A college statistics class conducted a survey of how students spend their money. They asked 25 students to estimate how much money they typically spend each week on fast food. They determined that the mean amount spent on fast food is $31.52 with a standard deviation of $21.60.

Later they realized that a value entered as $3 should have been $30. They recalculate the mean and standard deviation. The mean is now $32.60.

Which of the following is true about the standard deviation?

Group of answer choices

The standard deviation will decrease, because this change moved a data point closer to the mean.

The standard deviation will increase, because we have increased the value of a data point.

The standard deviation will stay the same, because the standard deviation is not affected by a change in a single measurement.

To understand how the standard deviation will be affected by the change in the data point, let's analyze the situation:

### **Original Situation:**
- **Mean:** $31.52
- **Standard Deviation:** $21.60
- A data point was originally entered as $3.

### **Correction:**
- The data point should have been $30 instead of $3.
- The new mean after correction is $32.60.

### **Impact on Standard Deviation:**

1. **Understanding Standard Deviation:**
   The standard deviation measures the spread or dispersion of data points around the mean. When a data point is closer to the mean, it reduces the spread, thus decreasing the standard deviation. Conversely, if a data point is far from the mean, it increases the spread, leading to a higher standard deviation.

2. **Effect of the Change:**
   - Originally, the value $3 was far from the mean ($31.52), contributing significantly to the standard deviation.
   - After correcting the value to $30, this value is much closer to the new mean ($32.60). 

   Since this data point is now closer to the mean, the overall spread of the data decreases.

### **Conclusion:**
- **The standard deviation will decrease** because this change moved a data point closer to the mean, reducing the overall spread of the data.

So, the correct answer is:

**"The standard deviation will decrease, because this change moved a data point closer to the mean."**

Would you like more details or have any other questions?

### 5 Related Questions:
1. How does changing the mean affect the standard deviation?
2. What is the difference between variance and standard deviation?
3. Can you give an example where removing an outlier decreases the standard deviation?
4. How does the range of a dataset relate to the standard deviation?
5. What happens to the standard deviation if all data points are increased by the same amount?

### Tip:
Remember that the standard deviation is sensitive to changes in individual data points, especially when those changes move points closer to or further from the mean.

Understand how correcting a data point affects standard deviation in statistics. Learn with an example from a college survey where a data point changed from $3 to $30, impacting the mean and standard deviation calculations.

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