Frequency Tables

Grade 8 · statistics · 100 practice problems · read aloud

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📊 Frequency Tables

What is a Frequency Table?

A frequency table is a simple way to organize data. It shows how often (the "frequency") each item or number appears in a set. This helps you see patterns and understand the data quickly, instead of looking at a long, messy list of numbers.

How to Make a Frequency Table

  1. Tally: Go through your data list. Make a tally mark (|) for each item next to its category.
  2. Count: Add up the tally marks for each category. This is the frequency.
  3. Create the Table: Make a table with two columns: one for the category and one for the frequency.

Example 1: Favorite Colors

Data: Blue, Red, Blue, Green, Red, Red, Blue, Yellow, Green, Red

  1. Tally & Count:
    • Blue: ||| (3)
    • Red: |||| (4)
    • Green: || (2)
    • Yellow: | (1)
  2. Final Table:
    ColorFrequency
    Blue3
    Red4
    Green2
    Yellow1

Example 2: Test Scores

Data: 85, 92, 78, 85, 90, 92, 85, 78

  1. Group Data: Sometimes you group numbers into intervals, like "70-79", "80-89", "90-99".
  2. Tally & Count:
    • 70-79: || (2)
    • 80-89: ||| (3)
    • 90-99: ||| (3)
  3. Final Table:
    Score RangeFrequency
    70-792
    80-893
    90-993

⚠️ Common Mistakes

  • Forgetting to Count Tallies: Don't just copy the tally marks. Write the final number in the frequency column.
  • Unequal Intervals: When grouping numbers, make sure the ranges are the same size (e.g., 0-9, 10-19, not 0-9, 10-14).
  • Missing Data: Double-check that your frequencies add up to the total number of data points.

💡 Tips & Tricks

  • Group Tallies: Remember, every fifth tally mark is a diagonal line through four others (||||) to make counting easier.
  • Check Your Total: Always add all your frequencies. The sum should equal the total number of items in your original data set.
  • Order Your Table: List categories in a logical order (alphabetically, numerically, or from smallest to largest).

Practice This Skill

1. Survey Your Class: Ask friends their favorite snack or hobby and make a frequency table.

2. Use Sports Data: Look at the final scores of basketball games and group them into ranges.

3. Online Games: Find interactive quizzes online that let you build frequency tables from random data sets.

Practice problems

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

1 √(81) + 3² = ?

Hint: Remember to evaluate each mathematical operation separately before combining the results. Consider what number multiplied by itself gives the value under the square root symbol.

Show the answer

Answer: 18

  1. Identify the operations in the problem. We have the square root of 81 and 3 squared: √(81) + 3².
  2. Calculate the square root of 81. The square root of 81 is the number that, when multiplied by itself, gives 81. Since 9 × 9 = 81, √(81) = 9.
  3. Calculate 3 squared. 3² means 3 × 3 = 9.
  4. Add the two results. 9 + 9 = 18.
  5. State the final answer.

The correct answer is 18.

2 √(64) + 4³ - 5² = ?

Hint: Remember to evaluate each term in the correct order: square root, exponent, then addition and subtraction from left to right.

Show the answer

Answer: 47

  1. Evaluate the square root: √(64) = 8
  2. Evaluate the exponent: 4³ = 4 × 4 × 4 = 64
  3. Evaluate the other exponent: 5² = 25
  4. Substitute back into the expression: 8 + 64 - 25
  5. Perform addition first: 8 + 64 = 72
  6. Perform subtraction: 72 - 25 = 47

The answer is 47.

3 (3x + 2)(2x - 5) = ?

Hint: Use the distributive property (FOIL method) to multiply each term in the first binomial by each term in the second binomial.

Show the answer

Answer: 6x² - 11x - 10

  1. Apply the distributive property (FOIL method): First: (3x)(2x) = 6x² Outer: (3x)(-5) = -15x Inner: (2)(2x) = 4x Last: (2)(-5) = -10
  2. Combine all terms: 6x² - 15x + 4x - 10
  3. Combine like terms: -15x + 4x = -11x
  4. Write the final expression: 6x² - 11x - 10

The answer is 6x² - 11x - 10.

4 (2x + 5)² - (3x - 2)² = ?

Hint: Remember to expand each squared binomial using the formula (a + b)² = a² + 2ab + b², then combine like terms carefully.

Show the answer

Answer: -5x² + 32x + 21

  1. Recall the formula for a² - b²** a² - b² = (a - b)(a + b) Here, a = (2x + 5) and b = (3x - 2). So: (2x + 5)² - (3x - 2)² = [(2x + 5) - (3x - 2)] × [(2x + 5) + (3x - 2)] --- **
  2. Simplify the first bracket (a - b)** (2x + 5) - (3x - 2) = 2x + 5 - 3x + 2 = (2x - 3x) + (5 + 2) = -x + 7 --- **
  3. Simplify the second bracket (a + b)** (2x + 5) + (3x - 2) = 2x + 5 + 3x - 2 = (2x + 3x) + (5 - 2) = 5x + 3 --- **
  4. Multiply the two brackets** (-x + 7)(5x + 3) Use distributive property (FOIL): First: (-x)(5x) = -5x² Outer: (-x)(3) = -3x Inner: (7)(5x) = 35x Last: (7)(3) = 21 --- **
  5. Combine like terms** -5x² + (-3x + 35x) + 21 = -5x² + 32x + 21 --- **Final Answer:** -5x² + 32x + 21

Let's solve step-by-step. We have: (2x + 5)² - (3x - 2)² --- **

5 (3x - 2)(2x + 5) - (x + 3)² = ?

Hint: Expand each part using the distributive property, then combine like terms

Show the answer

Answer: 5x² + 5x - 19

  1. Expand (3x - 2)(2x + 5) using FOIL 3x × 2x = 6x² 3x × 5 = 15x -2 × 2x = -4x -2 × 5 = -10 So (3x - 2)(2x + 5) = 6x² + 15x - 4x - 10 = 6x² + 11x - 10
  2. Expand (x + 3)² using (a + b)² = a² + 2ab + b² x² + 2(x)(3) + 3² = x² + 6x + 9
  3. Subtract the second expression from the first (6x² + 11x - 10) - (x² + 6x + 9) = 6x² + 11x - 10 - x² - 6x - 9
  4. Combine like terms 6x² - x² = 5x² 11x - 6x = 5x -10 - 9 = -19
  5. Final result is 5x² + 5x - 19

6 (3x² - 2x + 7) - (x² + 4x - 5) = ?

Hint: Remember to distribute the negative sign to all terms in the second polynomial before combining like terms.

Show the answer

Answer: 2x² - 6x + 12

  1. Write the expression: (3x² - 2x + 7) - (x² + 4x - 5)
  2. Distribute the negative sign: 3x² - 2x + 7 - x² - 4x + 5
  3. Combine x² terms: 3x² - x² = 2x²
  4. Combine x terms: -2x - 4x = -6x
  5. Combine constant terms: 7 + 5 = 12
  6. Write the final expression: 2x² - 6x + 12

The answer is 2x² - 6x + 12.

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