Find the mean, median, and mode of the following data. If necessary, round to one more decimal place than the largest number of decimal places given in the data.
0.40
0.30
0.64
0.66
0.30
0.53
0.74
0.61
0.28
0.68
0.70
0.70
0.76
0.79
0.53
0.40
0.58
0.51
0.37
0.37

To find the mean, median, and mode of the given data set, we can follow these steps:

### Data Set:
0.40, 0.30, 0.64, 0.66, 0.30, 0.53, 0.74, 0.61, 0.28, 0.68, 0.70, 0.70, 0.76, 0.79, 0.53, 0.40, 0.58, 0.51, 0.37, 0.37

### 1. **Mean**:
The mean is the average of all the numbers. It is calculated by summing all the numbers and then dividing by the number of values.

\[
\text{Mean} = \frac{\sum \text{(all values)}}{\text{number of values}}
\]

Sum of all values:
\[
0.40 + 0.30 + 0.64 + 0.66 + 0.30 + 0.53 + 0.74 + 0.61 + 0.28 + 0.68 + 0.70 + 0.70 + 0.76 + 0.79 + 0.53 + 0.40 + 0.58 + 0.51 + 0.37 + 0.37 = 11.55
\]

Number of values:
\[
20
\]

\[
\text{Mean} = \frac{11.55}{20} = 0.5775
\]

### 2. **Median**:
The median is the middle value when the data set is ordered from smallest to largest. If there is an even number of observations, the median is the average of the two middle numbers.

First, order the dataTo find the mean, median, and mode of the given data set, we can follow these steps:

### Data Set:
0.40, 0.30, 0.64, 0.66, 0.30, 0.53, 0.74, 0.61, 0.28, 0.68, 0.70, 0.70, 0.76, 0.79, 0.53, 0.40, 0.58, 0.51, 0.37, 0.37

### 1. **Mean**:
The mean is the average of all the numbers. It is calculated by summing all the numbers and then dividing by the number of values.

\[
\text{Mean} = \frac{\sum \text{(all values)}}{\text{number of values}}
\]

Sum of all values:
\[
0.40 + 0.30 + 0.64 + 0.66 + 0.30 + 0.53 + 0.74 + 0.61 + 0.28 + 0.68 + 0.70 + 0.70 + 0.76 + 0.79 + 0.53 + 0.40 + 0.58 + 0.51 + 0.37 + 0.37 = 11.55
\]

Number of values:
\[
20
\]

\[
\text{Mean} = \frac{11.55}{20} = 0.5775
\]

### 2. **Median**:
The median is the middle value when the data set is ordered from smallest to largest. If there is an even number of observations, the median is the average of the two middle numbers.

First, order the 
\[
0.28, 0.30, 0.30, 0.37, 0.37, 0.40, 0.40, 0.51, 0.53, 0.53, 0.58, 0.61, 0.64, 0.66, 0.68, 0.70, 0.70, 0.74, 0.76, 0.79
\]

Since there are 20 values, the median is the average of the 10th and 11th values:
\[
\text{Median} = \frac{0.53 + 0.58}{2} = \frac{1.11}{2} = 0.555
\]

### 3. **Mode**:
The mode is the value that appears most frequently in the data set.

From the ordered list, we can see that:
- 0.30 appears 2 times
- 0.37 appears 2 times
- 0.40 appears 2 times
- 0.53 appears 2 times
- 0.70 appears 2 times

Since these values all appear with the same frequency, the data set is **multimodal** with modes: **0.30, 0.37, 0.40, 0.53, 0.70**.

### Final Summary:
- **Mean**: 0.5775
- **Median**: 0.555
- **Mode**: 0.30, 0.37, 0.40, 0.53, 0.70

Would you like further details or have any questions?

Here are five related questions for further exploration:
1. How would the mean change if the data set included an additional value of 1.00?
2. What is the significance of having a multimodal distribution?
3. How would the median change if one of the data points was removed?
4. What are the implications of the mean being higher than the median in this data set?
5. How does the mode help in understanding the distribution of the data?

**Tip:** When data is multimodal, it can indicate that there are multiple common values, which may suggest distinct subgroups within the data.

Learn how to calculate the mean, median, and mode of a data set with a step-by-step solution. This problem covers key statistical concepts, including mean calculation, median determination, and identifying modes in a data set. Suitable for students in Grades 9-12.

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