Question
1. Here is a set of data. \( 25 ; 19 ; 39 ; 20 ; 3 ; 19 ; 10 ; 12 ; 8 ; 36 ; 12 ; 41 ; 30 ; 24 ; 25 ; 25 \) a. Find the mode of this data. b. What is the range of the data? c. Would the mean be typical of this data set? Explain why or why not.
Ask by Chan Ward. in South Africa
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**a.** The mode is **25**.
**b.** The range is **38**.
**c.** The mean is **21.75**, which is not typical because it's influenced by extreme values, making it less representative of the data set compared to the mode of **25**.
Solution
Let's analyze the given data set step by step to answer each of your questions.
### **Given Data Set:**
\[
25,\ 19,\ 39,\ 20,\ 3,\ 19,\ 10,\ 12,\ 8,\ 36,\ 12,\ 41,\ 30,\ 24,\ 25,\ 25
\]
---
### **a. Finding the Mode**
**Definition:** The **mode** is the value that appears most frequently in a data set.
**Steps to Find the Mode:**
1. **List all the numbers:**
\[
25,\ 19,\ 39,\ 20,\ 3,\ 19,\ 10,\ 12,\ 8,\ 36,\ 12,\ 41,\ 30,\ 24,\ 25,\ 25
\]
2. **Count the frequency of each number:**
| Number | Frequency |
|--------|-----------|
| 3 | 1 |
| 8 | 1 |
| 10 | 1 |
| 12 | 2 |
| 19 | 2 |
| 20 | 1 |
| 24 | 1 |
| 25 | 3 |
| 30 | 1 |
| 36 | 1 |
| 39 | 1 |
| 41 | 1 |
3. **Identify the number with the highest frequency:**
- **25** appears **3 times**, which is more frequent than any other number.
**Answer:**
**a.** The **mode** of the data set is **25**.
---
### **b. Calculating the Range**
**Definition:** The **range** is the difference between the largest and smallest values in a data set.
**Steps to Find the Range:**
1. **Identify the smallest and largest numbers in the data set:**
- **Smallest number:** 3
- **Largest number:** 41
2. **Calculate the range:**
\[
\text{Range} = \text{Largest number} - \text{Smallest number} = 41 - 3 = 38
\]
**Answer:**
**b.** The **range** of the data set is **38**.
---
### **c. Evaluating Whether the Mean is Typical**
**Definition:** The **mean** is the average of all numbers in a data set, calculated by summing all values and dividing by the count of values.
**Steps to Evaluate the Mean:**
1. **Calculate the mean:**
- **Sum of all numbers:**
\[
25 + 19 + 39 + 20 + 3 + 19 + 10 + 12 + 8 + 36 + 12 + 41 + 30 + 24 + 25 + 25 = 348
\]
- **Number of values:** 16
- **Mean:**
\[
\text{Mean} = \frac{348}{16} = 21.75
\]
2. **Assessing Typicality:**
- **Mode vs. Mean:** The mode is **25**, which is higher than the mean (**21.75**).
- **Presence of Outliers:**
- **Low outlier:** 3
- **High outliers:** 36, 39, 41
These extreme values can skew the mean, making it less representative of the central tendency for most of the data points.
- **Distribution Shape:** Given the presence of both low and high outliers, the data distribution is likely **skewed**, which means the mean is influenced by these extreme values and may not accurately reflect the typical value of the data set.
**Answer:**
**c.** The **mean** of **21.75** is **not typical** for this data set because it is influenced by extreme values (like **3**, **36**, **39**, and **41**). These outliers skew the average, making it less representative of the central tendency compared to the mode, which better reflects the most common value in the data set.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
The mode of the data set is 25, as it appears most frequently, occurring three times. When taking a look at the range, it's calculated by subtracting the smallest number (3) from the largest number (41), giving us a range of 38. Now, when it comes to the mean, it may not be typical for this data set due to the presence of outliers, especially the high value of 41. The mean might be skewed upwards and not reflect the central tendency of the data effectively. It's often better to rely on the mode or median in such cases to represent the typical value.
