Download - SP 225 Lecture 8
![Page 1: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/1.jpg)
SP 225Lecture 8
Measures of Variation
![Page 2: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/2.jpg)
Challenge Question
A randomized, double-blind study of 50 subjects shows daily administration of Echinacea supplements shortens the average duration of an Upper Respiratory Infection (URI) from 14 to 13 days.
Based on this study, is Echinacea an effective treatment for URI’s?
![Page 3: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/3.jpg)
Roll of the Dice
All outcomes are equally likely The probability of any outcome is 1/6 or
16.7%
![Page 4: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/4.jpg)
Casinos Patrons: Risky Fun
Red, White and Blue Slots 82% chance of loss on any spin Prizes for a dollar bet range
from $2400 to $1 Patrons are expected to lose
$0.10 for each dollar bet
![Page 5: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/5.jpg)
Casinos: False Risk
Soaring Eagle 4300 slot machines 25 spins per hour Open 24/7/365 94,170,000 possible spins
![Page 6: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/6.jpg)
Statistics vs. Parameters
Statistics: numerical description of a sample
Parameter: numerical description of a population
Statistics are calculated randomly selected members of a population
![Page 7: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/7.jpg)
Differences Between Statistics and Parameters
Population: All PeopleParameter: 5 of 15 or 33% wear glasses
Sample: 3 Randomly Selected People
Statistic: 0 of 3 or 0% wear glasses
![Page 8: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/8.jpg)
Random Sampling Activity
Number of siblings of each student in the freshman class of Powers Catholic High school
Take 3 samples, with replacement, of sizes 1, 5 and 10
Calculate the sample mean Record results in class data chart
![Page 9: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/9.jpg)
Challenge Question
A randomized, double-blind study of 50 subjects shows daily administration of Echinacea supplements shortens the average duration of an Upper Respiratory Infection (URI) from 14 to 13 days.
Based on this study, is Echinacea an effective treatment for URI’s?
![Page 10: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/10.jpg)
Why Do We Need Measures of Variation?
What is the average height of a male child? How many children are that tall? When is a child unusually tall or short?
![Page 11: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/11.jpg)
Range
Difference between the maximum and minimum value
Quick to Compute Not Comprehensive
Range = (maximum value) – (minimum value)
![Page 12: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/12.jpg)
Quartiles
Often used in the education field Can be used with any data distribution Measures distance in relation to the
MEDIAN not MEAN
![Page 13: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/13.jpg)
Quartiles
Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.
Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.
Q3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.
![Page 14: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/14.jpg)
Quartiles (2)
Q1, Q2, Q3 divide ranked scores into four equal parts
25% 25% 25% 25%
Q3Q2Q1(minimum) (maximum)
(median)
![Page 15: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/15.jpg)
Quartile Statistics
Interquartile Range (or IQR): Q3 - Q1
![Page 16: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/16.jpg)
Example
Given the following data calculate Q1, Q2 and Q3
4.2, 4.4, 5.1, 5.6, 6.0, 6.4, 6.8, 7.1, 7.4, 7.4, 7.9, 8.2, 8.2, 8.7, 9.1, 9.6, 9.6, 10.0, 10.5, 11.6
![Page 17: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/17.jpg)
Example Continued
http://www.maths.murdoch.edu.au/units/statsnotes/samplestats/boxplot.html
![Page 18: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/18.jpg)
Standard Deviation for a Population
Calculated by the following formula:
Used to show distance from the mean Tells how usual, or unusual a measurement is
(x - x)2
n - 1s = =
![Page 19: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/19.jpg)
Standard Deviation for a Sample
(x - x)2
n - 1s =
![Page 20: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/20.jpg)
Standard Deviation - Important Properties
Standard Deviation is always positive Increases dramatically with outliers The units of standard deviation s are
the same as the units of the mean
![Page 21: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/21.jpg)
Calculating the Standard Deviation of a SAMPLE
Data points 1, 3, 5, 7, 9
![Page 22: SP 225 Lecture 8](https://reader035.vdocuments.us/reader035/viewer/2022062315/56816107550346895dd04e20/html5/thumbnails/22.jpg)
Variance
A measure of variation equal to the square of the standard deviation
Sample Variance = s Population Variance = 2
2