Basic notation
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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

BASIC NOTATION


Summation

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

X= 1,2,3,2,1

 Xi = ???

 Xi2 = ???

( Xi)2 = ???

Summation ()

9

19

81


Qualitative variables

Nominal

Political affiliation

Republican

Democrat

Independent

Gender

Female

Male

Qualitative Variables


Quantitative variables

Ordinal

Categories have relative value/order

Example

Very Depressed

Depressed

Slightly depressed

Not depressed

Quantitative Variables


Quantitative variables1

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


Quantitative variables2

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

Statistical Analyses


Frequency distributions

Tables

Ungrouped (list of scores)

Grouped (grouped by ranges)

Graphs

histograms

frequency polygons

Frequency Distributions


Table 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


Distribution characteristics

Modality - Peaks

Symmetry – Mirror Reflection

Asymptoticness – Extreme Values on both Sides

Distribution Characteristics


Normal distributions

USA

Unimodal

Symmetric

Asymptotic

Normal Distributions


Inflection points

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

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 tendency modes

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

Modes in Populations

  • Unimodal

  • Bimodal

  • Trimodal

  • Amodal ?


Central tendency medians

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

Medians in Populations


Central tendency means

CENTRAL TENDENCYMeans


More modes medians and means

More modes, medians and means


The spread of distributions

How different are scores from central tendency?

Range

Standard Deviation

The Spread of Distributions


Measure of spread range

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

Why ‘range’ is weak


The spread of distributions1

How different are scores from central tendency?

Always, by definition of the mean

The Spread of Distributions


Population standard deviation

Population Standard Deviation


Sample variance and standard deviation

Sample Variance and Standard deviation

Also known as

“Estimated Population Standard Deviation”


Sample variance and standard deviation1

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

Sample Variance

 SUM OF SQUARED DEVIATIONS

 DEGREES OF FREEDOM

STANDARD DEVIATION


Differences between sample and population 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

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 deviation1

Population Standard Deviation


Sample standard deviation

Sample Standard Deviation


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