What is the mean of the range, median and mode of the data:
2,3,1,1,3,4,2,1,1,4,5,2,4,2,2,1,3,3,2,5

Options

A. `3 (1)/(3)`
B. `4(1)/(2)`
C. `2(2)/(3)`
D. 4

Answer

To solve the problem, we need to find the mean of the range, median, and mode of the given data set:

**Data Set:** 2, 3, 1, 1, 3, 4, 2, 1, 1, 4, 5, 2, 4, 2, 2, 1, 3, 3, 2, 5

### Step 1: Arrange the Data in Ascending Order
First, we need to arrange the data in ascending order.

**Ordered Data:** 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5

### Step 2: Calculate the Range
The range is calculated as the difference between the highest and lowest values in the data set.

- **Highest Value:** 5
- **Lowest Value:** 1

**Range = Highest Value - Lowest Value = 5 - 1 = 4**

### Step 3: Calculate the Median
To find the median, we need to determine the middle value of the ordered data. Since there are 20 observations (even number), the median will be the average of the 10th and 11th values.

- **10th Value:** 2
- **11th Value:** 2

**Median = (10th Value + 11th Value) / 2 = (2 + 2) / 2 = 2**

### Step 4: Calculate the Mode
The mode is the value that appears most frequently in the data set.

- **Frequency of Values:**
- 1 appears 5 times
- 2 appears 6 times
- 3 appears 4 times
- 4 appears 3 times
- 5 appears 2 times

The mode is the number that appears the most, which is 2.

**Mode = 2**

### Step 5: Calculate the Mean of Range, Median, and Mode
Now we need to find the mean of the range, median, and mode.

- **Range:** 4
- **Median:** 2
- **Mode:** 2

**Mean = (Range + Median + Mode) / 3 = (4 + 2 + 2) / 3 = 8 / 3 = 2.67 (approximately)**

### Final Answer
The mean of the range, median, and mode is approximately **2.67**.

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