The document provides instructions and examples for calculating z-scores, which measure how many standard deviations a data point is from the mean. It gives the formula for calculating z-scores and walks through examples of finding the z-scores for individuals' test scores when given the mean and standard deviation. The examples are used to demonstrate that z-scores standardize data so the distribution has a mean of 0 and standard deviation of 1.

1. Section 3-9
z-Scores

2. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.
2. The score of 68 is how many standard deviations
below the mean?
3. The score of 90 is how many standard deviations
above the mean?

3. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.

4. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.

5. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.

6. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.

7. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.
x = 82

8. Warm-up
1. Calculate the mean and standard deviation for these
scores: 68, 73, 84, 90, 95.
x = 82 s ≈ 11.33578405

9. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
3. The score of 90 is how many standard deviations
above the mean?

10. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
82 - 68 =
3. The score of 90 is how many standard deviations
above the mean?

11. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
82 - 68 = 14
3. The score of 90 is how many standard deviations
above the mean?

12. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
14
82 - 68 = 14 =?
s
3. The score of 90 is how many standard deviations
above the mean?

13. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
14
82 - 68 = 14 =?
s
3. The score of 90 is how many standard deviations
above the mean?

14. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
14
82 - 68 = 14 =?
s
3. The score of 90 is how many standard deviations
above the mean?

15. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
14
82 - 68 = 14 =?
s
above
the mean
3. The score of 90 is how many standard deviations
above the mean?

16. Warm-up
2. The score of 68 is how many standard deviations
below the mean?
14
82 - 68 = 14 =?
s
above
the mean
3. The score of 90 is how many standard deviations
above the mean?
About .7057297462 standard deviations below the
mean

17. z-Score

18. z-Score
Finds out how many standard deviations an individual
data point is from the mean

19. z-Score
Finds out how many standard deviations an individual
data point is from the mean
x−x
z=
s

20. Example 1
Matt Mitarnowski took a test at the fourth month of
sixth grade. The mean score of students taking that test
is theoretically 6.4, and the standard deviation is 1.0.
Matt scored a 7.8. What is his z-score?

21. Example 1
Matt Mitarnowski took a test at the fourth month of
sixth grade. The mean score of students taking that test
is theoretically 6.4, and the standard deviation is 1.0.
Matt scored a 7.8. What is his z-score?
x−x
z=
s

22. Example 1
Matt Mitarnowski took a test at the fourth month of
sixth grade. The mean score of students taking that test
is theoretically 6.4, and the standard deviation is 1.0.
Matt scored a 7.8. What is his z-score?
x−x 7.8 − 6.4
z= =
1
s

23. Example 1
Matt Mitarnowski took a test at the fourth month of
sixth grade. The mean score of students taking that test
is theoretically 6.4, and the standard deviation is 1.0.
Matt scored a 7.8. What is his z-score?
x−x 7.8 − 6.4
= 1.4
z= =
1
s

24. Example 1
Matt Mitarnowski took a test at the fourth month of
sixth grade. The mean score of students taking that test
is theoretically 6.4, and the standard deviation is 1.0.
Matt scored a 7.8. What is his z-score?
x−x 7.8 − 6.4
= 1.4
z= =
1
s
Matt’s score was 1.4 standard deviations above the
mean.

25. Raw Data/Scores:
Standardized Data/Scores:

26. Raw Data/Scores: The original data
Standardized Data/Scores:

27. Raw Data/Scores: The original data
Standardized Data/Scores:
The transformed data; the z-scores

28. Theorem:

29. Theorem:
The mean of the z-scores of a data set is 0, and the
standard deviation of the z-scores is 1

30. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?

31. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
Test 1:

32. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x
Test 1: z =
s

33. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90
Test 1: z = =
s 6

34. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
Test 1: z = = =
s 6 6

35. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
≈ −1.17
Test 1: z = = =
s 6 6

36. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
≈ −1.17
Test 1: z = = =
s 6 6
Test 2:

37. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
≈ −1.17
Test 1: z = = =
s 6 6
x−x
Test 2: z =
s

38. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
≈ −1.17
Test 1: z = = =
s 6 6
37 − 45
x−x
Test 2: z = =
5
s

39. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
≈ −1.17
Test 1: z = = =
s 6 6
37 − 45
x−x −8
Test 2: z = = =
5 5
s

40. Example 2
Fuzzy Jeff scored an 83 on a test with a mean of 90
and a standard deviation of 6. He scored a 37 on a
test with a mean of 45 and a standard deviation of 5.
On which test did he score in a lower percentile and
how do you know?
x−x 83 − 90 −7
≈ −1.17
Test 1: z = = =
s 6 6
37 − 45
x−x −8
Test 2: z = ≈ −1.6
= =
5 5
s

41. Homework

42. Homework
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