Download Median - Gr 12 Essentials

3.3
Median
Try These
You will need
Order from least to greatest.
i) 2298, 2196, 2279, 2275 2196 , 2275 , 2279 , 2298
29 , 28 , 26 , 0 , 1 , 10
ii) 1, 0, 29, 10, 26, 28
iii) 9.8, 9.05, 9.75, 9.5, 9.25 9.05 , 9.25 , 9.5 , 9.75 , 9.8
• paper for tracing
• a millimetre ruler
median
the middle term
in a set of data
arranged in
order; if there is
an even number
of numbers, the
median is halfway
between the
middle values
Keira is a repair technician. She uses text messages to
communicate with clients and co-workers. What is the median
number of daily text messages that Keira sent last week?
Day
Monday
Tuesday
Wednesday
Thursday
Friday
101
117
108
95
104
Number of text messages
1
Order the numbers from least to greatest. Circle the middle
number.
,
95
2
101 ,
104 ,
108 ,
117
What is the median number of daily text messages that Keira
sent last week?
The median number of daily text messages is
Reflecting
Does the median
change if the
numbers are
ordered from
greatest to least?
Hint
When determining
the median for a
data set, use the
number of times
they are in the
data set.
104 .
Example 1
A school basketball team scored these numbers of points in their
first 8 games: 86, 61, 72, 86, 74, 59, 91, and 80
What is the median number of points the team scored?
Solution
A. Order the numbers of points from least to greatest. Circle the
two middle numbers.
59 ,
61 ,
72 ,
74 ,
80 ,
86 ,
86 ,
91
B. What is the median number of points for these games?
Median: mean of two middle numbers
5 ( 77
) 4 2, or
77
The median number of points is
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74
1
80
.
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Example 2
Liam is a real-estate agent in
Winnipeg. Last month, he sold
nine homes. The prices are in the
chart at the right.
Selling Prices of Homes
$264 000
$223 000
$272 000
$319 000
$248 000
$288 000
$257 000
$188 000
$784 500
Is the median or the mean a better
measure of central tendency for Liam’s typical selling price?
Solution
A. What is the mean selling price for these homes?
Mean price ($): 2 843 500 4 9 5 315 944.44
measure of
central tendency
a way to
summarize the
values in a data
set with a single
value
The mean selling price is $315 944.44 .
B. Order the selling prices from least to greatest. Circle the
median selling price.
$ 188 000 , $ 223 000 , $ 248 000 , $ 257 000 , $ 264 000 ,
$ 272 000 , $ 288 000 , $ 319 000 , $ 784 500
The median selling price is $ 264 000 .
C. Which selling price is an outlier? $ 784 500
D. Is the median or the mean a better measure of central
tendency for Liam’s typical selling price? Explain.
e.g., The median is a better measure of central tendency for
Reflecting
How would
removing the
outlier affect the
mean and the
median?
Liam’s typical selling price. Seven of the nine homes that Liam
sold cost less than the mean.
Practice
1. Jacob uses a spring caliper to measure the width of a
metal part on his bicycle. To reduce measurement error, he
measures seven times. The measurements, in centimetres,
are 7.2, 7.0, 7.1, 7.1, 6.9, 7.3, and 7.2.
Order the measurements from least to greatest. What is the
median width?
6.9, 7.0, 7.1, 7.1, 7.2, 7.2, 7.3
The median width is 7.1 cm.
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2. Erin wants to buy a new car. She has compared the fuel
economy of several vehicles. This list gives the number of
litres of fuel that is used by these vehicles for 100 km of
city driving:
10.3, 14.1, 11.9, 4.6, 11.6, 10.7, 10.8, 11.1
a) What is the median fuel economy for these cars?
For 100 km (L): 4.6, 10.3, 10.7, 10.8, 11.1, 11.6, 11.9, 14.1
Median: (10.8 L 1 11.1 L) 4 2 5 10.95 L
The median fuel economy is 10.95 L/100 km.
b) One car that Erin likes is a hybrid. It uses electricity for
some city driving, so it uses less fuel. How does the
median change if this outlier is removed?
Outlier:
4.6
L
Median without outlier (L): 10.3, 10.7, 10.8, 11.1, 11.6, 11.9, 14.1
The median increases from 10.95 L/100 km to
11.1 L/100 km.
c) How does removing the outlier affect the mean?
Original mean: 85.1 L 4 8 5 10.6375 L
Mean without outlier: 80.5 L 4 7 5 11.5 L
The mean increases from about 10.6 L/100 km to
11.5 L/100 km.
3. Hector completed his carpentry apprenticeship. He applied
for six jobs in construction. This list is the hourly wages for
these jobs:
$27.48, $29.30, $28.96, $28.80, $33.85, and $28.80.
a) What is the median hourly wage?
$27.48, $28.80, $28.80, $28.96, $29.30, $33.85
($28.80 1 $28.96) 4 2 5 $28.88
The median hourly wage is $28.88.
b) What is the mean hourly wage? $177.19 4 6 5 $29.53
c) Suppose that the lowest wage increased from $27.48 to
$28.48. How would this affect the median and the mean?
e.g., The median would stay the same. The mean would
increase.
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4. Whitney received these donations for a charity run.
Donations Received ($)
a) What error did Whitney make when she determined the
median? Describe and correct her error.
10
5
20
30
Donations($): 5, 10, 20, 30, 40, 50, 100, 200
10
50
50
20
40
40
20
200
20
100
5
($30 1 $40) 4 2 5 $35 The median donation was $35.
Whitney’s error: She left out the repeated values.
Correct list of donations ($):
5, 5, 10, 10, 20, 20, 20, 20, 30, 40, 40, 50, 50, 100, 200
The median donation was $20.
b) What was the mean donation?
$620 4 15 5 $41.33
The mean donation was $41.33.
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c) Which donations could be considered outliers? e.g., $100, $200
d) Do you think the median or the mean is a better measure
of a typical donation in this situation? Explain.
e.g., The median is a better measure because four people gave
this much and it would not change if you removed the outliers.
5. Environment Canada keeps track of weather data for Canadian
cities. What is the median daily maximum temperature in
Calgary?
Typical Daily Maximum Temperatures in Calgary (8C)
Jan
Feb
Mar
22.8
20.1
4.0
Apr
AW12
May
Jun
0176519637
11.3 Figure
16.4
20.2
Number
Company
Jul
Aug
Sep
Oct
Nov
Dec
22.9
22.5
17.6
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12.1
2.8
21.3
MPS
Technical
Temperatures (8C): 22.8, 21.3,
20.1, 2.8, 4.0, 11.3, 12.1, 16.4, 17.6, 20.2, 22.5, 22.9
Pass
(11.3 8C 1 12.1 8C) 4 2 5 11.7 8C
1st pass
Approved
The median daily maximum Not
temperature
is 11.7 8C.
Approved
6. Trace your fingers. Then measure and record the widths of
your fingers. What is each measure of central tendency?
Your median finger width (cm): e.g., 1.2, 1.6, 1.6, 1.8, 2.1
Your mean finger width (cm): e.g., 1.66
Your trimmed mean finger width (cm): e.g., 1.666…
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