Find the Q1 for the following distribution. 11, 10, 24, 19, 13, 21, 20, 14, 25, 15, 32, 30, 36

Answer

To find the first quartile (Q1) for the given distribution, we will follow these steps:

### Step 1: Arrange the data in ascending order
We start with the given data:
11, 10, 24, 19, 13, 21, 20, 14, 25, 15, 32, 30, 36.

Arranging this data in ascending order, we get:
10, 11, 13, 14, 15, 19, 20, 21, 24, 25, 30, 32, 36.

### Step 2: Count the number of observations
Next, we count the total number of observations (n):
- The total number of observations, n = 13.

### Step 3: Find the median
To find the first quartile, we first need to find the median. The median is the middle value of the ordered data. Since n = 13 (which is odd), the median is the value at position (n + 1) / 2:
- Median position = (13 + 1) / 2 = 7.

Looking at the ordered data, the 7th value is:
- Median = 20.

### Step 4: Determine the lower half of the data
To find Q1, we need to consider the lower half of the data, which consists of all values below the median. The lower half includes:
10, 11, 13, 14, 15, 19.

### Step 5: Find Q1
Now we find Q1, which is the median of the lower half of the data. The lower half has 6 values (even number), so we take the average of the two middle values:
- The two middle values are 13 and 14 (3rd and 4th values in the lower half).

Calculating Q1:
\[ Q1 = \frac{13 + 14}{2} = \frac{27}{2} = 13.5. \]

### Final Answer
Thus, the first quartile (Q1) for the given distribution is:
**Q1 = 13.5.**

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