BASIC NOTATION

Summation ()

Xi = The number of meals I have on day “i”X= 1,2,3,2,1

Xi = ???

Xi2 = ???

( Xi)2 = ???

9

19

81

Qualitative Variables

Nominal •Political affiliation•Republican•Democrat• Independent

•Gender•Female•Male

Quantitative Variables

Ordinal•Categories have relative value/order•Example•Very Depressed•Depressed•Slightly depressed•Not depressed

Quantitative Variables

Interval•Categories have relative value/order•Difference in measurement = Difference in characteristic•Example•Temperature Fahrenheit, 83,84,85 …Difference from 83 to 84 = Difference from 84 to 85

Quantitative Variables

Ratio•Categories have relative value/order•Difference in measurement = Difference in characteristic •True zero (0) point exists•Example•Temperature Kelvin, 0,1,2,…343,345,346 …•Height 0 inches, 1 inch, …. 86 inches (Shaq)

Statistical Analyses

Design Interval/Ratio Nominal

One Sample One sample z or t test Goodness of fit χ2

Two Indepenedent samples Independent t test χ2 test of independence

Two dependent Samples Dependent t test

McNemar test for significance of change

K independent samples ANOVA χ2 test of independence

Correlation Pearson R  

Prediction Regression  

Frequency Distributions

Tables

Ungrouped (list of scores)

Grouped (grouped by ranges)

Graphs

histograms

frequency polygons

Table DistributionsThe variable: Time (in minutes) between getting out of bed this morning and eating your first bite of food.

Time (min) Ungrouped : (6, 28, 27, 7, 7, 24, 39, 55, 13, 17, 13, 13, 3, 23, 18, 37, 2, 8, 11, 18, 22, 2, 21, 31, 12)Bad Grouped Frequency Distribution

X f 0-10 7 11-20 8 21-30 6 31-40 3 41-50 0 51-60 1 25

Good Grouped Frequency Distribution

X f 1-10 7 11-20 8 21-30 6 31-40 3 41-50 0 51-60 1 25

Grouped Frequency Distribution

X f 51-60 1 41-50 0 31-40 3 21-30 6 11-20 8 1-10 7

25

Distribution Characteristics

Modality - Peaks

Symmetry – Mirror Reflection

Asymptoticness – Extreme Values on both Sides

Terms that Describe Distributions

Term Features Example

"Symmetric" left side is mirror image of right side

"Positively skewed"

right tail is longer then the left

"Negatively skewed"

left tail is longer than the right

"Unimodal" one highest point

"Bimodal" two high points

"Normal" unimodal, symmetric, asymptotic

Normal Distributions

USA

Unimodal

Symmetric

Asymptotic

Inflection points

Where curve changes from convex to concave or concave to convex

Also = 1 standard deviation from the mean

CENTRAL TENDENCY WHAT IS A TYPICAL SCORE LIKE?

Mode: Most common value; number of peaks; always an observed value

Median: Middle of distribution; not affected much by outliers

Mean: Average; greatly affected by outliers

- Most common score(s)

1,2,2,2,3,4,5,6,7 Unimodal Mode=2

1,3,3,4,4,5,6,7,8 BimodalModes=3,4

1,3,3,4,4,5,6,6,8 TrimodalModes=3,4,6

1,2,3,4,5,6,7,8,9 Amodal

CENTRAL TENDENCY Modes

Modes in Populations - Unimodal

- Bimodal

- Trimodal

- Amodal ?

- Middle score in distribution

- Odd number of scores

5-point data set: 2,3,5,9,12Median=51,2,5,5,7,9,500,700,999Median=?

- Even number of scores

4-point data set: 3,5,8,9Median=(5+8)/2=6.5

1,2,5,5,7,9,500,700,999,1122

Median=?

CENTRAL TENDENCY Medians

Medians in Populations

CENTRAL TENDENCY Means

More modes, medians and means

Measure Definition

Level of measurement Disadvantage

Modemost frequent

value nom., ord., int./rat. Crude

Median middle value ord., int./rat.Only two points

contribute

Meanbalancing

point int./rat. Affected by skew

The Spread of Distributions

- How different are scores from central tendency?

-Range-Standard Deviation

Measure of SpreadRANGE

- Highest value – Lowest Value

- Affected only by end points-Data set 1-1,1,1,50,99,99,99

-Data set 2-1,50,50,50,50,50,99

Why ‘range’ is weak

The Spread of Distributions

- How different are scores from central tendency?

- Always, by definition of the mean

( ) 0X X

Population Standard Deviation

2( )X

N

Sample Variance and Standard deviation

Also known as“Estimated Population Standard Deviation”

2( )

1

X XS

n

Sample Variance and Standard deviation

Why do we use N-1 for sample? Because sample means are closer to sample

mean than to population mean, which underestimates the estimate

Population 2,4,6,and 8, σ = (2+4+6+8)/4 = 5Scores 2 and 6

σ2 = (2-5)2 +(6-5)2 = 9 + 1 = 10

Scores 2 and 6, = (2+6)/2 = 4 S2 = (2-4)2 +(6-4)2 = 4 + 4 = 8

N-1 adjusts for bias

Sample Variance

SUM OF SQUARED DEVIATIONS 2

2 ( )

1

X XS

n

DEGREES OF FREEDOM

2( )

1

X XS

n

STANDARD DEVIATION

2( )

1

X XS

n

Differences BetweenSample and Population Standard Deviation

2( )X

N

1) Sigma vs. S2) Population mean versus Sample mean3) N vs. N-1

Super Important Relationship

Standard Deviation is square root of variance

2S SSAMPLE STANDARD DEVIATION = SQUARE ROOT OF THE SAMPLE VARIANCE

2

POPULATION STANDARD DEVIATION = SQUARE ROOT OF THE POPULATION VARIANCE

Population Standard Deviation

Sample Standard Deviation