\begin{tabular}{l} Distances of Stars Of the 25 brightest stars, the distances from earth (in light-years) for some of those with distances le \ than 100 light-years are found below. Find the mean, median, mode, and midrange for the data. \ $$ 36.7 \quad 11.4 \quad 8.6 \quad 16.8 \quad 42.2 \quad 33.7 \quad 25.1 \quad 4.4 \quad 51.5 \quad 25.3 $$ \ Send data to Excel \ Part 1 of 4 \ Find the mean. \ Rounding rule for the mean: round to one more decimal place than the data as needed. \ Mean: 25.57 \ Part 2 of 4 \ Find the mode(s). \ Find the median. \ Mere is more than one mode, separate them with commas. Select "None" if applicable. \ Part 3 of 4 \ Alternate Answer: \ Part: 25.20 \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

Sure, let's analyze the given data set to find the **mean**, **median**, **mode**, and **midrange**. Here's a step-by-step breakdown:

### **Given Data (Distances in Light-Years):**
$$ 36.7, \, 11.4, \, 8.6, \, 16.8, \, 42.2, \, 33.7, \, 25.1, \, 4.4, \, 51.5, \, 25.3 $$

### **1. Mean**
The **mean** (average) has already been calculated as **25.57**.

### **2. Mode**
The **mode** is the value that appears most frequently in the data set.

- **Frequency of Each Value:**
  - 4.4: 1
  - 8.6: 1
  - 11.4: 1
  - 16.8: 1
  - 25.1: 1
  - 25.3: 1
  - 33.7: 1
  - 36.7: 1
  - 42.2: 1
  - 51.5: 1

- **Conclusion:**  
  All values appear **once**. Therefore, there is **no mode**.

**Mode:** **None**

### **3. Median**
The **median** is the middle value when the data set is ordered from smallest to largest.

- **Ordered Data:**
  $$ 4.4, \, 8.6, \, 11.4, \, 16.8, \, 25.1, \, 25.3, \, 33.7, \, 36.7, \, 42.2, \, 51.5 $$

- **Number of Data Points:** 10 (even number)
  
- **Finding the Median:**
  - The median is the average of the 5th and 6th values.
  - 5th value: **25.1**
  - 6th value: **25.3**
  
  $$
  \text{Median} = \frac{25.1 + 25.3}{2} = \frac{50.4}{2} = 25.2
  $$

**Median:** **25.20**

### **4. Midrange**
The **midrange** is the average of the smallest and largest values in the data set.

- **Smallest Value:** 4.4
- **Largest Value:** 51.5

$$
\text{Midrange} = \frac{4.4 + 51.5}{2} = \frac{55.9}{2} = 27.95
$$

**Midrange:** **27.95**

### **Summary of Results**

- **Mean:** 25.57
- **Median:** 25.20
- **Mode:** None
- **Midrange:** 27.95

If you need to input these results into Excel or require further assistance with any of the parts, feel free to ask!
**Summary of Data Analysis:** - **Mean:** 25.57 - **Median:** 25.20 - **Mode:** None - **Midrange:** 27.95

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