Z-scores, Empirical Rules, and Percentiles

At a local high school, GPAs are normally distributed with a mean of 2.9 and standard deviation of 0.6. What is the z-score for the 80th percentile? What GPA falls at the 80th percentile?

• z= -0.84, A GPA of 3.40

• z= -0,84, A GPA of 0.8

• z= 0.84, A GPA of 3.80

• z= 0.84, A GPA of 3.40

Answer explanation

The z-score for the 80th percentile is 0.84. The GPA that falls at the 80th percentile is 3.40.

Tags

CCSS.HSS.ID.A.4

The percentile represents the....

• percent of data values that are below.

• percent of data values that are above.

• number of data values that are below.

• number of data values that are below.

Answer explanation

The percentile represents the percentage of data values that are below.

Tags

CCSS.6.SP.B.5C

If I tell you that you scored at the 55th percentile on your final exam, you would know:

• 55% of the class scored as good or worse than I did 

• I earned a 55% on the exam

• I failed the exam

• 45% of the class scored worse than I did 

Answer explanation

Scoring at the 55th percentile means that 55% of the class scored as good or worse than you did.

Tags

CCSS.6.SP.B.5C

What is the Z-score for Paula if she got 1400 on the SAT, the mean was 1100 and the SD was 275?

• -1.90

• -1.09

• +1.09

• +1.90

Answer explanation

The Z-score for Paula is +1.09 because it represents how many standard deviations above the mean her score is. A positive Z-score indicates a score above the mean.

A data point has a z-score of -2.50. How should be interpret this information?

• This data point is 2.5 standard deviations above the mean

• This data point is 2.5 standard deviations above the median

• This data point is 2.5 standard deviations below the mean

• This data point is 2.5 standard deviations below the median

Answer explanation

A z-score of -2.50 indicates that the data point is 2.5 standard deviations below the mean. This means it is significantly lower than the average value.

A data set has a mean of 290 and a standard deviation of 20. Calculate the z-score for 265.

Answer explanation

The z-score for 265 is -1.25. It indicates that the value is 1.25 standard deviations below the mean of 290.

Tags

CCSS.HSS.ID.A.4

Steve took his math test and scored an 88. If the class average was a 78 with a standard deviation of 5, what percent of students earned a grade that was higher than Steve's grade?

• 97.5%

• 93%

• 5%

• 2.5%

Answer explanation

The correct answer is 2.5%. Steve's grade is higher than 97.5% of the class, as it is one standard deviation above the mean.

Tags

CCSS.HSS.ID.A.4

• It means you did better than zero people.

• It means you scored the same grade as the average grade.

• It means you earned a zero on the quiz.

• It means the average score was a zero.

Answer explanation

A z-score of 0 means scoring the same grade as the average grade.

Use the image of the normal graph provided. What is the value of the mean?

• 6

• 34

• 52

• 70

Answer explanation

The value of the mean in the normal graph is 52, which is the correct choice.

Tags

CCSS.6.SP.A.3

Use the image of the normal graph provided. What is the value of the standard deviation?

• 6

• 12

• 36

• 18

Answer explanation

The value of the standard deviation is 6. It is calculated as the square root of the variance, which measures the spread of data around the mean.