \right)
\]
If the position is a fraction, we interpolate between the two nearest values.

---

#### Step 3: Calculate the Quartiles

##### 1st Quartile (Q₁):
The 1st Quartile corresponds to the 25th percentile. The position is calculated as:
\[
\text{Position of } Q_1 = \frac{1(N + 1)}{4} = \frac{1(9 + 1)}{4} = \frac{10}{4} = 2.5
\]
Since the position is 2.5, we interpolate between the 2nd and 3rd values in the ordered data set:
- The 2nd value is 10.
- The 3rd value is 15.

The interpolation formula is:
\[
Q_1 = \text{Value at position } 2 + 0.5 \times (\text{Value at position } 3 - \text{Value at position } 2)
\]
\[
Q_1 = 10 + 0.5 \times (15 - 10) = 10 + 0.5 \times 5 = 10 + 2.5 = 12.5
\]

##### Median (Q₂):
The Median corresponds to the 50th percentile. The position is calculated as:
\[
\text{Position of } Q_2 = \frac{2(N + 1)}{4} = \frac{2(9 + 1)}{4} = \frac{20}{4} = 5
\]
Since the position is 5, the Median is the 5th value in the ordered data set:
\[
Q_2 = 20
\]

##### 3rd Quartile (Q₃):
The 3rd Quartile corresponds to the 75th percentile. The position is calculated as:
\[
\text{Position of } Q_3 = \frac{3(N + 1)}{4} = \frac{3(9 + 1)}{4} = \frac{30}{4} = 7.5
\]
Since the position is 7.5, we interpolate between the 7th and 8th values in the ordered data set:
- The 7th value is 32.
- The 8th value is 33.

The interpolation formula is:
\[
Q_3 = \text{Value at position } 7 + 0.5 \times (\text{Value at position } 8 - \text{Value at position } 7)
\]
\[
Q_3 = 32 + 0.5 \times (33 - 32) = 32 + 0.5 \times 1 = 32 + 0.5 = 32.5
\]

---

#### Step 4: Calculate the 7th Decile (D₇)

The 7th Decile corresponds to the 70th percentile. The position is calculated as:
\[
\text{Position of } D_7 = \frac{7(N + 1)}{10} = \frac{7(9 + 1)}{10} = \frac{70}{10} = 7
\]
Since the position is 7, the 7th Decile is the 7th value in the ordered data set:
\[
D_7 = 32
\]

---

#### Step 5: Calculate the 50th Percentile

The 50th Percentile is the same as the Median (Q₂). From Step 3, we already calculated:
\[
\text{50th Percentile} = Q_2 = 20
\]

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Final Answers:

\[
\boxed{Q_1 = 12.5, \, Q_2 = 20, \, Q_3 = 32.5, \, D_7 = 32, \, \text{50th Percentile} = 20}
\]