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BASIC NOTATION. X i = The number of meals I have on day “ i ” X= 1,2,3,2,1  X i = ???  X i 2 = ??? ( X i ) 2 = ???. Summation (). 9. 19. 81. Nominal Political affiliation Republican Democrat Independent Gender Female Male. Qualitative Variables.

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BASIC NOTATION

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BASIC NOTATION


Xi = The number of meals I have on day “i”

X= 1,2,3,2,1

 Xi = ???

 Xi2 = ???

( Xi)2 = ???

Summation ()

9

19

81


Nominal

Political affiliation

Republican

Democrat

Independent

Gender

Female

Male

Qualitative 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)

Quantitative Variables


Statistical Analyses


Tables

Ungrouped (list of scores)

Grouped (grouped by ranges)

Graphs

histograms

frequency polygons

Frequency Distributions


The 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)

Table Distributions

Bad Grouped Frequency DistributionXf 0-10 7 11-20 8 21-30 6 31-40 3 41-50 0 51-60 1 25

Good Grouped Frequency DistributionXf1-10 7 11-20 8 21-30 6 31-40 3 41-50 0 51-60 1 25


Modality - Peaks

Symmetry – Mirror Reflection

Asymptoticness – Extreme Values on both Sides

Distribution Characteristics


USA

Unimodal

Symmetric

Asymptotic

Normal Distributions


Inflection points

Where curve changes from

convex to concave or

concave to convex

Also = 1 standard deviation from the mean


CENTRAL TENDENCYWHAT 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


CENTRAL TENDENCYModes

  • Most common score(s)

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

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

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

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


Modes in Populations

  • Unimodal

  • Bimodal

  • Trimodal

  • Amodal ?


CENTRAL TENDENCYMedians

  • Middle score in distribution

  • Odd number of scores

    5-point data set: 2,3,5,9,12Median=5

    1,2,5,5,7,9,500,700,999Median=?

  • 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=?


Medians in Populations


CENTRAL TENDENCYMeans


More modes, medians and means


How different are scores from central tendency?

Range

Standard Deviation

The Spread of Distributions


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

Measure of SpreadRANGE


Why ‘range’ is weak


How different are scores from central tendency?

Always, by definition of the mean

The Spread of Distributions


Population Standard Deviation


Sample Variance and Standard deviation

Also known as

“Estimated Population Standard Deviation”


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 = 5

Scores 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

 DEGREES OF FREEDOM

STANDARD DEVIATION


Differences BetweenSample and Population Standard Deviation

1) Sigma vs. S

2) Population mean versus Sample mean

3) N vs. N-1


Super Important Relationship Standard Deviation is square root of variance

SAMPLE STANDARD DEVIATION =

SQUARE ROOT OF THE SAMPLE VARIANCE

POPULATION STANDARD DEVIATION =

SQUARE ROOT OF THE POPULATION VARIANCE


Population Standard Deviation


Sample Standard Deviation


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