To find \( D_7 \) (the 7th decile) of the given distribution, we will follow these steps:
### Step 1: Arrange the data in ascending order
The given data is:
\[ 18, 20, 9, 15, 21, 26, 14, 13, 27, 22, 16, 28 \]
Arranging this in ascending order gives us:
\[ 9, 13, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28 \]
**Hint:** Always start by organizing your data from the smallest to the largest value.
### Step 2: Determine the number of observations (n)
Count the number of values in the arranged data:
\[ n = 12 \]
**Hint:** The number of observations is crucial for calculating deciles, so make sure to count them accurately.
### Step 3: Use the formula for the \( i \)-th decile
The formula for the \( i \)-th decile \( D_i \) is given by:
\[
D_i = \frac{n + 1}{10} \times i
\]
For \( D_7 \), we will substitute \( i = 7 \) and \( n = 12 \):
\[
D_7 = \frac{12 + 1}{10} \times 7 = \frac{13}{10} \times 7 = 9.1
\]
**Hint:** Remember to substitute the correct values into the formula for accurate results.
### Step 4: Identify the position of \( D_7 \)
The result \( 9.1 \) indicates that \( D_7 \) is located between the 9th and 10th terms of the ordered data. Since we cannot have a fractional term, we round \( 9.1 \) to the nearest whole number, which is 9.
**Hint:** When you get a decimal in the position, round it to the nearest integer to find the corresponding term.
### Step 5: Find the 9th term in the ordered data
Now, we look for the 9th term in our ordered list:
\[ 9, 13, 14, 15, 16, 18, 20, 21, 22, 26, 27, 28 \]
The 9th term is:
\[ 22 \]
**Hint:** Always refer back to your ordered data to find the exact term corresponding to your calculated position.
### Conclusion
Thus, the 7th decile \( D_7 \) is:
\[
D_7 = 22
\]
