To calculate the mean and standard deviation for the given data, we will follow these steps:

1. Identify Midpoints for Each Savings Range

For each savings range, we approximate the value by using the midpoint. The midpoint for each class is calculated by averaging the lower and upper boundaries of the savings ranges.

$\text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

2. Multiply Midpoints by Frequencies

Next, we multiply the midpoint of each class by the frequency of that class to get the total savings for that class.

3. Calculate the Mean

The sample mean is calculated by dividing the sum of the total savings by the total number of respondents (i.e., sum of the frequencies).

4. Calculate the Standard Deviation

The standard deviation involves calculating the variance first, then taking the square root of the variance. The variance is computed by: $\sigma^2 = \frac{\sum{f(x-\bar{x})^2}}{N}$ Where:

• $f$ is the frequency for each class
• $x$ is the midpoint of each class
• $\bar{x}$ is the sample mean
• $N$ is the total number of observations

Let me compute this for you.The approximate mean amount of savings is $233 (rounded to the nearest dollar), and the standard deviation is approximately $230.

Would you like further details on the calculation, or any other questions?

Here are 5 related questions:

1. How is the variance related to the standard deviation in data analysis?
2. Why do we use midpoints to approximate data in grouped frequency distributions?
3. What does the standard deviation tell us about the distribution of savings in this sample?
4. How would changing the savings range affect the mean and standard deviation?
5. What are other ways to represent data variability besides standard deviation?

Tip: A smaller standard deviation indicates that the data points tend to be closer to the mean.