Mean, Median, Mode - Algebra-1

1. Fundamental Concepts

Mean: It is the arithmetic average of a set of data, obtained by summing all the data and then dividing by the number of data points. It reflects the average level of the data.
Median: It is the value in the middle position after arranging a set of data in ascending (or descending) order. If the number of data points is odd, the middle number is the median; if the number of data points is even, the median is the average of the two middle numbers. It is not affected by extreme values and can reflect the middle level of the data.
Mode: It is the value that appears most frequently in a set of data. A set of data may have one mode, multiple modes, or no mode. It reflects the most common value in the data.

Mean, median, and mode are all statistics that describe the central tendency of data and are used to summarize the overall characteristics of the data.
The mean is easily affected by extreme values. When there are extreme values in the data, the mean may not well reflect the central tendency of the data, and the median may be more appropriate in such cases.
The mode is mainly used to describe the central tendency of categorical or discrete data and may not exist or be obvious in continuous data.
Using the three together can help people understand the distribution of data more comprehensively.

Easy Level
- Data: 2, 4, 4, 6, 8
- Mean: (2 + 4 + 4 + 6 + 8) ÷ 5 = 24 ÷ 5 = 4.8
- Median: Arrange the data in ascending order as 2, 4, 4, 6, 8. The number of data points is 5 (odd), so the middle number, which is 4, is the median.
- Mode: 4 appears the most frequently, 2 times, so the mode is 4.
Medium Level
- Data: 3, 5, 7, 9, 11, 13
- Mean: (3 + 5 + 7 + 9 + 11 + 13) ÷ 6 = 48 ÷ 6 = 8
- Median: Arrange the data in ascending order as 3, 5, 7, 9, 11, 13. The number of data points is 6 (even), so the median is the average of the two middle numbers, (7 + 9) ÷ 2 = 8.
- Mode: All data points appear once, so there is no mode.
Difficult Level
- Data: 10, 15, 12, 10, 200, 13, 10
- Mean:
  - Including the outlier: (10 + 15 + 12 + 10 + 200 + 13 + 10) ÷ 7 = 270 ÷ 7 ≈ 38.57
  - Excluding the outlier (the outlier is 200): (10 + 15 + 12 + 10 + 13 + 10) ÷ 6 = 70 ÷ 6 ≈ 11.67
- Median: Arrange the data in ascending order as 10, 10, 10, 12, 13, 15, 200. The number of data points is 7 (odd), so the middle number, 12, is the median.
- Mode: 10 appears the most frequently, 3 times, so the mode is 10.

Calculating the Mean:
1. First, sum all the data to get the total.
2. Count the number of data points.
3. Divide the total by the number of data points to get the mean. If there are extreme values, the mean with and without the extreme values can be calculated separately to better understand the data.
Calculating the Median:
1. Arrange the data in ascending (or descending) order.
2. Determine the median based on whether the number of data points is odd or even: if the number of data points is odd, the number in the middle position is the median; if even, the average of the two middle numbers is the median.
Calculating the Mode:
1. Count the frequency of each data point in the set of data.
2. The data point with the highest frequency is the mode. If multiple data points have the same highest frequency, they are all modes; if all data points have the same frequency, there is no mode.