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Reasoning in Psychology Using Statistics Psychology 138 2015 Lab Ex1, mean = 68.7/75 = 91.6% Lecture Ex1, mean = 57.9/75 = 77.2% Combined (Lab + Lecture) Ex1, mean = 126.6/150 = 84.4% Reasoning in Psychology Using Statistics Exam 1(s) • Summaries or pictures of the distribution • Numeric descriptive statistics – Shape: modality, and skew (and kurtosis, not cover much) – Measures of Center: Mode, Median, Mean – Measures of Variability (Spread): Range, Inter-Quartile Range, Standard Deviation (& variance) Reasoning in Psychology Using Statistics Descriptive statistics • Useful to summarize or describe distribution with single numerical value. – Value most representative of the entire distribution, that is, of all of the individuals – Central Tendency: 3 main measures – Mean (M) – Median (Mdn) – Mode • Note: “Average” may refer to each of these three measures, but it usually refers to Mean. Reasoning in Psychology Using Statistics Measures of Center • Most commonly used measure of center • Arithmetic average – Computing the mean – Formula for population mean (a parameter): – Formula for sample mean (a statistic): –Note: Mean is mathematical result, not necessarily score on scale (e.g., average of 2.5 children) Reasoning in Psychology Using Statistics The Mean åX m= N åX M= X = n Divide by the total number in the population Add up all of the X’s Divide by the total number in the sample – Conceptualizing the mean As the center of As the representative the distribution score in the distribution Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution Balancing point Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution 1 2 3 4 5 6 7 8 9 10 1+10 = 11 Mean = 11/2 = 5.5 Reasoning in Psychology Using Statistics Balancing point The Mean – Conceptualizing the mean As center of As representative score distribution in distribution 1 2 3 4 5 6 7 8 9 10 Balancing points Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution 1 2 3 4 5 6 7 8 9 10 Reasoning in Psychology Using Statistics The Mean What happens if we add an observation to our distribution? – Conceptualizing the mean As center of As representative score distribution in distribution 1 2 3 4 5 6 7 8 9 10 Reasoning in Psychology Using Statistics The Mean What happens if we add an observation to our distribution? – Conceptualizing the mean As center of As representative score distribution in distribution What happens if we add an observation to our distribution? Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution What happens if we add an observation to our distribution? Reasoning in Psychology Using Statistics The Mean – Conceptualizing the mean As center of As representative score distribution in distribution 1+10+7 = 18 Mean = 18/3 = 5.5 Reasoning in Psychology Using Statistics What happens if we add an Balancing observation to our distribution? point The Mean – Conceptualizing the mean As center of As representative score distribution in distribution 1+10+7 = 18 Mean = 18/3 = 6.0 Reasoning in Psychology Using Statistics The Mean What happens if we add an observation to our distribution? – Conceptualizing the mean As center of As representative score distribution in distribution 1+10+7 = 18 Mean = 18/3 = 6.0 Reasoning in Psychology Using Statistics The Mean What happens if we add an observation to our distribution? – Conceptualizing the mean As center of As representative score distribution in distribution 1+10+7 = 18 Mean = 18/3 = 6.0 Reasoning in Psychology Using Statistics The Mean What happens if we add an observation to our distribution? – Conceptualizing the mean As center of As representative score distribution in distribution 1 2 3 4 5 6 7 8 9 10 1+10+7 = 18 Mean = 18/3 = 6.0 Reasoning in Psychology Using Statistics What happens if we add an observation to our New Balancing distribution? point The Mean – Conceptualizing the mean As center of As the representative To be fair, let’s give distribution score in the distribution everybody the Girl Scout bake sale for camping trip same amount. 1 2 3 4 5 6 7 8 9 10 $12 $25 $30 $6 $18 $15 12+25+30+6+18+15+13=119 Reasoning in Psychology Using Statistics The Mean $13 119/7 = 17 – Conceptualizing the mean As center of As representative score distribution in distribution Girl Scout bake sale for camping trip 1 2 3 4 5 6 7 8 9 10 $17 $17 $17 $17 $17 $17 12+25+30+6+18+15+13=119 $17 119/7 = 17 So everybody is represented by same score, the mean is the “standard” 17+17+17+17+17+17+17=119 119/7 = 17 Reasoning in Psychology Using Statistics The Mean • Suppose that you combine 2 groups together. – How do you compute new group mean? Average the 2 averages 91.6 + 77.2 X= = 84.4 2 But it only works this way when the two groups have exactly the same number of scores Reasoning in Psychology Using Statistics A weighted mean • Suppose that you combine 2 groups together. $205!? – How do you compute new group mean? I only have $191 Group 1 Group 2 New Group X1 = 17 Reasoning in Psychology Using Statistics X2 = 24 17 + 24 XN = = 20.5 2 A weighted mean • Suppose that you combine 2 groups together. – How do you compute new group mean? Group 1 Group 2 New Group 12+25+30+6+18+15+13+25+17+30=191 X1 = 17 X2 = 24 Mean = 191/10 = 19.1 $30 $12 $25 $6 $17 $30 $12 $30 $25 $13 $18 $18 $15 $13 $17 $15 Reasoning in Psychology Using Statistics A weighted mean $6 $30 $25 $25 The mean thecombine 2 groups together. • Suppose that is you score in new group mean? –representative How do you compute the distribution Group 1 Group 2 New Group X1n1 + X 2 n 2 X1 = 17 X2 = 24 XN = n1 + n 2 (17 * 7) + (24 * 3) = = 19.1 7+3 $17 $17 $17 $17 $24 $24 $24 $17 $17 Reasoning in Psychology Using Statistics $17 A weighted mean • Change/add/delete a given score, then the mean will change. – Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7=112 112/7 = $16 (instead of $17) 5 10 15 20 25 30 17 Reasoning in Psychology Using Statistics Characteristics of a mean • Change/add/delete a given score, then the mean will change. – Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7=112 112/7 = $16 (instead of $17) 17 Reasoning in Psychology Using Statistics Characteristics of a mean • Change/add/delete a given score, then the mean will change. – Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7=112 112/7 = $16 (instead of $17) 5 10 15 20 25 30 16 Reasoning in Psychology Using Statistics Characteristics of a mean • Change/add/delete a given score, then the mean will change. • Add/subtract a constant to each score, then the mean will change by adding(subtracting) that constant. – Suppose that you want to factor out a $2 camping fee for each girl scout. Subtract 2 from each amount. Now the total is $105, so the mean is 105/7 = $15. – But notice you could have just subtracted $2 from the previous mean of $17 and arrived at the same answer. Reasoning in Psychology Using Statistics Characteristics of a mean • Change/add/delete a given score, then the mean will change. – Suppose that the troop sponsor agreed to match the money made by each girl scout (they give each girl scout an additional amount • Add/subtract a constant to each score, then the of money equal to however much each made on the sale). So now mean willischange adding(subtracting) the total $238, andby the mean for each girl is 238/7 =that $34 – Which is 2 times the original mean constant. • Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant. Reasoning in Psychology Using Statistics Characteristics of a mean • Median divides distribution in half: 50% of individuals in distribution have scores at or below the median. – Case1: Odd number of scores Step1: put scores in order $12 $25 Reasoning in Psychology Using Statistics $30 $6 $18 $15 The median $13 • Median divides distribution in half: 50% of individuals in distribution have scores at or below the median. – Case1: Odd number of scores Step1: put scores in order Step2: find middle score $6 $12 $13 $15 $18 $25 $30 That’s the median, a score on scale Reasoning in Psychology Using Statistics The median • Median divides distribution in half: 50% of individuals in distribution have scores at or below the median. – Case2: Even number of scores Step1: put scores in order Step2: find middle 2 scores $6 $12 $13 $15 $18 $18 $25 15 + 18 = 16.5 2 Reasoning in Psychology Using Statistics The median $30 Step3: find arithmetic average of 2 middle scores That’s the median Note: mathematical result not a score on scale • Mode: score or category with greatest frequency. – Pick variable in frequency table or graph with highest frequency (mode always a score on scale). Mode = 5 Modes = 2, 8 T-shirt size 30 Mode = Medium 25 20 15 10 5 0 Small Reasoning in Psychology Using Statistics The mode Medium Large X-large • Depends on a number of factors, like scale of measurement and shape. – The mean is the most preferred measure and it is closely related to measures of variability – However, there are times when the mean is not the appropriate measure. Reasoning in Psychology Using Statistics Which center when? • If data on nominal scale: Mode only – – – – Unranked categories (e.g. eye color) Not a numeric scale Can not do arithmetic operations on values Can not calculate cumulative percentages Eye color Mode = Brown Median = Green Reasoning in Psychology Using Statistics Which center when? • If data on ordinal scale: Median (plus Mode) – Not a numeric scale (e.g., T-shirt size) – Can not do arithmetic operations on values – Can calculate cumulative percentages on frequencies (median is score at 50th percentile) Median of T-shirt size = Medium Mode of T-shirt size = Medium Reasoning in Psychology Using Statistics Which center when? • If data on interval or ratio scale BUT: – Distributions open-ended • Response category like “5 or more” • Extreme values unknown, so can not calculate mean – Distributions skewed with long tails • Extreme values over influence mean • E.g., income sample of 50 – 47 middle income ($60,000-$100,000) and 3 millionaires or billionaires – Median = $80,000 – Mean = $135,000 or $60,000,000 • Median (plus Mode) Reasoning in Psychology Using Statistics Which center when? • If data on interval or ratio scale AND no exclusionary conditions: Mean (plus Median) (plus Mode) – Numeric scale – Can do arithmetic calculations on values – Have benefit of other statistics using the mean, such as standard deviation Reasoning in Psychology Using Statistics Which center when? • Impact of shape on center (interval or ratio scale) mean = median = mode Positively skewed distribution mean > median > mode mean = median, 2 modes Negatively skewed distribution mean < median < mode Mean & median pulled toward tail Reasoning in Psychology Using Statistics Which center when? Mode Median Mean Reasoning in Psychology Using Statistics 0-10,000 45,734 ? 175-200,000 261,600 325,212 Chicago distributions Check out your hometown: http://www.city-data.com/ price of Mode The average 0-10,000 houses in this Medianneighborhood 45,734 is … Mean ? buying 175-200,000 261,600 325,212 selling When you say “average” are you talking about the median or the mean? Buyer beware: Know your distribution Reasoning in Psychology Using Statistics • Today’s lab – Compute mean, median, & mode both by hand & using SPSS • Questions? Reasoning in Psychology Using Statistics Wrap up