measures of centre and spread
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Measures of Centre and Spread
Sections 2.5 and 2.6
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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
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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?
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Mean (arithmetic mean), sometimes referred to as the average
Population Mean Sample Mean
70, 70, 81, 83, 90€
μ =x ini=1
n
∑
€
x =x ini=1
n
∑
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Median
The middle most number when numbers are arranged from lowest to highest:
70, 70, 81, 83, 90
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Mode
The most common number. 70, 70, 81, 83, 90
Note: There can be multiple modes, and there can also be no mode.
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Eg) Find the mean, median and mode of 2, 2, 4, 6, 8, 8
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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
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Or this…..
• Example 2) A class has the following marks on their tests:
30, 30, 31, 80, 83, 88, 90
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Or even this…
• Example 3) A class has the following marks on their tests: 30, 30, 35, 95, 96
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What is the moral of the story?
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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
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Variance: A Measure of Spread / Consistency
• Population Variance is calculated as:
€
σ 2 =(x −μ)2∑N
Applicant 1: 71, 73, 75, 75, 77, 79
Applicant 2: 50, 62.5, 75, 75, 87.5, 100
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Standard Deviation
• Population Standard Deviation is Calculated as follows:
€
σ =(x −μ)2∑N
Applicant 1: 71, 73, 75, 75, 77, 79
Applicant 2: 50, 62.5, 75, 75, 87.5, 100
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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:
€
s =(x −μ)2∑n −1
€
s2 =(x −μ)2∑N
Standard Deviation of a Sample
Variance of a Sample
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Practice:
Pg. 127 Read Example 1Pg. 133 1, 2, 3, 4, 9, 16
Pg. 138 1, 6a
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Write a response to yourself:
• Why do we have different measures of centre?
• What do variance and standard deviation tell us?