Recently, a random sample of 25-34 year olds was​ asked, "How much do you currently have in​ savings, not including retirement​ savings?" The data in the table represent the responses to the survey. Approximate the mean and standard deviation amount of savings.

Savings		Frequency
$0-$199	       348
$200-$399    	94
$400-$599     49
$600-$799		22
$800-$999		13
$1000-$1199   7
$1200-$1399	 2

The sample mean amount of savings is ​$

​(Round to the nearest dollar as​ needed.)

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.

Recently, a random sample of 25-34 year olds was asked, 'How much do you currently have in savings, not including retirement savings?' The data in the table represent the responses to the survey. Approximate the mean and standard deviation amount of savings.

Learn how to approximate the mean and standard deviation of savings for 25-34 year olds using frequency distribution data. This problem covers key statistical concepts, including mean, variance, and standard deviation, and demonstrates how to handle grouped data. Suitable for students in Grades 10-12.

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