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
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
70, 70, 81, 83, 90€
μ =x ini=1
n
∑
€
x =x ini=1
n
∑
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:
€
σ 2 =(x −μ)2∑N
Applicant 1: 71, 73, 75, 75, 77, 79
Applicant 2: 50, 62.5, 75, 75, 87.5, 100
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
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
Practice:
Pg. 127 Read Example 1Pg. 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?