mean, median, and mode of 40, 32, 56, 60, 80, 68, 86, 88

To find the mean, median, and mode of the given data set \(\{40, 32, 56, 60, 80, 68, 86, 88\}\), we need to follow these steps:

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

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

First, sum all the values:

\[
40 + 32 + 56 + 60 + 80 + 68 + 86 + 88 = 510
\]

There are 8 values in total, so the mean is:

\[
\text{Mean} = \frac{510}{8} = 63.75
\]

### Median
The median is the middle value when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers.

First, arrange the numbers in ascending order:

\[
32, 40, 56, 60, 68, 80, 86, 88
\]

Since there are 8 numbers (even), the median is the average of the 4th and 5th numbers:

\[
\text{Median} = \frac{60 + 68}{2} = \frac{128}{2} = 64
\]

### Mode
The mode is the number that appears most frequently in the data set. If no number repeats, the data set has no mode.

In this data set, each number appears exactly once, so there is no mode.

### Summary
- **Mean:** 63.75
- **Median:** 64
- **Mode:** None

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How would the mean, median, and mode change if an additional number, say 72, was added to the data set?
2. What are the implications of having no mode in a data set?
3. How do outliers affect the mean and median of a data set?
4. What is the impact of a symmetrical distribution on the mean, median, and mode?
5. Can a data set have more than one mode? If so, what is it called?

**Tip:** In data sets with outliers, the median is often a better measure of central tendency than the mean.

Calculate the mean, median, and mode of the data set 40, 32, 56, 60, 80, 68, 86, 88 with step-by-step explanations. Understand key statistical concepts and formulas, such as mean, median, and mode, suitable for Grades 6-8 students.

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