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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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**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=?**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**How different are scores from central tendency?**Always, by definition of the mean The Spread of Distributions**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

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