Transcript Lesson 2 Measures of Spread and Centre

Measures of Centre and Spread
Sections 2.5 and 2.6
Population vs. Sample
• A population refers to an entire group that is
being studied. A sample is a selection of
people or things from that group.
Eg) We survey students in our classroom to
determine the opinion of students at NACI
Eg) We take our top 5 marks to represent all of
our marks
Measures of Central Tendency
• You have decided on your top 5 marks to submit for your
post-secondary applications.
Your marks are 70, 70, 81, 83, 90
What best describes this data,
Mean, median or mode?
Mean (arithmetic mean), sometimes
referred to as the average
Population Mean
Sample Mean
n
n
xi

i1 n
xi
x 
i1 n
70, 70, 81, 83, 90

Median
The middle most number when numbers are
arranged from lowest to highest:
70, 70, 81, 83, 90
Mode
The most common number.
70, 70, 81, 83, 90
Note: There can be multiple modes, and there
can also be no mode.
Eg) Find the mean, median and mode
of 2, 2, 4, 6, 8, 8
So why do we have three? Which is
the best measure?
Example 1) Suppose a class has the following
marks on a test:
43, 43, 48, 49, 100
Or this…..
• Example 2) A class has the following marks on
their tests:
30, 30, 31, 80, 83, 88, 90
Or even this…
• Example 3) A class has the following marks on
their tests: 30, 30, 35, 95, 96
What is the moral of the story?
Measures of Spread
• Suppose two students are applying for a position at a
university, and they submit all of their marks. What is
different about their marks?
• Applicant 1: 71, 73, 75, 75, 77, 79
• Applicant 2: 50, 62.5, 75, 75, 87.5, 100
Variance: A Measure of Spread /
Consistency
• Population Variance is calculated as:
Applicant 1: 71, 73, 75, 75, 77, 79

Applicant 2: 50, 62.5, 75, 75, 87.5, 100
2
(x  )


N
2
Standard Deviation
• Population Standard Deviation is Calculated as
follows:
(x  )2

N
Applicant 1: 71, 73, 75, 75, 77, 79

Applicant 2: 50, 62.5, 75, 75, 87.5, 100

Sample Deviation and Sample Variance
• When we take a sample, the spread of the data is usually
underestimated. To adjust for this, we simply change the
formula for deviation and variance:
Standard Deviation of
a Sample
s
Variance of a Sample
2
(x


)

s2 
n 1

2
(x


)

N
Practice:
Pg. 127 Read Example 1
Pg. 133 1, 2, 3, 4, 9, 16
Pg. 138 1, 6a
Write a response to yourself:
• Why do we have different measures of
centre?
• What do variance and standard deviation tell
us?