Median, Q1, Q3 - Algebra-1

1. Fundamental Concepts

Median: It is the value in the middle of a set of data when the data is sorted in ascending (or descending) order. It divides the data into two equal parts, each accounting for 50%. If the number of data points is odd, the median is the middle number; if even, it is the average of the two middle numbers.
Q1 (Lower Quartile): When the sorted data is divided into four equal parts, Q1 is the value at the 1/4 position, with 25% of the data below it.
Q3 (Upper Quartile): Similarly, when the sorted data is divided into four equal parts, Q3 is the value at the 3/4 position, with 75% of the data below it.

All three are calculated based on sorted data and are key statistical measures describing the position and distribution of data.
The median reflects the central tendency of the data and is not affected by extreme values; Q1 and Q3 are used to divide the distribution range of the data, helping to analyze the degree of data dispersion.

1.
Data: 3, 5, 7, 9, 11
- Median: The middle number, 7
- Q1: The median of the first half (3, 5), i.e., (3 + 5) ÷ 2 = 4
- Q3: The median of the second half (9, 11), i.e., (9 + 11) ÷ 2 = 10
2.
Data: 6, 4, 9, 6, 2, 8, 4, 10
Sorted: 2, 4, 4, 6, 7, 8, 9, 10
- Median: The average of the two middle numbers (6, 7), i.e., (6 + 7) ÷ 2 = 6.5
- Q1: The median of the first half (2, 4, 4, 6), i.e., (4 + 4) ÷ 2 = 4
- Q3: The median of the second half (7, 8, 9, 10), i.e., (8 + 9) ÷ 2 = 8.5
3.
Data: 0, 2, 3, 5, 4, 4, 6, 7, 9, 8, 17
Sorted: 0, 2, 3, 4, 4, 5, 6, 7, 8, 9, 17
- Median: The 6th number, 5
- Q1: The median of the first half (0, 2, 3, 4, 4), i.e., the 3rd number, 3
- Q3: The median of the second half (6, 7, 8, 9, 17), i.e., the 3rd number, 8

Prioritize Sorting: The data must be sorted in ascending (or descending) order before calculation, which is the prerequisite for accurately finding the median, Q1, and Q3.
Locate Key Positions:
- Median position: If the number of data points is n, the position is (n + 1) ÷ 2 (for an odd n, take the value at this position directly; for an even n, take the average of the values at the adjacent two positions).
- Q1 position: The median position of the first half of the data, i.e., (number of data points in the first half + 1) ÷ 2.
- Q3 position: The median position of the second half of the data, calculated in the same way as Q1.
Distinguish Between Odd and Even Counts: When the number of data points is even, the first half and the second half each contain n/2 data points; when odd, the median is a separate middle value, and the first half and the second half each contain (n - 1)/2 data points.