Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall, 2014 Room 120 Integrated Learning Center (ILC) 10: :50 Mondays, Wednesdays & Fridays.

Hand in your homework & Correlation worksheet

Remember: Bring electronic copy of your data (flash drive or it to yourself) Your data should have correct formatting See Lab Materials link on class website to double- check formatting of excel is exactly consistent Labs

Schedule of readings Before next exam (September 26 th ) Please read chapters in Ha & Ha textbook Please read Appendix D, E & F online On syllabus this is referred to as online readings 1, 2 & 3 Please read Chapters 1, 5, 6 and 13 in Plous Chapter 1: Selective Perception Chapter 5: Plasticity Chapter 6: Effects of Question Wording and Framing Chapter 13: Anchoring and Adjustment

Reminder A note on doodling

By the end of lecture today 9/17/14 Use this as your study guide Characteristics of a distribution Central Tendency Dispersion Primary types of “measures of central tendency”? Mean Median Mode Measures of variability Range Standard deviation Variance Memorizing the four definitional formulae

Homework due – Monday (September 22 nd ) On class website: please print and complete homework worksheet # 6 & 7

Overview Frequency distributions The normal curve Mean, Median, Mode, Trimmed Mean Standard deviation, Variance, Range Mean Absolute Deviation Skewed right, skewed left unimodal, bimodal, symmetric Challenge yourself as we work through characteristics of distributions to try to categorize each concept as a measure of 1) central tendency 2) dispersion or 3) shape

Another example: How many kids in your family? Number of kids in family

Measures of Central Tendency (Measures of location) The mean, median and mode Mean: The balance point of a distribution. Found by adding up all observations and then dividing by the number of observations Mean for a sample: Mean for a population: ΣX / N = mean = µ (mu) Note: Σ = add up x or X = scores n or N = number of scores Σx / n = mean = x Measures of “location” Where on the number line the scores tend to cluster

Measures of Central Tendency (Measures of location) The mean, median and mode Mean: The balance point of a distribution. Found by adding up all observations and then dividing by the number of observations Mean for a sample: Note: Σ = add up x or X = scores n or N = number of scores Σx / n = mean = x Number of kids in family / 10 = mean = 4.1

How many kids are in your family? What is the most common family size? Number of kids in family Median: The middle value when observations are ordered from least to most (or most to least)

How many kids are in your family? What is the most common family size? Median: The middle value when observations are ordered from least to most (or most to least) 1, 3, 1, 4, 2, 4, 2, 8, 2, 14 1, 2, 3, 4, 8, 14 Number of kids in family

Number of kids in family , 4, 2, 1, How many kids are in your family? What is the most common family size? Number of kids in family Median: The middle value when observations are ordered from least to most (or most to least) 1, 3, 1, 4, 2, 4, 2, 8, 2, , 3, 1, 2, 4, 2, 4,8, 1, 14 2, 3, 1, Median always has a percentile rank of 50% regardless of shape of distribution µ = 2.5 If there appears to be two medians, take the mean of the two

Mode: The value of the most frequent observation Number of kids in family Score f Please note: The mode is “2” because it is the most frequently occurring score. It occurs “3” times. “3” is not the mode, it is just the frequency for the value that is the mode Bimodal distribution: If there are two most frequent observations

What about central tendency for qualitative data? Mode is good for nominal or ordinal data Median can be used with ordinal data Mean can be used with interval or ratio data

Overview Frequency distributions The normal curve Mean, Median, Mode, Trimmed Mean Challenge yourself as we work through characteristics of distributions to try to categorize each concept as a measure of 1) central tendency 2) dispersion or 3) shape Skewed right, skewed left unimodal, bimodal, symmetric

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

A little more about frequency distributions An example of a normal distribution

Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Normal distribution In a normal distribution: mode = mean = median In all distributions: mode = tallest point median = middle score mean = balance point

Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Positively skewed distribution In a positively skewed distribution: mode < median < mean In all distributions: mode = tallest point median = middle score mean = balance point Note: mean is most affected by outliers or skewed distributions

Measure of central tendency: describes how scores tend to cluster toward the center of the distribution Negatively skewed distribution In a negatively skewed distribution: mean < median < mode In all distributions: mode = tallest point median = middle score mean = balance point Note: mean is most affected by outliers or skewed distributions

Mode: The value of the most frequent observation Bimodal distribution: Distribution with two most frequent observations (2 peaks) Example: Ian coaches two boys baseball teams. One team is made up of 10-year-olds and the other is made up of 16-year-olds. When he measured the height of all of his players he found a bimodal distribution

Overview Frequency distributions The normal curve Mean, Median, Mode, Trimmed Mean Standard deviation, Variance, Range Mean Absolute Deviation Skewed right, skewed left unimodal, bimodal, symmetric

6’ 7’ 5’ 5’6” 6’6” 6’ 7’ 5’ 5’6” 6’6” 6’ 7’ 5’ 5’6” 6’6” Dispersion: Variability Some distributions are more variable than others Range: The difference between the largest and smallest observations Range for distribution A? Range for distribution B? Range for distribution C? A B C The larger the variability the wider the curve tends to be The smaller the variability the narrower the curve tends to be

