Name: __________________________
What You’ll Learn



Mean, Median and Mode
Trimmed and Weighted Mean
Percentile
Why It’s Important
Statistics are used by:




Governments
Researchers
Advertisers
Businesses
Key Formulas
𝐖𝐞𝐢𝐠𝐡𝐭𝐞𝐝 𝐌𝐞𝐚𝐧 =
data(weight) + data(weight) + data (weight) + ⋯
total weight
Percentile Rank =
MAE40S
b
n
x 100
Mrs. Graham
Grade 12 Essentials - Statistics
Measure of Central Tendency: Notes
Statistics is all about
data set is a collection of related
and
sets of data. A
.
To start us off, let’s collect a set of data from our own class.
Question: How many minutes do you usually spend getting ready in the
morning?
Class results:
With this data we can calculate:
Mean: Determined by
data set by the number of values in the set.
of all the values in a
Median: Represented by the
of an ordered data set. If the
data set has more than one middle value, the median is the mean of the
Mode: Represented by the value that occurs
set.
in a data
The above methods are all a measure of central tendency. This is when a single
value attempts to describe a
by identifying the
within the data set.
2
Grade 12 Essentials - Statistics
Mean
Median
_____________________________________
______________________________________
How to find the mean:
How to find the median:
1. Add all the numbers in the set.
2. Divide your sum but the quantity of
numbers added together
1. List the numbers from smallest to largest.
2. The middle number is the median
*if there are two numbers in the middle, the
mean of those numbers is the median
Mode
______________________________________
How to find the mode:
1. List the numbers from smallest to largest.
2. The number that occurs most often is the
mode.
*there can be more than one mode
Example 1) What is the mean, median, and mode for this number set?
[13, 14, 12.5, 14, 15, 16]
Example 2) A Grade 12 class with 15 students was polled to determine the
number of television sets they had in their homes. These are the results:
3, 4, 1, 3, 2, 2, 1, 1, 4, 5, 8, 3, 2, 7, 4
What is the mean, median, and mode of this set of data?
3
Grade 12 Essentials - Statistics
Central Tendency: Practice
Find the mean, median, and mode for the following data sets.
1. Five test scores were 60, 67, 73, 63, and 67.
2. Seven people were asked how many kilometres they lived from work. Their
responses were 15, 7, 14, 21, 5, 9, and 13.
3. At a pet store, a survey was taken asking how many pets each person
had. The results were 2, 5, 3, 1, 0, 4, 2, 7, 0, 2, 7, 3.
4. A sample of eight students were randomly selected and asked, “How
many times did you check your email yesterday?” The responses were
3, 0, 8, 7, 2, 6, 12.
4
Grade 12 Essentials - Statistics
5. Chloe’s soccer team has the following goals per game record:
1, 5, 2, 0, 2, 6, 7, 1, 8, 1, 2, 1, 0.
6. Barrie Gwillim operates a farm in Strasbourg, SK. He keeps meticulous
records of his yield per acre for his different crops. One of the crops he has
tracked is hard red spring wheat. The yields in bushels per acre from 2000
to 2010 are shown in the table below.
YIELD OF HARD RED SPRING WHEAT PER YEAR, 2000-2010
a)
Year
Yield (bu/ac)
2000
36
2001
30
2002
22
2003
40
2004
36
2005
40
2006
33
2007
50
2008
51
2009
55
2010
35
What is the mean yield of Barrie’s crop per year?
b)
What is the median yield of hard red spring wheat per year?
c)
Does the data have a mode? If so, what is it?
d)
Which measure of central tendency better represents Barrie’s expected
yields? Explain your reasoning.
5
Grade 12 Essentials - Statistics
Outliers and Trimmed Mean: Notes
An outlier is a value in a data set that is very
values in the set.
from other
For example, in the scores [3, 25, 27, 28, 29, 32, 33, 85] both ______________ are
considered “outliers.”
Example 1) There are six people in a group that are 61, 61, 63, 64, 66, and 90
inches tall.
a) Determine the mean, median, and mode.
b) What is the outlier?
c) If you remove the outlier, which measure of central tendency is affected
most?
6
Grade 12 Essentials - Statistics
Trimmed Mean is the mean calculated when a certain number of the
and
scores are discarded.
