Working with z-scores

z by sandcastlematt

Let's apply what we've learned ...

The frequency distribution table at right shows the midterm marks of 85 Grade 12 math students. The first column shows the mark interval, the second column the average mark within each mark interval, and the third column the number of students at each mark.

(a) Calculate the mean and std. dev. to two decimal places.

(b) Calculate the number of students that have marks within one std. dev. of the mean.

(c) What percent of students have marks within one std. dev. of the mean?

58 or 70

68% or 82%

Mean = 66.99 σ = 13.27

mark interval mark # of students 29 to 37 33 1 38 to 46 42 4 47 to 55 51 12 56 to 64 60 18 65 to 73 69 24 74 to 82 78 16 83 to 91 87 7 92 to 100 96 3

Total 85

This is the correct way to do this.

An experiment was performed to determine the approximate mass of a penny. Three hundred pennies were weighed, and the weights were recorded in the frequency distribution table shown below.

Mass (grams) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4Frequency 2 4 34 71 94 74 17 4

Determine the mean, median, mode, range, and standard deviation of the data. Create a histogram that shows the frequencies of different masses of this set of pennies.

HOMEWORK

Four hundred people were surveyed to find how many videos they had rented during the last month. Determine the mean and median of the frequency distribution shown below, and draw a probability distribution histogram. Also, determine the mode by inspecting the frequency distribution and the histogram.

HOMEWORK

1 28 2 102 3 160 4 70 5 25 6 13 7 0 8 2

No. of Videos Returned No. of Persons

The table shows the weights (in pounds) of 125 newborn infants. The first column shows the weight interval, the second column the average weight within each weight interval, and the third column the number of newborn infants at each weight.(a) Calculate the mean weight and standard deviation.

(d) What percent of the infants have weights that are within one standard deviation of the mean weight?

(c) Determine the number of infants whose weights are within one standard deviation of the mean weight.

(b) Calculate the weight of an infant at one standard deviation below the mean weight, and one standard deviation above the mean.

HOMEWORK weight interval mean interval # of infants 3.5 to 4.5 4 4 4.5 to 5.5 5 11 5.5 to 6.5 6 19 6.5 to 7.5 7 33 7.5 to 8.5 8 29 8.5 to 9.5 9 17 9.5 to 10.5 10 8 10.5 to 11.5 11 4

TOTAL 125

Standard Score AKA z-score: A z-score indicates how many standard deviations a specific score is from the mean of a distribution.

For example, the following table shows some z-scores for a distribution with:

• a mean of 150• a standard deviation of 10

actual score 130 140 150 160 170 174 180standard score -2.0 -1.0 0.0 1.0 2.0 2.4 3.0

A z-score may also be described by the following formula:

z = the z-score (standardized score)x = a number in the distributionμ = the population meanσ = standard deviation for the population

z-scores

Student A from Parkland High and Student B from Metro Collegiate both had a final math mark of 95 percent. The awards committee must select the student with the 'highest' mark for the annual math award. The table shows the mean mark and standard deviation for each school.

Calculate the z-score for each student. Which student should receive the award?

School Mean Standard DeviationParkland High 75 8Metro Collegiate 77 6

The Canadian Armed Forces used to have a height requirement of 158 cm to 194 cm for men. The mean height of Canadian men at that time was 176 cm with a standard deviation of 8 cm. What was the z-score range for allowable heights?

Standardizing Two Sets of ScoresTwo consumer groups, one in Vancouver and one in Halifax, recently tested five brands of breakfast cereal for taste appeal. Each consumer group used a different rating system.

Vancouver HalifaxCereal Brand Rating Cereal Brand Rating A 1 P 25 B 10 Q 35 C 15 R 45 D 21 S 50 E 28 T 70

Use z-scores to determine which cereal has the higher taste appeal rating.

HOMEWORK

The table shows the lengths in millimetres of 52 arrowheads.

16 16 17 17 18 18 18 18 19 20 20 21 2121 22 22 22 23 23 23 24 24 25 25 25 2626 26 26 27 27 27 27 27 28 28 28 28 2930 30 30 30 30 30 31 33 33 34 35 39 40

(a) Calculate the mean length and the standard deviation.

(d) What percent of the arrowheads are within one standard deviation of the mean length?

(c) How many arrowheads are within one standard deviation of the mean?

(b) Determine the lengths of arrowheads one standard deviation below and one standard deviation above the mean.

HOMEWORK

Using Z-Score, Mean, and Standard Deviation to Calculate the Real Score

Numerous packages of raisins were weighed. The mean mass was 1600 grams, and the standard deviation was 40 grams. Trudy bought a package that had a z-score of -1.6. What was the mass of Trudy's package of raisins? HOMEWORK

A survey was conducted at DMCI to determine the number of music CDs each student owned. The results of the survey showed that the average number of CDs per student was 73 with a standard deviation of 24. After the scores were standardized, the people doing the survey discovered that DJ Chunky had a z-score rating of 2.9. How many CDs does Chunky have?

HOMEWORK