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