NOTE: A trimmed mean is less susceptible to the effects of extreme data
(outliers)
Example 2) Find the trimmed mean of 2, 35, 46, 47, 51, 51, 59, 61, 121 by
removing the highest and lowest data.
Example 3) The manager of the Manitoba Ski Hill rental shop collects data on
how many snowboards were rented during the 20 Saturdays of winter. The
numbers of snowboards rented each Saturday were as follows:
100, 99, 32, 87, 74, 76, 95, 53, 69, 80, 91, 105, 156, 109, 93, 83, 92, 82, 94, 102
Calculate the trimmed mean by removing the two highest and two lowest
scores.
7
Grade 12 Essentials - Statistics
Outliers and Trimmed Mean: Practice
1. Find the outliers for the following data sets. State the new data sets after
removing these outliers.
a) 5, 8, 10, 13, 7, 9, 66
b) 12, 14, 16, 15, 14, 43, 12, 18, 9, 0, 13, 11, 23
2. Calculate the trimmed mean of the following data by removing the
highest and lowest scores.
a) 4, 7, 3, 8, 12, 34, 23, 12, 34, 23, 41, 73, 46, 14, 94.
b) 12, 18, 17, 8, 0, 23, 15, 8, 9, 5.
3. A billionaire is in a room with 10 farm workers. The billionaire’s yearly
income is $40 000 000. The 10 farm workers each earn a yearly income of
approximately $30 000.
a) What is the mean income of all the people in the room?
b) Is the billionaire’s income classified as an outlier? Why or why not?
c) How does the outlier affect the mean income? Is the mean an
accurate representation of the typical income in the room?
d) Which measure of central tendency (mean, median, or mode) would
be the best representative of the typical income in the room?
8
Grade 12 Essentials - Statistics
4. Maddox wants to run the 100 m race at his school’s track event. To find his
typical race time, his physical education teacher is deciding whether or
not to use the mean or the trimmed mean. Over the course of the month,
Maddox has run 20 races to determine if he will qualify for the track team.
Maddox’s race times are measured in seconds and are as follows:
12.37
12.27
12.46
12.61
12.45
13.51
13.64
12.63
13.35
12.84
11.99
11.67
12.00
11.95
13.75
12.52
12.75
12.63
12.73
13.52
a) Find the mean of Maddox’s race times.
b) Find the trimmed mean of Maddox’s race times by removing the
highest and lowest times.
c) Which value should the teacher use to determine whether Maddox
makes the track team? Explain.
9
Grade 12 Essentials - Statistics
Weighted Mean: Notes
Weighted mean is similar to the mean, except some values contribute
and some values contribute
.
Formula to calculate weighted mean:
weighted mean =
𝐝𝐚𝐭𝐚(𝐰𝐞𝐢𝐠𝐡𝐭) + 𝐝𝐚𝐭𝐚(𝐰𝐞𝐢𝐠𝐡𝐭) + 𝐝𝐚𝐭𝐚 (𝐰𝐞𝐢𝐠𝐡𝐭) + ⋯
𝐭𝐨𝐭𝐚𝐥 𝐰𝐞𝐢𝐠𝐡𝐭
Example 1) The teachers of two Grade 12 English classes want to find the mean
final exam score for both classes. The first class has 24 students and its mean final
exam score was 62%. The second class has 30 students and its mean final exam
score was 78%. What is the mean final exam score of both classes?
10
Grade 12 Essentials - Statistics
Example 2) Fifty students taking Grade 12 Essential Mathematics were asked
how many hours they spent studying for the exam. 20 students said they spent 6
hours studying, 13 students said they spent four hours studying, 9 students said
they spent two hours studying, and 8 students said they spent no time studying.
Calculate the mean time these students spend studying.
Example 3) You want to determine your final mark in Grade 12 Essential
Mathematics. Assignments are worth 75%, the midterm exam is work 12.5%, and
your final exam is worth 12.5%. You received a mean mark of 76 on all of your
assignments, a mark of 68 on your midterm exam, and 80 on your final exam.
What is your final mark?
11
Grade 12 Essentials - Statistics
Weighted Mean: Practice
1. In one high school psychology class, the marks are weighted as follows:
Category
Weight
Janie’s Mark
Scotty’s Mark
Test 1
15%
80%
79%
Test 2
25%
73%
84%
Test 3
10%
69%
68%
Assignments
20%
83%
52%
Participation
5%
100%
97%
Final Exam
25%
65%
93%
a) What is Janie’s final mark?
b) What is Scotty’s final mark? Round your answer to the nearest whole
number.
c) Scotty earned lower marks in every item except for two. Why is Scotty’s
final mark higher than Janie’s final mark if Janie did better in more
items?
