Reasoning in Psychology Using Statistics Psychology 138 2019

• Quiz 3 due Friday, Feb. 22 at 11: 59 pm – Covers

• In addition to pictures of the distribution, numerical summaries are also typically

• Quantitative measure of how spread out or clustered scores are – the

• Simplest measure of variability – Range = Maximum - Minimum – Disadvantage:

• IQR: Alternative measure of variability – Middle 50% of scores Inter-Quartile Range

• IQR: Alternative measure of variability – Middle 50% of scores – Median

• IQR: Alternative measure of variability – IQR = 3 rd quartile -

• IQR: Alternative measure of variability - IQR more representative than Range -

• Most utilized & important measure of variability – Standard deviation measures how

• Step 1: For measure of deviation, subtract population mean from every score.

• Characteristic of average of deviation scores: Total of the deviations must equal

• Step 2: Preserve all deviations, positive & negative. How? – Square the

• Step 3: Compute Variance. – To get mean squared deviation, divide SS

• Step 4: Compute the Standard Deviation – Variance = average squared deviation

• A good idea: check your answer, does it appear to be in

• To review (computing standard deviation for a population): – – Step 1:

• For a sample: The basic procedure is the same. – – Step

• Step 1: Compute deviation scores – subtract sample mean from every score

• Step 2: Compute Sum of Squared deviations (SS). X - X =

• Step 3: Determine Variance Recall: Population variance = σ2 = SS/N Samples’

• Step 4: Determine the standard deviation Standard Deviation = s = (Sample)

• Change/add/delete a given score, then the standard deviation will change. – May

• Change/add/delete a given score, then the standard deviation will change. • Add/subtract

• Change/add/delete a given score, then the mean will change. • Add/subtract a

• Extreme scores: Range is most affected, IQR is least affected • Sample

• Today’s lab: compute several measures of variability both by hand using SPSS

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Reasoning in Psychology Using Statistics Psychology 138 2019

• Quiz 3 due Friday, Feb. 22 at 11: 59 pm – Covers • Tables and graphs • Measures of center • Measures of variability • You don’t need SPSS, but may want to have a calculator handy Announcements Reasoning in Psychology Using Statistics

• In addition to pictures of the distribution, numerical summaries are also typically presented. – Numeric Descriptive Statistics Today’s focus • Shape: skew and kurtosis • Measures of Center: mode, median, mean • Measures of Variability (Spread): Range, Inter-quartile range, Standard Deviation (& variance) Descriptive statistics Reasoning in Psychology Using Statistics

• Quantitative measure of how spread out or clustered scores are – the degree of “differentness” of the scores in the distribution. • High variability: scores differ a lot – SD = 8. 1 – 26 - 72 • Low variability: scores are all similar – SD = 5. 8 – 35 - 74 Variability of a distribution Reasoning in Psychology Using Statistics

• Simplest measure of variability – Range = Maximum - Minimum – Disadvantage: based solely on two most extreme values Range Reasoning in Psychology Using Statistics

• IQR: Alternative measure of variability – Middle 50% of scores Inter-Quartile Range Reasoning in Psychology Using Statistics Newly designed webpages

• IQR: Alternative measure of variability – Middle 50% of scores – Median (50%tile): 1/2 distribution on one side & 1/2 on other side. – 25%tile? – 75%tile? Median 25%tile 75%tile 25% Inter-Quartile Range Reasoning in Psychology Using Statistics 25%

• IQR: Alternative measure of variability – IQR = 3 rd quartile - 1 st quartile – Middle 50% of the scores Median 25%tile 75%tile 25% 25% IQR Inter-Quartile Range Reasoning in Psychology Using Statistics

• IQR: Alternative measure of variability – IQR = 3 rd quartile - 1 st quartile – Middle 50% of the scores – Works well for skewed distributions Inter-Quartile Range Reasoning in Psychology Using Statistics

• IQR: Alternative measure of variability - IQR more representative than Range - Extreme scores (i. e. , outliers) have less influence (robust) - Disadvantage: all scores not represented (only the middle 50%) IQR 25% 25% IQR Inter-Quartile Range Reasoning in Psychology Using Statistics 25%

• Most utilized & important measure of variability – Standard deviation measures how far off all of the scores are from a standard. • Computations for sample and population are (slightly) different – Standard deviation ~~ average of deviations* Typically mean is used as standard. μ Standard Deviation & Variance Reasoning in Psychology Using Statistics *Mean Absolute Differences vs. SD

• Step 1: For measure of deviation, subtract population mean from every score. Our population 2, 4, 6, 8 -3 1 2 3 4 5 6 7 8 9 10 X - μ = deviation scores μ 2 - 5 = -3 Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Step 1: For measure of deviation, subtract population mean from every score. Our population 2, 4, 6, 8 -1 1 2 3 4 5 6 7 8 9 10 X - μ = deviation scores μ 2 - 5 = -3 4 - 5 = -1 Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Step 1: For measure of deviation, subtract population mean from every score. Our population 2, 4, 6, 8 1 1 2 3 4 5 6 7 8 9 10 X - μ = deviation scores 2 - 5 = -3 4 - 5 = -1 μ 6 - 5 = +1 (balances off -1) Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Step 1: For measure of deviation, subtract population mean from every score. Our population 2, 4, 6, 8 3 1 2 3 4 5 6 7 8 9 10 X - μ = deviation scores 2 - 5 = -3 4 - 5 = -1 μ 6 - 5 = +1 8 - 5 = +3 Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Characteristic of average of deviation scores: Total of the deviations must equal 0 X - μ = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3 -3 + -1 + 3 = -4 + 4 = 0 Mean is the balancing point, so 2 sides are equal, that is, total deviations on each side are equal 1 2 3 4 5 6 7 8 9 10 μ Good check of computations Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Step 2: Preserve all deviations, positive & negative. How? – Square the deviations! SS = Σ (X – μ)2 – Add to get Sum of Squared deviations, MEMORIZE! “Sum of Squares” (SS) Note Order Of Operations! X - μ = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 SS = (-3)2 + (-1)2 + (+3)2 = 9 + 1 + 9 = 20 Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Step 3: Compute Variance. – To get mean squared deviation, divide SS by number of scores Variance = σ2 = SS/N Our population So N = 4 2, 4, 6, 8 MEMORIZE! (Population) = 20/4 = 5. 0 Computing standard deviation (population) Reasoning in Psychology Using Statistics

