URBP 204A QUANTITATIVE METHODS I Statistical Analysis Lecture I Gregory Newmark San Jose State University (This lecture accords with Chapters 2 & 3 of Neil Salkind’s Statistics for People who (Think They) Hate Statistics)

Descriptive Statistics Statistics that describe a data set Measures of Central Tendency – Mean (Average) – Median (Midpoint) – Mode (Most Prevalent) Measures of Variability – Range (Highest Value – Lowest Value) – Standard Deviation (Average Distance from Mean) – Variance (Average Distance from Mean Squared)

Central Tendency What is the central tendency? – Single number that best describes data set – Representative score in a set of scores – Possible measures: Mean, Median, Mode

Mean (Average) Mean – Average value – Sum of all values divided by the number of values What is the average housing value here? – Ms. Johnson’s House$600,000 – Mr. Wood’s House$400,000 – Ms. Brown’s House$500,000

Mean (Average) Mean = ($600,000 + $400,000 +$500,000)/3 = $1,500,000/3 = $500,000 The average house value here is $500,000 Problem: Mean is sensitive to extreme values “If Bill Gates walked into this classroom, on average, we would all be billionaires”

Mean (Average) What is the average housing value here? – Ms. Johnson’s House + Addition$2,400,000 – Mr. Wood’s House$400,000 – Ms. Brown’s House$500,000 Mean = ($2,400,000 + $400,000 +$500,000)/3 = $3,300,000/3 = $1,100,000 The average house value here is $1,100,000 Mean may no longer be a useful statistic

Median (Midpoint) Median – Midpoint value (half above, half below) What is the median housing value here? – Ms. Johnson’s Mansion$2,400,000 – Mr. Wood’s House$400,000 – Ms. Brown’s House$500,000

Median (Midpoint) To find the median: – Order the values [2,400,000; 500,000; 400,000] – Select the midpoint value [500,000] (If there are an even number of values, average the two middle values) Ms. Johnson’s addition is destroyed by a meteor and her house is worth $600,000 again. – What would the new median housing value be?

Median (Midpoint) What does this map tell us?

Median vs. Mean

Mode (Most Prevalent) Mode – Most frequently occurring value What is the mode housing value here? – Ms. Johnson’s Mansion$2,400,000 – Mr. Wood’s House$400,000 – Ms. Brown’s House$500,000 – Mr. Purple’s House$400,000

Mode (Most Prevalent) To find the mode – Count the frequencies of each value 2,400,000Once 500,000Once 400,000Twice – Select the value with the highest frequency [400,000]

Mode (Most Prevalent) The mode is very important with nominal data – Most voted for candidate – Most purchased beverage – Most common birth month – Favorite sports team – Most occurring M&M type Multimodalism? – If I ate all the red M&Ms, then what would the new mode be?

When do I use which Measure? Use the Mean when: – Values are interval or ratio measures – No values are extreme Use the Median when: – Values are interval or ratio measures – Some values are extreme Use the Mode when: – Values are nominal or ordinal measures

Variability Which set of data has most variability? – 20,20,20,20,20,20Mean = 20Median = 20 – 20,21,19,20,18,22Mean = 20Median = 20 – 2,7,8,20,26,33,44Mean = 20Median = 20 Variability (or Spread or Dispersion) – measures how values differ from each other – measures how different the values are from each other by measuring how different the values are from the mean.

Range Range = Highest Value – Lowest Value What are the ranges for the following sets? – 20,20,20,20,20,20Range = = 0 – 20,21,19,20,18,22Range = 22 – 18 = 4 – 2,7,8,20,26,33,44Range = 44 – 2 = 42

Standard Deviation – Average distance from the mean – Sometimes called “mean error” – Like the mean, the SD is sensitive to extreme values – Expressed in the same units as the underlying values (the following examples are made up) Mean Male Height: 5’10” with an SD of 3” Mean TV Winnings: $4,760 with an SD of $3,400 Mean Runs per Game: 7.8 runs with an SD of 3.2 runs – An SD of zero implies no variability

Standard Deviation Standard Deviation (σ) – Average distance from the mean – Example Mean = 50 SD = 20

Standard Deviation

Formula Wheres is the standard deviation Σ is sigma, which sums what follows X is each individual score Xbar is the mean of all the scores n is the sample size

Standard Deviation Formula “This formula finds the difference between each individual score and the mean (X – Xbar), squares each difference, and sums them all together. Then, it divides the sum by the size of the sample (minus 1) and takes the square root of the result.” (Salkind 2004)

Standard Deviation Formula Why do we square the differences? Why do we take the square root of everything? Why do we minus 1 from n?

Standard Deviation Why ‘n – 1’ and not just ‘n’: – This makes the resulting SD slightly larger – This is a conservative approach to apply the SD from a sample to an entire population – Unbiased Estimate (versus the Biased Estimate) – As n grows, the unbiased estimated approaches the biased estimate

Standard Deviation Example – Runs scored by A’s in last nine games: Runs: 3,3,4,5,5,7,8,9,10 – Formula

SD of A’s Runs Example GamesRunsAverage(X – Xbar)(X – Xbar) 2 Last Steps Sum540 54/(n-1) = 54/8 = 6.75 Square Root of 6.75 = 2.6 SD2.6 Runs

Standard Deviation Example: – Two curves with same μ but different σ – What does this say about the dispersion?

Variance Variance = (Standard Deviation) 2 – Not in same unit as original scores – Will become very relevant later in the class Formula