Calculate the range and its coefficient for the following data 100, 80, 200, 150, 250, 300

Answer

To solve the problem of calculating the range and its coefficient for the given data set \(100, 80, 200, 150, 250, 300\), we can follow these steps:

### Step 1: Identify the largest and smallest values in the data set.
- **Largest value (xM)**: This is the maximum value in the data set.
- **Smallest value (x0)**: This is the minimum value in the data set.

From the data provided:
- Largest value \(xM = 300\)
- Smallest value \(x0 = 80\)

### Step 2: Calculate the range.
The range is calculated using the formula:
\[ \text{Range} = xM - x0 \]
Substituting the values we identified:
\[ \text{Range} = 300 - 80 = 220 \]

### Step 3: Calculate the coefficient of range.
The coefficient of range is calculated using the formula:
\[ \text{Coefficient of Range} = \frac{xM - x0}{xM + x0} \]
Substituting the values:
\[ \text{Coefficient of Range} = \frac{300 - 80}{300 + 80} = \frac{220}{380} \]

### Step 4: Simplify the coefficient of range.
To simplify \(\frac{220}{380}\):
- Divide both the numerator and denominator by 20:
\[ \frac{220 \div 20}{380 \div 20} = \frac{11}{19} \]

### Step 5: Convert the fraction to decimal form.
To convert \(\frac{11}{19}\) to decimal:
\[ \frac{11}{19} \approx 0.5789 \quad (\text{rounded to } 0.58) \]

### Final Answer:
- **Range**: 220
- **Coefficient of Range**: 0.58

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