12
Grade 12 Essentials - Statistics
2. Residents from two towns in Manitoba were asked for their IQ scores. 250
people from A-Town reported and the mean IQ score was 120. 100
people from B-Town reported and the mean IQ score was 90.
a) What is the mean IQ of all the residents (combine both towns)? Round
your answer to the nearest whole number.
b) Is the weighted mean IQ closer to the IQ of A-town or B-town? Why do
you think this is?
3. One retail company wants to determine how long its employees have
been working for them. It found the following statistics:
8 people had worked at the company for 1 year.
5 people had worked at the company for 2 years.
4 people had worked at the company for 3 years.
7 people had worked at the company for 4 years.
5 people had worked at the company for 5 years.
a) Calculate the mean number of years that employees have worked for this
company using the weighted mean.
b) Consider what it would take to find the regular (or arithmetic) mean. Why
is this easier to calculate the weighted mean than the regular mean?
13
Grade 12 Essentials - Statistics
4. In a Brandon high school, chemistry is taught during the fall semester, the
spring semester, and summer school. The following chart displays the
percentage of students who passed the chemistry course.
If 700 people took the course in fall, 500 took
the course in winter, and 100 took the
course in summer, what percentage who
took this course passed? Round your answer
to the nearest tenth (one decimal place).
Term
Fall
Winter
Summer
Percentage
Who Passed
74%
80%
68%
5. A construction company is starting up and they have 25 new employees.
What is the weighted mean if the average starting salary for the 8
journeyman carpenters is $55 000 and the average starting salary for the
17 labourers is $33 000?
14
Grade 12 Essentials - Statistics
Percentile Rank: Notes
Percentile is a value below which a certain
falls.
Percentile rank is a number between
percent of cases that fall at or


of the data set
that indicates the
that score.
Percentile rank is used to compare one piece of data with the rest of the
data
It is denoted P#
Percentile Rank:
P=
b
n
x 100
b = number of scores below and equal to the given
score.
n= the total number of scores
Modifications on the formula:
b=
n=
P
100
xn
b × 100
P
* ROUND UP to the next whole number
Some examples of percentile ranks:
o a child is in the 80th percentile for weight, ____________
o a student’s exam mark was in the 45th percentile, ______________
The higher the percentile rank, the ____________________ the score when
compared with the other scores.
The lower the percentile rank, the _________________ the score when compared
with the other scores.
The median is always the _____________________________________________________
Example 1) You received a mark of 80% on an exam. 9 students out of 100 score
lower than you. How does your mark compare with the marks of the other
students who have written the exam? Use a percent bar to compare your mark
with the other student’s marks.
15
Grade 12 Essentials - Statistics
Example 2) The following is a set of 32
marks achieved by students on an
examination worth 100 marks.
Determine the percentile ranking of
each of the following marks.
a) 40
b) 83
Example 3) A survey is conducted in a community of 2000 families. The survey
yields that the 25th percentile for income is $23500 (P25 = $23 500). What
percentage of families in the community have a yearly income of $23 500 or
less? How many families is that?
Example 4) Jen finished a half marathon in 1 hour 45 minutes. There were 120
people who finished slower than her, and two people finished at the exact
same time as her. Her percentile rank was P95. How many racers were there?
16
Grade 12 Essentials - Statistics
Percentile Rank: Practice
1. Use a percent bar to represent the following percentile ranks of a new
baby, let’s call her Joy.
a) Joy’s weight is in the 95th percentile, P95.
b) Joy’s length is in the 25th percentile, P25.
c) From what you know, circle the best description of Joy.
Pudgy, short, and adorable.
A long, lean baby.
Teeny tiny.
A very large baby, in every way.
2. Marie scores 78% on her law exam. Three hundred other students wrote
the same exam. Fifty other students received the same score as she did,
205 received a lower score. What is her percentile rank?
3. Wendy is 1.7 m tall. She is taller than 65 of the students in her grade and
no one else is exactly her height. There are 139 students in her grade.
What is her percentile rank?
17
Grade 12 Essentials - Statistics
4. Using the following information about a class of chemistry test results,
determine the percentile rank for the students who achieved 72.
