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Z-Score worksheet

Grades 11–12 · CCSS HSS.ID.A.4

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Find: Value from a z-score
- Z-score (z): 1.5
- Mean (μ): 74
- Standard deviation (σ): 12
Find: Value (x)
Answer
2.
Problem 2.
- Find: Z-score from a value
- Value (x): 68
- Mean (μ): 68
- Standard deviation (σ): 9
Find: Z-score
Answer
3.
Problem 3.
- Find: Z-score from a value
- Value (x): 80
- Mean (μ): 65
- Standard deviation (σ): 12
Find: Z-score
Answer
4.
Problem 4.
- Find: Z-score from a value
- Value (x): 65
- Mean (μ): 58
- Standard deviation (σ): 11
Find: Z-score
Answer
5.
Problem 5.
- Find: Value from a z-score
- Z-score (z): 1.4
- Mean (μ): 100
- Standard deviation (σ): 9
Find: Value (x)
Answer
6.
Problem 6.
- Find: Z-score from a value
- Value (x): 110
- Mean (μ): 78
- Standard deviation (σ): 10
Find: Z-score
Answer
7.
Problem 7.
- Find: Z-score from a value
- Value (x): 80
- Mean (μ): 76
- Standard deviation (σ): 10
Find: Z-score
Answer
8.
Problem 8.
- Find: Z-score from a value
- Value (x): 85
- Mean (μ): 45
- Standard deviation (σ): 9
Find: Z-score
Answer
9.
Problem 9.
- Find: Z-score from a value
- Value (x): 100
- Mean (μ): 60
- Standard deviation (σ): 12
Find: Z-score
Answer
10.
Problem 10.
- Find: Z-score from a value
- Value (x): 100
- Mean (μ): 46
- Standard deviation (σ): 11
Find: Z-score
Answer

Answer key: Z-Score worksheet

1. Value (x): 92
1. Work backward: start at the mean (μ) and add z steps of one standard deviation (σ) each
  $x$ $= μ + z \times σ$
2. Put in your mean, z-score and standard deviation
  $x$ $= 74 + 1.5 \times 12$ $= 74 + 18$ $= 92$
3. Find the share of a bell curve that falls below z. That share, as a percent, is the percentile
  $Φ (1.5)$ $= 93.3193%$
2. Z-score: 0
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{68 - 68}{9}$ $= \frac{0}{9}$ $= 0$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (0)$ $= 50%$
3. Z-score: 1.25
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{80 - 65}{12}$ $= \frac{15}{12}$ $= 1.25$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (1.25)$ $= 89.435%$
4. Z-score: 0.64
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{65 - 58}{11}$ $= \frac{7}{11}$ $= 0.636364$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (0.636364)$ $= 73.773%$
5. Value (x): 112.6
1. Work backward: start at the mean (μ) and add z steps of one standard deviation (σ) each
  $x$ $= μ + z \times σ$
2. Put in your mean, z-score and standard deviation
  $x$ $= 100 + 1.4 \times 9$ $= 100 + 12.6$ $= 112.6$
3. Find the share of a bell curve that falls below z. That share, as a percent, is the percentile
  $Φ (1.4)$ $= 91.9243%$
6. Z-score: 3.2
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{110 - 78}{10}$ $= \frac{32}{10}$ $= 3.2$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (3.2)$ $= 99.9313%$
7. Z-score: 0.4
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{80 - 76}{10}$ $= \frac{4}{10}$ $= 0.4$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (0.4)$ $= 65.5422%$
8. Z-score: 4.44
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{85 - 45}{9}$ $= \frac{40}{9}$ $= 4.444444$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (4.444444)$ $= 99.9996%$
9. Z-score: 3.33
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{100 - 60}{12}$ $= \frac{40}{12}$ $= 3.333333$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (3.333333)$ $= 99.9571%$
10. Z-score: 4.91
1. Find how far your value (x) is from the mean (μ), then count that gap in standard deviations (σ)
  $z$ $= \frac{x - μ}{σ}$
2. Put in your value, mean and standard deviation
  $z$ $= \frac{100 - 46}{11}$ $= \frac{54}{11}$ $= 4.909091$
3. Find the share of a bell curve that falls below z, written Φ(z). Times 100, that's the percentile
  $Φ (4.909091)$ $= 100%$