Relative Frequencies

Grade 11 · statistics · 100 practice problems · read aloud

🔊 Listen to this explanation

Relative Frequencies 📊

What is it and Why is it Useful?

A relative frequency is the ratio (fraction or proportion) of the number of times a value occurs in a dataset to the total number of data points. Instead of just a count, it tells you how often something happens relative to the whole. This is crucial because it allows you to compare datasets of different sizes and understand probability and distribution.

How to Calculate Relative Frequency

  1. Step 1: Tally the frequency for the specific category or value you're interested in.
  2. Step 2: Find the total number of data points in the entire dataset.
  3. Step 3: Divide the category frequency by the total frequency.
  4. Step 4: (Optional) Multiply by 100 to express the relative frequency as a percentage.

Formula: Relative Frequency = (Frequency of Category) / (Total Frequency)

Worked Examples

Example 1: Simple Data Set

A survey asks 50 students their favorite subject: Math (15), Science (20), English (10), History (5). Find the relative frequency of students who prefer Science.

  1. Frequency of Science: 20
  2. Total Frequency: 15 + 20 + 10 + 5 = 50
  3. Relative Frequency = 20 / 50 = 0.4
  4. As a percentage: 0.4 × 100 = 40%

Example 2: Two-Way Table

This table shows data from 100 people on pet ownership and allergies.

Has a Pet | No Pet
Allergies: 15 | 25
No Allergies: 45 | 15

Question: What is the relative frequency of people who have a pet?

  1. Frequency of "Has a Pet": 15 (with allergies) + 45 (no allergies) = 60
  2. Total Frequency: 100
  3. Relative Frequency = 60 / 100 = 0.6 or 60%

Common Mistakes to Avoid ⚠️

  • Forgetting the Total: Always double-check that your denominator is the grand total of all data points, not just a subtotal.
  • Sum Check: The sum of all relative frequencies in a dataset should always equal 1 (or 100%). If it doesn't, you've made a calculation error.
  • Confusing Frequency Types: Don't mix up frequency (the count) with relative frequency (the proportion). Pay attention to what the question is asking for.

Tips & Tricks

  • Memory Aid: Think "Part over Whole".
  • Decimal & Percentage: Get comfortable converting between decimals, fractions, and percentages, as all three are used to represent relative frequency.
  • Probability Link: Relative frequency is an empirical probability. If 40% of students prefer science, the probability a randomly selected student prefers science is 0.4.

How to Practice

  • Create your own two-way tables from real-world scenarios (e.g., class data on handedness and eye color).
  • Find statistics articles online and try to calculate the relative frequencies from the raw numbers they provide.
  • Practice problems where you must find missing frequencies in a table when given some relative frequencies.
  • Always check your work by ensuring all relative frequencies add up to 1.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 In a survey of 161 students, Sophia found that 96 preferred online learning. The relative frequency is 0.596. Interpret this value in context.

Hint: Consider what the relative frequency represents as a proportion of the total group surveyed.

Show the answer

Answer: Approximately 59.6% of the surveyed students prefer online learning.

  1. The relative frequency is calculated as (number who prefer online learning) / (total surveyed)
  2. 96 / 161 = 0.596
  3. This means 59.6% of the 161 surveyed students prefer online learning
  4. In context, this indicates that slightly more than half of the students in this survey prefer online learning over other options.

2 A circle is inscribed in a right triangle with legs of length 6 cm and 8 cm. The circle touches all three sides of the triangle. Calculate the radius of the inscribed circle.

Hint: Consider the relationship between the triangle's area, its perimeter, and the radius of the inscribed circle. A different right triangle with legs 5 and 12 would have an inscribed circle radius that can be found using the same geometric principle.

Show the answer

Answer: 2

  1. Calculate the hypotenuse of the right triangle using the Pythagorean theorem. Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm
  2. Calculate the area of the triangle. Area = (1/2) * base * height = (1/2) * 6 * 8 = 24 cm²
  3. Use the formula for the radius of an inscribed circle in a triangle: r = (2 * Area) / Perimeter Perimeter = 6 + 8 + 10 = 24 cm
  4. Calculate the radius. r = (2 * 24) / 24 = 48 / 24 = 2 cm The radius of the inscribed circle is 2 cm.

3 A circle is inscribed in a right triangle with legs of length 6 cm and 8 cm. The circle touches all three sides of the triangle. Calculate the exact radius of the inscribed circle.

Hint: Consider the relationship between the triangle's area, its perimeter, and the inscribed circle's radius. A different triangle with legs 5 and 12 would have an area equal to half the product of its legs.

Show the answer

Answer: 2

  1. Calculate the area of the triangle. For a right triangle, area = (1/2) * leg1 * leg2 = (1/2) * 6 * 8 = 24 cm².
  2. Calculate the hypotenuse using the Pythagorean theorem: hypotenuse = sqrt(6² + 8²) = sqrt(36 + 64) = sqrt(100) = 10 cm.
  3. Calculate the perimeter (semi-perimeter): perimeter = 6 + 8 + 10 = 24 cm, so semi-perimeter s = 24 / 2 = 12 cm.
  4. For any triangle, area = r * s, where r is the inradius and s is the semi-perimeter.
  5. Substitute known values: 24 = r * 12.
  6. Solve for r: r = 24 / 12 = 2 cm. The exact radius is 2 cm.

4 Aroha surveyed 175 students about their favorite subject. 63 preferred Mathematics. What is the relative frequency of students who prefer Mathematics, and what does this value mean in context?

Hint: Relative frequency is calculated by dividing the number of times an outcome occurs by the total number of observations. Think about what this ratio represents in terms of the whole group.

Show the answer

Answer: 0.36

  1. Identify the number of students who prefer Mathematics: 63
  2. Identify the total number of students surveyed: 175
  3. Calculate the relative frequency: 63 ÷ 175
  4. Perform the division: 63 ÷ 175 = 0.36
  5. Interpretation: This means that 36% of the students surveyed prefer Mathematics. It represents the proportion of the total group that has this preference.

5 In a survey of 180 students, Hana found that 54 students prefer online learning. What is the relative frequency of students who prefer online learning, and what does this value mean in context?

Hint: Relative frequency is calculated by dividing the number of students with a specific preference by the total number of students surveyed. Consider what this ratio represents about the population.

Show the answer

Answer: 0.3

  1. Identify the number of students who prefer online learning: 54
  2. Identify the total number of students surveyed: 180
  3. Calculate the relative frequency: 54 ÷ 180 = 0.3
  4. Interpret the meaning: A relative frequency of 0.3 means that 30% of the surveyed students prefer online learning, or that the proportion of students who prefer online learning in this sample is 0.3.

6 In a survey of 180 students, Hana found that 54 students preferred online learning. What is the relative frequency of students who prefer online learning, and what does this value represent in context?

Hint: Relative frequency is calculated by dividing the number of students with a specific preference by the total number of students surveyed. This value represents a proportion of the whole group.

Show the answer

Answer: 0.3

  1. Identify the number of students who prefer online learning: 54
  2. Identify the total number of students surveyed: 180
  3. Calculate relative frequency: 54 ÷ 180 = 0.3
  4. Interpretation: This means 30% of the students surveyed prefer online learning, or the proportion of students who prefer online learning is 0.3.
Practise this topic — 10 free problems, no signup →