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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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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
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)

1,2,2,2,3,4,5,6,7 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
Modes in Populations
  • Unimodal
  • Bimodal
  • Trimodal
  • Amodal ?
central tendency medians
CENTRAL TENDENCYMedians
  • Middle score in distribution
  • Odd number of scores

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

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
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