• Step 4: Compute the Standard Deviation – Variance = average squared deviation score – Get average deviation score by taking square root of variance MEMORIZE! Standard Deviation = σ = (Population) Computing standard deviation (population) Reasoning in Psychology Using Statistics

• A good idea: check your answer, does it appear to be in the right ballpark? Our population 2, 4, 6, 8 +2. 24 -2. 24 1 2 3 4 5 6 7 8 9 10 X - μ = deviation scores 2 - 5 = -3 4 - 5 = -1 μ 6 - 5 = +1 8 - 5 = +3 Computing standard deviation (population) Reasoning in Psychology Using Statistics

• To review (computing standard deviation for a population): – – Step 1: compute deviation scores Step 2: compute SS Step 3: determine variance ( σ2 squared standard deviation) Step 4: determine standard deviation (square root of variance) Standard Deviation = σ Computing standard deviation (population) Reasoning in Psychology Using Statistics

• For a sample: The basic procedure is the same. – – Step 1: Step 2: Step 3: Step 4: compute deviation scores compute SS determine variance This step is different determine standard deviation Computing standard deviation (sample) Reasoning in Psychology Using Statistics

• Step 1: Compute deviation scores – subtract sample mean from every score Our sample 2, 4, 6, 8 Notational differences 1 2 3 4 5 6 7 8 9 10 X - X = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 X Numerically everything is the same as before Computing standard deviation (sample) Reasoning in Psychology Using Statistics

• Step 2: Compute Sum of Squared deviations (SS). X - X = deviation scores 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 SS = Σ (X - X)2 = (-3)2 + (-1)2 + (+3)2 = 9 + 1 + 9 = 20 Apart from notational differences, procedure same as before Computing standard deviation (sample) Reasoning in Psychology Using Statistics

• Step 3: Determine Variance Recall: Population variance = σ2 = SS/N Samples’ variability smaller than population’s X 4 X 1 μ X 3 X 2 Computing standard deviation (sample) Reasoning in Psychology Using Statistics

• Step 3: Determine Variance Recall: Population variance = σ2 = SS/N Samples’ variability smaller than population’s To estimate population variance (what we are interested in) from sample, correct by dividing by (n-1) instead of just n Sample variance = s 2 Notational differences Computing standard deviation (sample) Reasoning in Psychology Using Statistics

• Step 4: Determine the standard deviation Standard Deviation = s = (Sample) MEMORIZE! MOST IMPORTANT!! NOTE: SPSS, Excel, other spreadsheets, and calculators use this formula for standard deviation. Computing standard deviation (sample) Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the standard deviation will change. – May change the mean and (if adding or subtracting) the number of scores (n or N) Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the standard deviation will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. – All of the scores change by the same constant. Xold Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the standard deviation will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. – All of the scores change by the same constant. – But so does the mean Xnew Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the standard deviation will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the standard deviation will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. – It is as if you just pick up the distribution and move it over, but the spread (variability) stays the same Xold Xnew Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the standard deviation will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. – Looking at a numerical example. (subtract 1 from every score) Original sample Original mean 2, 4, 6, 8 5 New sample New mean 1, 3, 5, 7 4 2 - 5 = -3 6 - 5 = +1 Original SS 4 - 5 = -1 8 - 5 = +3 20 1 - 4 = -3 5 - 4 = +1 Original SS 3 - 4 = -1 7 - 4 = +3 20 Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the mean will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. • Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant. 20 21 22 23 24 21 - 22 = -1 23 - 22 = +1 (-1)2 (+1)2 s= X Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Change/add/delete a given score, then the mean will change. • Add/subtract a constant to each score, then the standard deviation will NOT change. • Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant. 40 42 44 46 48 42 - 44 = -2 46 - 44 = +2 s= X (-2)2 (+2)2 Sold=1. 41 Characteristics of a standard deviation Reasoning in Psychology Using Statistics

• Extreme scores: Range is most affected, IQR is least affected • Sample size: Range tends to increase as n increases, IQR & SD do not • With open-ended distributions, one cannot even compute the Range or SD, so the IQR is the only option • Range is unstable when you repeatedly sample from the same population, but the IQR & SD are stable and tend not to fluctuate. When to use which Reasoning in Psychology Using Statistics

• Today’s lab: compute several measures of variability both by hand using SPSS • Questions? • • • Variability Defined: part 1 (~4 mins) | part 2 (~7 mins) Jbstatistics: Measures of Variability (~12 mins) Kahn. Academy: Why divide by n-1? (~10 mins) Brief Tutorial on Statistics: Measures of Variability (~17 mins) SPSS Range, IQR, Standard Deviation (~3 mins) Wrap up Reasoning in Psychology Using Statistics