Range: The difference between the largest and smallest scores 84” – 71” = 13” Tallest player = 84” (same as 7’0”) (Kaleb Tarczewski and Dusan Ristic) Shortest player = 71” (same as 5’11”) (Parker Jackson-Cartwritght) Wildcats Basketball team: Range is 13” Fun fact: Mean is 78 x max - x min = Range

Range: The difference between the largest and smallest score 77” – 70” = 7” Tallest player = 77” (same as 6’5”) (Austin Schnabel) Shortest player = 70” (same as 5’10”) (Five players are 5’10”) Wildcats Baseball team: Range is 7” (77” – 70” ) Fun fact: Mean is 72 x max - x min = Range Please note: No reference is made to numbers between the min and max Baseball

Frequency distributions The normal curve

Variability Some distributions are more variable than others 6’ 7’ 5’ 5’6” 6’6” Let’s say this is our distribution of heights of men on U of A baseball team Mean is 6 feet tall 6’ 7’ 5’ 5’6” 6’6” 6’ 7’ 5’ 5’6” 6’6” What might this be?

6’ 7’ 5’ 5’6” 6’6” 6’ 7’ 5’ 5’6” 6’6” 6’ 7’ 5’ 5’6” 6’6” The larger the variability the wider the curve the larger the deviations scores tend to be The smaller the variability the narrower the curve the smaller the deviations scores tend to be Variability

Standard deviation: The average amount by which observations deviate on either side of their mean Mean is 6’ Generally, (on average) how far away is each score from the mean?

Let’s build it up again… U of A Baseball team Diallo Diallo is 6’0” Diallo is 0” Deviation scores 6’0” – 6’0” = 0 5’8” 5’10” 6’0” 6’2” 6’4” Diallo’s deviation score is 0

Preston Preston is 6’2” 6’2” – 6’0” = 2 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Preston is 2” Deviation scores Preston’s deviation score is 2” Diallo is 6’0” Diallo’s deviation score is 0 Let’s build it up again… U of A Baseball team

Mike Hunter Mike is 5’8” Hunter is 5’10” 5’8” – 6’0” = -4 5’10” – 6’0” = -2 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Preston is 2” Deviation scores Preston is 6’2” Preston’s deviation score is 2” Diallo is 6’0” Diallo’s deviation score is 0 Mike’s deviation score is -4” Hunter’s deviation score is -2” Let’s build it up again… U of A Baseball team

David Shea Shea is 6’4” David is 6’ 0” 6’4” – 6’0” = 4 6’ 0” – 6’0” = 0 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores Preston’s deviation score is 2” Diallo’s deviation score is 0 Mike’s deviation score is -4” Hunter’s deviation score is -2” Shea’s deviation score is 4” David’s deviation score is 0 Let’s build it up again… U of A Baseball team

David Shea 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores Preston’s deviation score is 2” Diallo’s deviation score is 0 Mike’s deviation score is -4” Hunter’s deviation score is -2” Shea’s deviation score is 4” David’s deviation score is 0” Let’s build it up again… U of A Baseball team

5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores Let’s build it up again… U of A Baseball team

Standard deviation: The average amount by which observations deviate on either side of their mean 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores

Standard deviation: The average amount by which observations deviate on either side of their mean 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores

Standard deviation: The average amount by which observations deviate on either side of their mean 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores

How do we find the average height? 5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Standard deviation: The average amount by which observations deviate on either side of their mean Σ(x - x) = 0 Σ (x - µ ) = ? Diallo Mike Hunter Preston 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores Σ(x - µ ) = 0 = average height N ΣxΣx = average deviation Σ(x - µ ) N How do we find the average spread?

5’8” - 6’0” = - 4” 5’9” - 6’0” = - 3” 5’10’ - 6’0” = - 2” 5’11” - 6’0” = - 1” 6’0” - 6’0 = 0 6’1” - 6’0” = + 1” 6’2” - 6’0” = + 2” 6’3” - 6’0” = + 3” 6’4” - 6’0” = + 4” Standard deviation: The average amount by which observations deviate on either side of their mean Σ(x - x) = 0 Σ x - x = ? 5’8” 5’10” 6’0” 6’2” 6’4” Diallo is 0” Mike is -4” Hunter is -2 Shea is 4 David is 0” Preston is 2” Deviation scores Σ(x - µ ) = 0 Big problem Σ(x - x) 2 Square the deviations Σ(x - µ ) 2 N ΣxΣx N Big problem 2

Mean: The average value in the data Standard deviation: The average amount scores deviate on either side of their mean Standard deviation is typical “spread” (typical size of deviations or distance from mean) – can never be negative Mean is a measure of typical “value” (where the typical scores are positioned on the number line)

Standard deviation: The average amount by which observations deviate on either side of their mean These would be helpful to know by heart – please memorize these formula

Standard deviation: The average amount by which observations deviate on either side of their mean What do these two formula have in common?

Standard deviation: The average amount by which observations deviate on either side of their mean What do these two formula have in common? n-1 is “Degrees of Freedom” More, next lecture