Relative Frequency - Algebra-1

1. Fundamental Concepts

Relative Frequency: It refers to the ratio of a specific subcategory (which can be a joint frequency or a marginal frequency) to the total number of the entire population. It is used to indicate the proportion of that subcategory within the overall population. It can more intuitively reflect the distribution characteristics of data and is usually presented in the form of decimals, fractions, or percentages. For example, in a survey on students' sports preferences, "the proportion of 11th-grade students who play golf among the total number of surveyed students" is a type of relative frequency.

2. Key Concepts

The calculation of relative frequency is based on "the number of a subcategory ÷ the total number of the population", and its value ranges between 0 and 1 (or between 0% and 100%).
Relative frequency can be divided into Joint Relative Frequency and Marginal Relative Frequency:
- Joint Relative Frequency: Obtained by dividing a joint frequency by the total number of the population, reflecting the proportion of the combination of two variables in the overall population.
- Marginal Relative Frequency: Obtained by dividing a marginal frequency by the total number of the population, reflecting the proportion of a single variable category in the overall population.
The sum of all relative frequencies is 1 (or 100%), which is an important basis for verifying the correctness of relative frequency calculations.

3. Examples

The following examples are based on a two-way frequency table of "11th-grade and 12th-grade students participating in golf (G) and tennis (T)" (with a total of 45 students, and the data is as follows):

	Golf (G)	Tennis (T)	Row Total
11th Grade	10	15	25
12th Grade	8	12	20
Column Total	18	27	45 (Total number of people)

Easy Level

Calculate the joint relative frequency of "11th-grade students playing golf": 10 ÷ 45 ≈ 0.22 (or 22%).
Calculate the marginal relative frequency of "all students playing tennis": 27 ÷ 45 = 0.6 (or 60%).

Medium Level

Given that the marginal frequency of 12th-grade students is 20, find its marginal relative frequency and determine whether the proportion of students in this grade among the total population exceeds 40%: 20 ÷ 45 ≈ 0.44 (or 44%), and 44% > 40%, so it exceeds.
Calculate the sum of the joint relative frequency of "11th-grade students not playing golf" (i.e., 11th-grade students playing tennis) and the joint relative frequency of "12th-grade students playing golf": 15 ÷ 45 + 8 ÷ 45 = 23 ÷ 45 ≈ 0.51 (or 51%).

Hard Level

If the total number of people increases by 5 (all new people are 12th-grade students playing tennis), find the new joint relative frequency of "12th-grade students playing tennis" and the marginal relative frequency of the "tennis column":
- New joint frequency of 12th-grade students playing tennis: 12 + 5 = 17;
- New total number of people: 45 + 5 = 50;
- New joint relative frequency of "12th-grade students playing tennis": 17 ÷ 50 = 0.34 (or 34%);
- New marginal frequency of the tennis column: 27 + 5 = 32;
- New marginal relative frequency of the "tennis column": 32 ÷ 50 = 0.64 (or 64%).
Analyze why the sum (95.6%) of the marginal relative frequency of "11th-grade students" (25 ÷ 45 ≈ 55.6%) and the marginal relative frequency of the "golf column" (18 ÷ 45 = 40%) does not reach 100%: Because the overlapping part (the relative frequency of 11th-grade students playing golf, which is 22%) is counted twice. The actual proportion in the overall population needs to subtract the overlapping part (55.6% + 40% - 22% = 73.6%), and the remaining part is the relative frequency of 12th-grade students playing tennis (26.4%), with the total being 100%.

4. Problem-Solving Techniques

Clarify the calculation object: Distinguish between calculating joint relative frequency (based on joint frequency) and marginal relative frequency (based on marginal frequency) to avoid confusing subcategories.
Keep the calculation formula in mind: Relative frequency = number of subcategory ÷ total number of population. When calculating, ensure the accuracy of the number of the subcategory and the total number of the population (especially when the data is updated or adjusted).
Verify the sum relationship: Check the correctness of the calculation results by checking whether "the sum of all relative frequencies is 1 (or 100%)". If not, recheck the data and the calculation process.
Analyze in combination with reality: The core of relative frequency is to reflect the "proportional relationship". When solving problems, you can compare the relative frequencies of different categories to judge the distribution differences of data (such as which category has a higher proportion, what the change trend is, etc.).
Flexibly convert forms: Convert relative frequency into decimals, fractions, or percentages according to the problem requirements (for example, percentages are more convenient for intuitively understanding the size of the proportion).