Section 3-9z-Scores

Warm-up1. 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?

Warm-up1. Calculate the mean and standard deviation for these

scores: 68, 73, 84, 90, 95.

Warm-up1. Calculate the mean and standard deviation for these

scores: 68, 73, 84, 90, 95.

Warm-up1. Calculate the mean and standard deviation for these

scores: 68, 73, 84, 90, 95.

Warm-up1. Calculate the mean and standard deviation for these

scores: 68, 73, 84, 90, 95.

Warm-up1. Calculate the mean and standard deviation for these

scores: 68, 73, 84, 90, 95.

x = 82

Warm-up1. Calculate the mean and standard deviation for these

scores: 68, 73, 84, 90, 95.

x = 82 s ≈ 11.33578405

Warm-up2. 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?

Warm-up2. 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?

82 - 68 =

Warm-up2. 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?

82 - 68 = 14

Warm-up2. 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?

82 - 68 = 1414s = ?

Warm-up2. 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?

82 - 68 = 1414s = ?

Warm-up2. 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?

82 - 68 = 1414s = ?

Warm-up2. 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?

82 - 68 = 1414s = ?

above the mean

Warm-up2. 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?

82 - 68 = 1414s = ?

About .7057297462 standard deviations below the mean

above the mean

z-Score

z-Score

Finds out how many standard deviations an individual data point is from the mean

z-Score

Finds out how many standard deviations an individual data point is from the mean

z =

x − xs

Example 1Matt 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?

Example 1Matt 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?

z =

x − xs

Example 1Matt 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?

z =

x − xs

=7.8 − 6.4

1

Example 1Matt 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?

z =

x − xs

=7.8 − 6.4

1 =1.4

Example 1Matt 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?

z =

x − xs

=7.8 − 6.4

1 =1.4

Matt’s score was 1.4 standard deviations above the mean.

Raw Data/Scores:

Standardized Data/Scores:

Raw Data/Scores:

Standardized Data/Scores:

The original data

Raw Data/Scores:

Standardized Data/Scores:

The original data

The transformed data; the z-scores

Theorem:

Theorem:

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

Example 2Fuzzy 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?

Example 2Fuzzy 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:

Example 2Fuzzy 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?

z =

x − xs

Test 1:

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6Test 1:

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1:

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1: ≈ −1.17

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1:

Test 2:

≈ −1.17

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1:

Test 2: z =

x − xs

≈ −1.17

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1:

Test 2: z =

x − xs

=37 − 45

5

≈ −1.17

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1:

Test 2: z =

x − xs

=37 − 45

5 =−85

≈ −1.17

Example 2Fuzzy 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?

z =

x − xs

=83 − 90

6 =−76

Test 1:

Test 2: z =

x − xs

=37 − 45

5 =−85

≈ −1.17

≈ −1.6

Homework

Homework

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