5. Kadeesha recently ran in a marathon with 350 competitors (including
herself). 313 people finished slower than her. No one tied her time. What is
her percentile rank?
6. A survey is conducted in a community of 1000 families to determine the
incomes of the families living in the community. This is the information
collected:
P25 = $40 000
P50 = $60 000
P75 = $85 000
P90 = $110 000
a) What is the median family income in the community?
b) What percentage of the families earn less than $85 000? How many
families is that?
c) What percentage of the families earn less than $110 000? How many
families is that?
18
Grade 12 Essentials - Statistics
7. Ron plays 2 hours of video games every day. He surveyed his classmates
to find out how he compared to them in terms of time gaming. He found
that 15 people gamed less than him and his percentile rank was P75. How
many people are in his class?
8. Jane was comparing the circumference of her head to that of other
students at her school. The circumference of her head is 54 cm. She found
that 12 students had a circumference less than 54 cm, not including her.
She is ranked in the 15th percentile for head circumference. How many
students are in her school?
19
Grade 12 Essentials - Statistics
Chapter Review
Multiple Choice
1. Which of the following data sets has a median of 30?
a. 38, 34, 38, 50, 23, 20, 60
c. 20, 31, 60, 28, 60, 30, 55
b. 29, 30, 56, 60, 20, 45, 24
d. 26, 39, 20, 60, 21, 33, 26
2. What is the Smith family’s mean heating bill for the months shown?
a. $91.90
b. $97.40
c. $97.40
d. $94.47
Month
November
December
January
February
March
Heating bill
$94.85
$95.90
$85.02
$89.25
$94.47
3. What is the mode of the following data set?
3, 6, 7, 7, 8, 8, 9, 10, 10, 11, 11, 11, 31
a. 7
b. 8
c. 10
d. 11
4. Bao knows he scored in the 65th percentile on a test, and that there were
375 people in total taking the test. How many people had a lower score
than him?
a. 244
b. 150
c. 131
d. 268
20
Definitions - You will be expected to describe two terms used in this unit AND
include an example. You do not need to memorize a specific definition. Just
make sure you have a basic understanding of the following terms:
 mean
 median
 mode
 percentile rank
 outlier
 trimmed mean
 weighted mean
Short Answer/Problem Solving
5. Jane has worked a number of different jobs. Below are the wages she has
earned per hour.
$22, $10, $21, $24, $22, $14, $23, $23, $21, $25
6.
a. Find the mean, median, and mode for her wages.
b. Are there any outliers? If yes, list it/them.
c. Calculate the trimmed mean by removing the two highest and two
lowest data.
Grade 12 Essentials - Polygons
6. The list below gives the student population at each high school in a school
district.
810, 1014, 900, 1071, 1900, 925, 1186, 821, 849, 1215, 126
Calculate the trimmed mean by removing the highest and the lowest
numbers.
7. A paintball facility offers three different birthday packages. They sold:
 15 “Bronze” packages at $1450.00 each;
 7 “Silver” packages at $2000.00 each; and
 27 “Gold” packages at $2500.00.
Calculate the mean cost of the party packages sold.
22
Grade 12 Essentials - Statistics
8. There are 3 Grade 12 English classes. Class A’s average mark is 87%, Class
B’s average marks is 72%, and Class C’s average mark is 68%. Class A has
12 students, Class B has 18 students, and Class C has 21 students. Use the
weighted mean to calculate the overall average of the English classes.
9. The following table shows the wages
made by students at summer jobs.
Calculate the percentile rank of a
student earning $10.00/hour.
Students’ hourly wages
Wage per hour
Number of
students
$8.00
29
$10.00
33
$12.00
51
$14.00
47
$16.00
33
10. A pumpkin growing competition at a county fair had 12 entries. If Farmer
Dale’s pumpkin placed 9th, what is the pumpkin’s percentile ranking?
23
Grade 12 Essentials - Polygons
11. There are 84 registered massage therapists in Kelowna, BC. If Maximum
Massage is in the 58th percentile, how many therapists have the same or
lower prices?
12. A study was done to test lifespan of a certain type of hamster. One
hamster named Ham lived for 13 months. 386 hamsters lived less than him
and 45 other hamsters also lived for 13 months. If Ham was in the 30th
percentile, how many hamsters were in this study?
24