sp 225 lecture 8
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SP 225 Lecture 8. Measures of Variation. 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. - PowerPoint PPT PresentationTRANSCRIPT
SP 225Lecture 8
Measures of Variation
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?
Roll of the Dice
All outcomes are equally likely The probability of any outcome is 1/6 or
16.7%
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
Casinos: False Risk
Soaring Eagle 4300 slot machines 25 spins per hour Open 24/7/365 94,170,000 possible spins
Statistics vs. Parameters
Statistics: numerical description of a sample
Parameter: numerical description of a population
Statistics are calculated randomly selected members of a population
Differences Between Statistics and Parameters
Population: All People
Parameter: 5 of 15 or 33% wear glasses
Sample: 3 Randomly Selected People
Statistic: 0 of 3 or 0% wear glasses
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
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?
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?
Range
Difference between the maximum and minimum value
Quick to Compute Not Comprehensive
Range = (maximum value) – (minimum value)
Quartiles
Often used in the education field Can be used with any data distribution Measures distance in relation to the
MEDIAN not MEAN
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%.
Quartiles (2)
Q1, Q2, Q3 divide ranked scores into four equal parts
25% 25% 25% 25%
Q3Q2Q1(minimum) (maximum)
(median)
Quartile Statistics
Interquartile Range (or IQR): Q3 - Q1
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
Example Continued
http://www.maths.murdoch.edu.au/units/statsnotes/samplestats/boxplot.html
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 = =
Standard Deviation for a Sample
(x - x)2
n - 1s =
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
Calculating the Standard Deviation of a SAMPLE
Data points 1, 3, 5, 7, 9
Variance
A measure of variation equal to the square of the standard deviation
Sample Variance = s Population Variance = 2
2