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.  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  Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.  Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.  Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.

Quartiles (2) Q 1, Q 2, Q 3 divide ranked scores into four equal parts 25% Q3Q3 Q2Q2 Q1Q1 (minimum)(maximum) (median)

Quartile Statistics  Interquartile Range (or IQR): Q 3 - Q 1

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

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 - 1 s =s =  =

Standard Deviation for a Sample  ( x - x ) 2 n - 1 s =s =

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