Variability

Quantitative Methods in HPELS

440:210

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Introduction

Statistics of variability: Describe how values are spread out Describe how values cluster around the middle

Several statistics Appropriate measurement depends on: Scale of measurement Distribution

Basic Concepts

Measures of variability:FrequencyRange Interquartile rangeVariance and standard deviation

Each statistic has its advantages and disadvantages

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Frequency

Definition: The number/count of any variable

Scale of measurement: Appropriate for all scalesOnly statistic appropriate for nominal data

Statistical notation: f

Frequency

Advantages:Ease of determinationOnly statistic appropriate for nominal data

Disadvantages: Terminal statistic

Calculation of the Frequency Instat Statistics tab Summary tab Group tab

Select groupSelect column(s) of interestOK

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Range

Definition: The difference between the highest and lowest values in a distribution

Scale of measurement: Ordinal, interval or ratio

Range

Advantages:Ease of determination

Disadvantages:Terminal statisticDisregards all data except extreme scores

Calculation of the Range Instat

Statistics tab Summary tab Describe tab

Calculates range automaticallyOK

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Interquartile Range

Definition: The difference between the 1st quartile and the 3rd quartile

Scale of measurement:Ordinal, interval or ratioExample: Figure 4.3, p 107

Interquartile Range

Advantages:Ease of determinationMore stable than range

Disadvantages:Disregards all values except 1st and 3rd

quartiles

Calculation of the Interquartile Range Instat Statistics tab Summary tab Describe tab

Choose additional statisticsChoose interquartile rangeOK

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Variance/SD Population Variance:

The average squared distance/deviation of all raw scores from the mean

The standard deviation squared Statistical notation: σ2

Scale of measurement: Interval or ratio

Advantages: Considers all data Not a terminal statistic

Disadvantages: Not appropriate for nominal or ordinal data Sensitive to extreme outliers

Variance/SD Population Standard deviation:

The average distance/deviation of all raw scores from the meanThe square root of the varianceStatistical notation: σ

Scale of measurement: Interval or ratio

Advantages and disadvantages: Similar to variance

Calculation of the Variance Population

Why square all values? If all deviations from the mean are

summed, the answer always = 0

Calculation of the Variance Population

Example: 1, 2, 3, 4, 5 Mean = 3 Variations:

1 – 3 = -2 2 – 3 = -1 3 – 3 = 0 4 – 3 = 1 5 – 3 = 2

Sum of all deviations = 0

Sum of all squared deviations

Variations: 1 – 3 = (-2)2 = 4 2 – 3 = (-1)2 = 1 3 – 3 = (0)2 = 0 4 – 3 = (1)2 = 1 5 – 3 = (2)2 = 4

Sum of all squared deviations = 10

Variance = Average squared deviation of all points 10/5 = 2

Calculation of the Variance Population

Step 1: Calculate deviation of each point from mean

Step 2: Square each deviation Step 3: Sum all squared deviations Step 4: Divide sum of squared deviations

by N

Calculation of the Variance Population

σ2 = SS/number of scores, where SS =Σ(X - )2

Definitional formula (Example 4.3, p 112) or

ΣX2 – [(ΣX)2] Computational formula (Example 4.4, p 112)

Computational formula

Step 4: Divide by N

Computation of the Standard Deviation Population

Take the square root of the variance

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Variance/SD Sample

Process is similar with two distinctions: Statistical notation Formula

Statistical Notation DistinctionsPopulation vs. Sample σ2 = s2

σ = s = M N = n

Formula DistinctionsPopulation vs. Sample s2 = SS / n – 1, where SS =

Σ(X - M)2

Definitional formula

ΣX2 - [(ΣX)2] Computational formula

Why n - 1?

N vs. (n – 1) First Reason

General underestimation of population variance

Sample variance (s2) tend to underestimate a population variance (σ2)

(n – 1) will inflate s2

Example 4.8, p 121

Actual population σ2 = 14

Average biased s2 = 63/9 = 7 Average unbiased s2 = 126/9 = 14

N vs. (n – 1) Second Reason

Degrees of freedom (df)df = number of scores “free” to varyExample:

Assume n = 3, with M = 5 The sum of values = 15 (n*M) Assume two of the values = 8, 3 The third value has to be 4 Two values are “free” to vary df = (n – 1) = (3 – 1) = 2

Computation of the Standard Deviation of Sample Instat Statistics tab Summary tab Describe tab

Calculates standard deviation automatically OK

Agenda

Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection

Selection

When to use the frequency Nominal data With the mode

When to use the range or interquartile range Ordinal data With the median

When to sue the variance/SD Interval or ratio data With the mean

Textbook Problem Assignment

Problems: 4, 6, 8, 14.