Z-Scores and Standard Deviation

Grade 10 · mathematics · 100 practice problems · read aloud

🔊 Listen to this explanation

Z-Scores and Standard Deviation

🎯 What is it and Why is it Useful?

A Z-score tells you how many standard deviations a specific data point is from the mean (average) of a data set. It's incredibly useful for comparing scores from different data sets and understanding if a value is typical or unusual.

📝 Step-by-Step Guide

To calculate a Z-score, use this formula:

z = (x - μ) / σ

  1. Identify your values: x (data point), μ (mean), σ (standard deviation)
  2. Subtract: (x - μ) — Find the difference between the data point and the mean
  3. Divide: Divide the result by the standard deviation (σ)

🔢 Visual Examples

Example 1: Test scores have a mean (μ) of 75 and standard deviation (σ) of 5. What is the z-score for a student who scored 82?

Step 1: z = (82 - 75) / 5

Step 2: z = 7 / 5

Step 3: z = 1.4

Interpretation: This score is 1.4 standard deviations above the mean.

Example 2: The same test, but a student scored 68.

Step 1: z = (68 - 75) / 5

Step 2: z = -7 / 5

Step 3: z = -1.4

Interpretation: This score is 1.4 standard deviations below the mean.

⚠️ Common Mistakes

  • Forgetting the order: Always subtract the mean FROM the data point (x - μ), not the other way around!
  • Misinterpreting negative z-scores: A negative z-score isn't bad—it just means the value is below average.
  • Using the wrong formula: Remember this is for population data. For samples, the formula changes slightly.

💡 Tips & Tricks

  • Memory Aid: "Your score minus the average, all over the spread"
  • Quick Check: About 68% of data falls between z = -1 and z = +1 in a normal distribution.
  • Interpretation: The farther the z-score is from 0, the more unusual the data point.

🏋️ Practice Suggestions

  • Create your own data sets with classmates' heights or mock test scores
  • Practice calculating both mean and standard deviation first, then z-scores
  • Use real-world scenarios: compare sports statistics, weather temperatures, or product prices
  • Try interpreting z-scores: Is a z-score of 2.5 unusual? What about -0.3?

Practice problems

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

1 z = (82 - 70) ÷ 4 = ?

Hint: Remember that the z-score formula measures how many standard deviations a value is from the mean. Focus on the order of operations within the parentheses first.

Show the answer

Answer: 3

  1. Calculate the difference between the data value and the mean: 82 - 70 = 12
  2. Divide this difference by the standard deviation: 12 ÷ 4 = 3
  3. The result is the z-score: z = 3

The answer is 3.

2 μ = 68, σ = 5, x = 78, z = ?

Hint: Recall the formula for calculating a z-score and substitute the given values appropriately.

Show the answer

Answer: 2

  1. Write the z-score formula: z = (x - μ) / σ
  2. Substitute the given values: z = (78 - 68) / 5
  3. Calculate the numerator: 78 - 68 = 10
  4. Divide by the standard deviation: 10 / 5 = 2
  5. The z-score is 2.

3 μ = 82, σ = 6, x = 97, z = ?

Hint: The z-score formula measures how many standard deviations a data point is from the mean. For a different scenario with mean 50 and standard deviation 5, a value of 60 would be 2 standard deviations above the mean.

Show the answer

Answer: 2.5

  1. Recall the z-score formula: z = (x - μ) / σ
  2. Substitute the given values: z = (97 - 82) / 6
  3. Calculate the numerator: 97 - 82 = 15
  4. Divide by the standard deviation: 15 ÷ 6 = 2.5
  5. The z-score is 2.5, meaning the data point is 2.5 standard deviations above the mean.

The answer is 2.5.

4 Given μ = 82, σ = 6, and x = 97, find z = ?

Hint: Recall the formula for converting a raw score to a standardized score. It involves subtracting the central value and then dividing by the spread.

Show the answer

Answer: 2.5

  1. Write the z-score formula: z = (x - μ) / σ
  2. Substitute the given values: z = (97 - 82) / 6
  3. Calculate the subtraction: 97 - 82 = 15
  4. Divide the result by the standard deviation: 15 / 6 = 2.5
  5. The z-score is 2.5.

5 A normal distribution has μ = 75 and σ = 8. If x = 91, then z = ?

Hint: To find a z-score, subtract the population mean from the data value and divide by the standard deviation. For example, if μ = 100 and σ = 15, and x = 130, then z = (130 - 100) / 15.

Show the answer

Answer: 2

  1. Substitute the given values into the formula. x = 91, μ = 75, σ = 8 z = (91 - 75) / 8
  2. Calculate the numerator (x - μ). 91 - 75 = 16
  3. Divide the result by σ. 16 / 8 = 2
  4. Interpret the result. The z-score is 2, which means x = 91 is 2 standard deviations above the mean. Final answer: z = 2

We are given a normal distribution with mean μ = 75 and standard deviation σ = 8. We have a data point x = 91, and we want to find the z-score. The formula for the z-score is: z = (x - μ) / σ

6 A normal distribution has μ = 82 and σ = 6. If x = 97, then z = ?

Hint: The z-score measures how many standard deviations a data point is from the mean. Consider the formula that relates the data value, mean, and standard deviation.

Show the answer

Answer: 2.5

  1. Recall the z-score formula: z = (x - μ) / σ
  2. Substitute the given values: z = (97 - 82) / 6
  3. Calculate the numerator: 97 - 82 = 15
  4. Divide by the standard deviation: 15 / 6 = 2.5
  5. The z-score is 2.5

The answer is 2.5.

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