Get out your Test Corrections!(and test corrections go on the back table under your class period)

You will be able to determine a percentile.You will be able to calculate a z-score.

Today’s Objectives:

Warm UpFind the mean, standard deviation, and 5# Summary of the following sets of numbers.1. {12, 32, 41 53, 73, 10, 85, 54, 36, 74, 8}2. {4, 5, 6, 2, 3, 4, 5, 8, 6, 7, 9, 10, 2, 1, 2}3. {43, 45, 36, 42, 35, 47, 46, 45. 32. 34}

Unit 2: Chapter 3Consider each of the following situations:• A student gets a test back with a score of 78 marked

clearly at the top. • A middle-aged man goes to his doctor to have his

cholesterol checked. His total cholesterol reading is 210 mg/dl.

• An employee in a large company earns an annual salary of $42,000.

• A 10th grader scores 46 on the PSAT Writing test.

Isolated numbers don’t always provide enough information.

OverviewWhen you first get a test back, you generally look for your grade. The next thing you want to know is how you did relative to the other students in the class. In this section, you will learn two ways to describe the location of an individual within the distribution of a quantitative variable:

1. measure location relative to the median.

2. measure location relative to the mean.

Where Do I Stand Activity1. Convert your height into inches.

2. At the appropriate time, each student should stand above the appropriate location along the number line in the front of the classroom based on your height (to nearest inch).

Where Do I Stand?3. Once everyone is on the number line,

count the number of people in the class that have heights less than or equal to your height. Remember this value.

4. Once everyone has their value, you may be seated.

Where Do I stand?1. Take out a sheet of paper.

2. Put your name at the top.

3. Number your paper 1-5.

Where Do I stand?1. Write down the number of students at

or below you. What percent of the students in the class have heights equal to or less than yours?

This is your in the distribution of heights.

Where Do I stand?2. Calculate the mean and standard

deviation of the class’s height distribution from the dot plot on the board. (use your calculator)

Where Do I stand?3. Where does your height fall relative to

the mean: above or below?

4. How far above or below the mean is it? (subtract the mean from your height—negatives are ok)

Where Do I stand?5. How many standard deviations above

or below the mean is it? (take your answer for #4 and divide by the standard deviation)

This is your corresponding to your height.

PercentileThe percentile of a distribution is the value with percent of the observations less than or equal to it. Percentiles are based off of the median.

Note: you will never see a standardized test score reported above the 99th percentile.

ExampleBelow is a list of test grades for a class of 24 AIG students.

79 81 80 77 73 8378 80 75 67 73 7783 86 90 79 85 8389 84 82 77 72 74

Determine the percentile of the following students.1. If Jenny scored an 86 on the test.2. Greg scored a 72 on the test.3. Christopher scored an 80%.4. Samantha scored a 67%.

Example: PSAT scores Nationally, 6 percent of test-takers earned a score higher than 65 on the Critical Reading test’s 20 to 80 scale.

Scott was one of 50 junior boys to take the PSAT at his school. He scored 65 on the Critical Reading test. This placed Scott at the 68th percentile within the group of boys.

Write a sentence or two comparing Scott’s percentile among the national group of test takers and among the 50 boys at his school.

CautionBeing at a higher percentile isn’t always better. For example, you really don’t want your doctor to tell you that your weight is at the 98th percentile for your height!

StandardizeConverting observations from original values (your heights) to standard deviation units is known as standardizing.

The standardized value of an observation, , is

Z-ScoreA standardized value is often called a -score.

A -score tells use how many standard deviations from the mean the observation falls, and in what direction.

Observations larger than the mean have positive -scores.

Observations smaller than the mean have negative -scores.

Examples1. A certain brand of automobile tire has a mean life span of

35,000 miles and a standard deviation of 2250 miles. If the life spans of three randomly selected tires are 34,000 miles, 37,000 miles, and 31,000 miles. Find the z-scores that correspond with each of these mileages. Would the life spans of any of the tires be considered unusual?

Examples2. A highly selective university will only admit students who

place at least 2-zcores above the mean on the ACT that has a mean of 18 and a standard deviation of 6. What is the minimum score that an applicant must obtain to be admitted to the university?

Example: PSAT scores (continued)

1. In October 2007, about 1.4 million college-bound high school juniors took the PSAT. The mean score on the Critical Reading test was 46.7 and the standard deviation was 11.3.

Scott scored 65 on the Critical Reading test. Looking at all 50 boys’ Critical Reading scores, the mean was 58.2 and the standard deviation was 9.4.

Calculate and write a sentence or two to compare Scott’s two z-scores based on his school and nationally.

SAT vs. ACTSofia scores 660 on the SAT Math test. The distribution of SAT scores in the population is roughly symmetric with a mean 500 and standard deviation 100. Jim takes the ACT Math test and scores 26. ACT scores are also symmetric with mean 18 and standard deviation 6.

Assuming that both tests measure the same kind of ability, who did better?

Ticket Out The DoorMrs. Munson is concerned about how her daughter’s height and weight compare with those of other girls her age. She uses an online calculator to determine that her daughter is at the 87th percentile for weight and the 67th percentile for height.

Explain to Mrs. Munson in paragraph form what this means.

Homework

Percentile & Z-score Worksheet

Due MONDAY!