For the data For the data : 28, 17, 12, 25, 26, 19, 13, 27, 21, 16
70th percentile is

Options

A. 25
B. 26
C. 25.25
D. 25.7

Answer

To find the 70th percentile of the given data set: 28, 17, 12, 25, 26, 19, 13, 27, 21, 16, follow these steps:

### Step 1: Arrange the data in ascending order
First, we need to sort the data from the smallest to the largest value.

**Sorted Data:** 12, 13, 16, 17, 19, 21, 25, 26, 27, 28

### Step 2: Determine the total number of observations
Count the total number of observations in the sorted data.

**Total Observations (N):** 10

### Step 3: Calculate the rank for the 70th percentile
To find the rank (position) of the 70th percentile (P70), use the formula:
\[ P_k = \frac{k}{100} \times (N + 1) \]
where \( k \) is the percentile you want to find (in this case, 70).

\[ P_{70} = \frac{70}{100} \times (10 + 1) = 0.7 \times 11 = 7.7 \]

### Step 4: Identify the observations around the rank
Since 7.7 is not a whole number, we need to find the 7th and 8th observations in the sorted data.

**7th Observation:** 25
**8th Observation:** 26

### Step 5: Interpolate to find the 70th percentile
To find the value at the 70th percentile, we will use interpolation between the 7th and 8th observations:

Using the formula:
\[ P_k = X_7 + (0.7 \times (X_8 - X_7)) \]

Where:
- \( X_7 \) = 25 (7th observation)
- \( X_8 \) = 26 (8th observation)

Substituting the values:
\[ P_{70} = 25 + (0.7 \times (26 - 25)) \]
\[ P_{70} = 25 + (0.7 \times 1) \]
\[ P_{70} = 25 + 0.7 \]
\[ P_{70} = 25.7 \]

### Final Answer
The 70th percentile of the data set is **25.7**.

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