For an exam given to a class, the students' scores ranged from 40 to 95, with a mean of 78. Which of the following is the most realistic value for the standard deviation: -14, 57, 12, 1, or 0? Clearly explain what's unrealistic about each of the other values.

To determine the most realistic value for the standard deviation, let’s first consider the key properties of standard deviation and the range of the students' scores.

### Important Facts:
- **Range of scores**: 40 to 95
- **Mean (average)**: 78

### Key points about standard deviation:
- **Standard deviation (SD)** measures how spread out the data points are from the mean. A higher SD means the data is more spread out, while a lower SD indicates that the data points are closer to the mean.
- SD is always **non-negative**.
- The SD is typically a value that reflects a reasonable portion of the range of data.

### Now, let's analyze each option:

#### 1. **SD = -14**:
   - **Unrealistic**: The standard deviation **cannot be negative** by definition. It measures distance from the mean, and distances are always non-negative.

#### 2. **SD = 57**:
   - **Unrealistic**: This is too large. The **entire range** of scores is from 40 to 95, which means the **range** is \( 95 - 40 = 55 \). The standard deviation being 57 would imply that the majority of the scores are outside of the range, which is impossible given the data.

#### 3. **SD = 12**:
   - **Realistic**: This seems reasonable. The standard deviation should be a moderate fraction of the range. Since the scores are spread across a range of 55 points, an SD of 12 suggests that most scores are within about 12 points of the mean (78), which is consistent with the data's spread.

#### 4. **SD = 1**:
   - **Unrealistic**: This is too small. An SD of 1 would mean that almost all of the students' scores are extremely close to the mean (78). Given that the scores range from 40 to 95, a very small standard deviation like 1 cannot explain such a wide spread.

#### 5. **SD = 0**:
   - **Unrealistic**: This would mean **all the students got the exact same score**, which is clearly not the case since the scores range from 40 to 95.

### Conclusion:
The most realistic value for the standard deviation is **12**. The other values are either impossible or unrealistic given the range of the data.

---

Would you like further details or have any other questions?

Here are 5 questions to expand on the concept of standard deviation:
1. How would the standard deviation change if the range of scores were larger?
2. Can the mean of a data set ever be higher than the highest score?
3. What is the relationship between variance and standard deviation?
4. How would the presence of outliers affect the standard deviation?
5. How do we calculate standard deviation step-by-step for a given set of data?

**Tip**: Standard deviation is a crucial indicator of spread in a data set. When analyzing data, always look at both the mean and the standard deviation for a full picture.

Discover the most realistic value for the standard deviation when analyzing student exam scores ranging from 40 to 95, with a mean of 78. This explanation covers key statistical concepts, the properties of standard deviation, and why certain values are unrealistic.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation