For the data For the data : 28, 17, 12, 25, 26, 19, 13, 27, 21, 16
6th decile is

Options

A. 22
B. 23.4
C. 24
D. 25

Answer

To find the 6th decile (D6) for the given data set, we will follow these steps:

### Step 1: Arrange the Data in Ascending Order
First, we need to arrange the given data in ascending order. The data provided is:
28, 17, 12, 25, 26, 19, 13, 27, 21, 16

Arranging it in ascending order gives us:
12, 13, 16, 17, 19, 21, 25, 26, 27, 28

### Step 2: Count the Number of Observations
Next, we count the number of observations (n) in the data set.
Here, n = 10 (since there are 10 values).

### Step 3: Calculate the Position of the 6th Decile
The formula to find the position of the k-th decile (Dk) is given by:
\[ D_k = \frac{k(n + 1)}{10} \]
For the 6th decile (D6), we substitute k = 6 and n = 10 into the formula:
\[ D_6 = \frac{6(10 + 1)}{10} = \frac{6 \times 11}{10} = 6.6 \]

### Step 4: Identify the 6th and 7th Observations
Since the position is 6.6, we need to find the 6th and 7th observations in our ordered data.
The 6th observation is:
1. 12
2. 13
3. 16
4. 17
5. 19
6. 21 (6th observation)
7. 25 (7th observation)

### Step 5: Interpolate to Find the 6th Decile
To find the value of D6, we will interpolate between the 6th and 7th observations:
- The 6th observation is 21.
- The 7th observation is 25.

We use the formula for interpolation:
\[ D_6 = \text{6th observation} + (0.6 \times (\text{7th observation} - \text{6th observation})) \]
Substituting the values:
\[ D_6 = 21 + (0.6 \times (25 - 21)) \]
\[ D_6 = 21 + (0.6 \times 4) \]
\[ D_6 = 21 + 2.4 \]
\[ D_6 = 23.4 \]

### Final Answer
The 6th decile (D6) is **23.4**.

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