Reasoning in Psychology Using Statistics 2017

Descriptive statistics 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) 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. Measures of Center

The Mean Most commonly used measure of center Arithmetic average Computing the mean Divide by the total number in the population Formula for population mean (a parameter): Add up all of the X’s Formula for sample mean (a statistic): Divide by the total number in the sample M= Note: Mean is mathematical result, not necessarily score on scale (e.g., average of 2.5 children) The Mean

The Mean Conceptualizing the mean As the center of the distribution As the representative score in the distribution The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution Balancing point The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 Balancing point 1+10 = 11 Mean = 11/2 = 5.5 The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 Balancing points The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? Balancing point 1+10+7 = 18 Mean = 18/3 = 5.5 The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? 1+10+7 = 18 Mean = 18/3 = 6.0 The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? 1+10+7 = 18 Mean = 18/3 = 6.0 The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? 1+10+7 = 18 Mean = 18/3 = 6.0 The Mean

The Mean Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution? New Balancing point 1+10+7 = 18 Mean = 18/3 = 6.0 The Mean

The Mean Conceptualizing the mean As center of distribution As the representative score in the distribution To be fair, let’s give everybody the same amount. Girl Scout bake sale for camping trip $30 $6 $15 $12 $25 $18 $13 1 2 3 4 5 6 7 8 9 10 12+25+30+6+18+15+13=119 119/7 = 17 The Mean

The Mean Conceptualizing the mean As center of distribution As representative score 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 $17 12+25+30+6+18+15+13=119 119/7 = 17 So everybody is represented by same score, the mean is the “standard” 17+17+17+17+17+17+17=119 119/7 = 17 The Mean

A weighted mean Suppose that you combine 2 groups together. How do you compute new group mean? Average the 2 averages But it only works this way when the two groups have exactly the same number of scores A weighted mean

A weighted mean Suppose that you combine 2 groups together. How do you compute new group mean? $205!? I only have $191 Group 1 Group 2 New Group A weighted mean

A weighted mean Suppose that you combine 2 groups together. How do you compute new group mean? Group 1 $12 $25 $30 $6 $18 $15 $13 Group 2 $25 $30 $17 New Group 12+25+30+6+18+15+13+25+17+30=191 Mean = 191/10 = 19.1 $12 $30 $6 $13 $30 $18 $17 $25 $25 $15 A weighted mean

The mean is the representative score in Suppose that you combine 2 groups together. How do you compute new group mean? The mean is the representative score in the distribution Group 1 $17 Group 2 $24 New Group A weighted mean

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 17 Characteristics of a mean

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 17 Characteristics of a mean

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 Characteristics of a mean

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. Characteristics of a mean

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 the troop sponsor agreed to match the money made by each girl scout (they give each girl scout an additional amount of money equal to however much each made on the sale). So now the total is $238, and the mean for each girl is 238/7 = $34 Which is 2 times the original mean Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant. 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 $30 $6 $18 $15 $13 The median

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 $15 $6 $25 $30 $12 $13 $18 Step2: find middle score That’s the median, a score on scale 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 $12 $25 $30 $18 $15 $13 $6 Step2: find middle 2 scores Step3: find arithmetic average of 2 middle scores That’s the median Note: mathematical result not a score on scale The median

The mode 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 Mode = Medium The mode

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. Which center when?

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 Green Mode = Brown Median = Which center when?

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 Which center when?

Which center when? Median 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) 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 Which center when?

In Summation Situation Most Representative Least Representative Can’t Use Nominal Mode Median/Mean Ordinal Median Mean Skewed interval or ratio Open ended interval or ratio Interval or Ratio In Summation

Which center when? Impact of shape on center (interval or ratio scale) mean median , 2 modes mode mean median = = = Positively skewed distribution Negatively skewed distribution median mode mode median mean mean > > < < Mean & median pulled toward tail Which center when?

Chicago distributions Mode 0-10,000 175-200,000 Median 45,734 261,600 Mean ? 325,212 Chicago distributions Check out your hometown: http://www.city-data.com/

Buyer beware: Know your distribution The average price of houses in this neighborhood is … Mode 0-10,000 175-200,000 Median 45,734 261,600 Mean ? 325,212 selling buying When you say “average” are you talking about the median or the mean? Buyer beware: Know your distribution

Wrap up Today’s lab Questions? Compute mean, median, & mode both by hand & using SPSS Questions? Wrap up