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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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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
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)

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

1,3,3,4,4,5,6,7,8 Bimodal Modes=3,4

1,3,3,4,4,5,6,6,8 Trimodal Modes=3,4,6

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

Modes in Populations
• Unimodal
• Bimodal
• Trimodal
• Amodal ?
CENTRAL TENDENCYMedians
• Middle score in distribution
• Odd number of scores

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

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

• Even number of scores

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

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

Median=?

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
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