Measures of Central Tendency and Dispresion

Content Analysis- Challenges Lose some nuance when coding How to select material from universe of possible material? Is material accurate? Unintentional problems Unintentional problems Purposeful distortion Purposeful distortion Ultimately a question of validity Ultimately a question of validity Are coders accurate? Can establish reliability Can establish reliability Harder to establish validity Harder to establish validity

Statistics Provides description of a sample or population Simplification Univariate- Only interested in one attribute at a time Bivariate- consider relationships between 2 attributes Multivariate- the sky is the limit

Percentages Useful for comparing groups with unequal numbers

Percentages

Percentages To Compute: (#with trait of interest/total #) X 100 (#with trait of interest/total #) X 100 Example 1- Sample of 4 cats, one is black (¼)X % Example 2-Sample of 750, 612 approve of the president (612/750)X100= 81.6%

What Constitutes the Denominator? Percentage of Total Percentage of Valid Cases Excludes missing cases Excludes missing cases Typically more appropriate Typically more appropriate Cumulative Percent-what percentage so far have reached this level

An Example

Measures of Central Tendency Mode Mean (Average) Median

Computing the Mean Requires At least ordinal data (Y 1 + Y 2 + Y 3 …. +Y i )/I Example have people with incomes of 10,000, 15,000, 25,000, 55,000, 32,000, 29,500 Mean=(10,000+15,000+25,000, +55, ,000+29,500)/6= 27,750

Mode Most common with nominal data Count frequencies, find most common Ask 30 1 st graders favorite color 7 blue 3 chartreuse 4 purple 2 yellow 10 red 3 green 1 Black Mode- Red

Frequencies

Computing the Median Requires at least Ordinal Data Put values in order If odd number, value half are above, half below If even number- Average of two middle cases Income Example: 10,000, 15,000, 25,000, 55,000, 32,000, 29,500 10,000, 15,000, 25,000, 55,000, 32,000, 29,500 10,000, 15,000, 25,000, 29,500, 32,000, 55,000 10,000, 15,000, 25,000, 29,500, 32,000, 55,000 Median=25,250 Median=25,250

When To Use Which? Mode- nominal data Better to actually give totals for all if few choices, e.g. 33% red, 10% green Better to actually give totals for all if few choices, e.g. 33% red, 10% green Mean- when appropriate data Median- with ordinal data, in cases where there are a few values that might cause a skew Outlier- Data point with extreme value

Median vs. Mean Created a fake town with 100 residents Incomes 19,00-138,000 Mean=57600, Median=49,500 Suppose one person with 30,000 moves away, replaced by Millionaire Mean=67,300, Median=55,000 Mean=67,300, Median=55,000 Replaced by 50,000,000 Replaced by 50,000,000 Mean=557,300 Median= 55,000 Mean=557,300 Median= 55,000 Replaced by Bill Gates (50 Billion) Mean=500Million, Median= 55,000 Mean=500Million, Median= 55,000

Measures of Dispersion Measure of Central Tendency loses something Income example? Dispersion Measure of how much divergence there is from the mean Measure of how much divergence there is from the meanHistogram Horizontal Axis breaks variable down into ranges Horizontal Axis breaks variable down into ranges Vertical Axis-count within each range Vertical Axis-count within each range

Quantifying Dispersion- Standard Deviation Find difference from mean for each observation Add them up Divide by the number of cases minus1

Standard Deviation from Previous cases Mean= 50,024, S.D=992.5 Min=46,834, Max=52,935

Mean=50,255 S.D.=4792 Min=35,671 Max=65,095

Mean=50,311 S.D.=10,124 Min=22,522 Max=78,642

Mean=50,982 S.D.=18,898 Min=1591 Max=105,957

Gore Thermometer Mean=57.4, S.D.=25.7 0=4.6%, 100= 5.6%

George W Bush Thermometer Mean=56.1 S.D.=24.9 0=4.4%100=4.7%

Clinton Thermometer Mean=55.2S.D.=29.7 0=9.5%100=7.1%

For Next Time The Normal Distribution Bivariate Relationships Get stats assignments