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Get out your Test Corrections!
(and test corrections go on the back table under your class period)
Today’s Objectives:
You will be able to determine a percentile.
You will be able to calculate a z-score.
Warm Up
Find 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 3
Consider 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.
Overview
When 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. 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒𝑠 measure location relative to the
median.
2. 𝑍 − 𝑆𝑐𝑜𝑟𝑒𝑠 measure location relative to the mean.
Where Do I Stand Activity
1. 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 𝒑𝒆𝒓𝒄𝒆𝒏𝒕𝒊𝒍𝒆 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 𝒛 − 𝒔𝒄𝒐𝒓𝒆
corresponding to your height.
Percentile
The percentile of a distribution is the value
with 𝑝 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.
Example
Below is a list of test grades for a class of 24 AIG students.
79
78
83
89
81
80
86
84
80
75
90
82
77
67
79
77
73
73
85
72
83
77
83
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.
Caution
Being 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!
Standardize
Converting observations from original
values (your heights) to standard deviation
units is known as standardizing.
The standardized value of an observation,
𝑥, is
𝑥 − 𝑚𝑒𝑎𝑛
𝑧=
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Z-Score
A standardized value is often called a 𝑧-score.
A 𝑧-score tells use how many standard deviations
from the mean the observation falls, and in what
direction.
Observations larger than the mean have positive
𝑧-scores.
Observations smaller than the mean have
negative 𝑧-scores.
Examples
1. 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?
Examples
2. 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. ACT
Sofia 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 Door
Mrs. 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!