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z-scores

Statistics · 20 problems, easy to harder

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How we do it

  1. Write the formula
  2. Put the numbers in
  3. Read it
  4. Check

The steps for Calculate z-score of 62, mean 50, sd 8. Every solution names its steps, so you always know what comes next.

Worked examples

Calculate z-score of 62, mean 50, sd 8 Easy

The question

Question: find the z-score of 62, when the mean is 50 and the SD is 8.

z-score of 62: mean 50, SD 8

A z-score tells how many SDs a number is from the mean.

z = how many SDs from the mean

1 Write the formula

z is x minus the mean, all over the SD.

z = x − meanSD

Careful!

Subtract first, then divide.

The whole top is divided by the SD.

Careful: not 62 − 508 ✗

2 Put the numbers in

The top first: 62 minus 50 is 12.

Then 12 divided by 8 is 1.5.

z = 62 − 508 = 128 = 1.5

3 Read it

62 on a number line with the SD marks.

The z numbers count the SDs.

So 62 is 1.5 SDs above the mean.

1.5 SDs above the mean

4 Check

Let's check: start at the mean and go 1.5 SDs.

50 plus 1.5 times 8 is 62.

So it is right!

50 + 1.5 × 8 = 62

Let's look back.

First: write the formula.

Then: put the numbers in.

Then: read it.

Last: check.

Answer
z = 62 − 508 = 128 = 1.5

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Calculate z-score of 34, mean 40, sd 4 Medium

The question

Question: find the z-score of 34, when the mean is 40 and the SD is 4.

z-score of 34: mean 40, SD 4

A z-score tells how many SDs a number is from the mean.

z = how many SDs from the mean

1 Write the formula

z is x minus the mean, all over the SD.

z = x − meanSD

Careful!

Subtract first, then divide.

The whole top is divided by the SD.

Careful: not 34 − 404 ✗

2 Put the numbers in

The top first: 34 minus 40 is −6.

Then −6 divided by 4 is −1.5.

z = 34 − 404 = −64 = −1.5

3 Read it

34 on a number line with the SD marks.

The z numbers count the SDs.

So 34 is 1.5 SDs below the mean.

1.5 SDs below the mean

4 Check

Let's check: start at the mean and go −1.5 SDs.

40 plus −1.5 times 4 is 34.

So it is right!

40 + (−1.5) × 4 = 34

Let's look back.

First: write the formula.

Then: put the numbers in.

Then: read it.

Last: check.

Answer
z = 34 − 404 = −64 = −1.5

▶ Watch it solved

Calculate z-score of 47, mean 40, sd 6 Harder

The question

Question: find the z-score of 47, when the mean is 40 and the SD is 6.

z-score of 47: mean 40, SD 6

A z-score tells how many SDs a number is from the mean.

z = how many SDs from the mean

1 Write the formula

z is x minus the mean, all over the SD.

z = x − meanSD

Careful!

Subtract first, then divide.

The whole top is divided by the SD.

Careful: not 47 − 406 ✗

2 Put the numbers in

The top first: 47 minus 40 is 7.

Then 7 divided by 6 is about 1.17.

z = 47 − 406 = 76 ≈ 1.17

3 Read it

47 on a number line with the SD marks.

The z numbers count the SDs.

So 47 is 1.17 SDs above the mean.

1.17 SDs above the mean

4 Check

Let's check: start at the mean and go 1.17 SDs.

40 plus 1.17 times 6 is about 47.02.

So it is right!

40 + 1.17 × 6 = 47.02

Let's look back.

First: write the formula.

Then: put the numbers in.

Then: read it.

Last: check.

Answer
z = 47 − 406 = 76 ≈ 1.17

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. z-score of 62, mean 50, sd 8
    Answer
    z = 62 − 508 = 128 = 1.5
  2. z-score of 70, mean 60, sd 5
    Answer
    z = 70 − 605 = 105 = 2
  3. z-score of 45, mean 50, sd 5
    Answer
    z = 45 − 505 = −55 = −1
  4. z-score of 80, mean 80, sd 6
    Answer
    z = 80 − 806 = 06 = 0
  5. value with z = 2, mean 50, sd 4
    Answer
    x = 50 + 2 × 4 = 50 + 8 = 58
  6. value with z = -1, mean 30, sd 3
    Answer
    x = 30 + (−1) × 3 = 30 − 3 = 27

Medium

  1. z-score of 34, mean 40, sd 4
    Answer
    z = 34 − 404 = −64 = −1.5
  2. z-score of 115, mean 100, sd 15
    Answer
    z = 115 − 10015 = 1515 = 1
  3. z-score of 72, mean 65, sd 4
    Answer
    z = 72 − 654 = 74 = 1.75
  4. value with z = -1.5, mean 60, sd 4
    Answer
    x = 60 + (−1.5) × 4 = 60 − 6 = 54
  5. value with z = 0.5, mean 200, sd 30
    Answer
    x = 200 + 0.5 × 30 = 200 + 15 = 215
  6. z-score of 140, mean 100, sd 16
    Answer
    z = 140 − 10016 = 4016 = 2.5
  7. compare 80 (mean 70, sd 5) and 85 (mean 75, sd 10)
    Answer
    = 80 is better (z = 2 > 1)
  8. compare 65 (mean 60, sd 10) and 58 (mean 50, sd 4)
    Answer
    = 58 is better (z = 2 > 0.5)

Harder

  1. z-score of 47, mean 40, sd 6
    Answer
    z = 47 − 406 = 76 ≈ 1.17
  2. z-score of 18, mean 25, sd 3
    Answer
    z = 18 − 253 = −73 ≈ −2.33
  3. value with z = 2.5, mean 150, sd 12
    Answer
    x = 150 + 2.5 × 12 = 150 + 30 = 180
  4. value with z = -0.75, mean 80, sd 8
    Answer
    x = 80 + (−0.75) × 8 = 80 − 6 = 74
  5. compare 88 (mean 80, sd 4) and 75 (mean 65, sd 8)
    Answer
    = 88 is better (z = 2 > 1.25)
  6. compare 30 (mean 35, sd 5) and 52 (mean 60, sd 4)
    Answer
    = 30 is better (z = −1 > −2)