Reasoning in Psychology Using Statistics 2017

Announcements Quiz 3 due Friday, Feb. 24 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

Exam 1s Lecture Exam 1 Lab Exam 1 Mean = 58.7 (78.3%) SD = 8.1 Min = 26; Max = 72 Lab Exam 1 Mean = 64.8 (86.4%) SD = 5.8 Min = 35; Max = 74 Exam 1s Final Exam schedule posted: Tue May 9 @ 10AM

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

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

Variability of a distribution 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 Low variability: scores are all similar SD = 8.1 26 - 72 SD = 5.8 35 - 74 Variability of a distribution

Range Simplest measure of variability Range = Maximum - Minimum Disadvantage: based solely on two most extreme values Range

IQR: Alternative measure of variability Inter-Quartile Range

Inter-Quartile Range IQR: Alternative measure of variability Median (50%tile): 1/2 distribution on one side & 1/2 on other side. 25%tile? 75%tile? Median 25%tile 75%tile 25% 25% Inter-Quartile Range

Inter-Quartile Range IQR: Alternative measure of variability IQR = 3rd quartile - 1st quartile Middle 50% of the scores Median 25%tile 75%tile 25% 25% IQR Inter-Quartile Range

Inter-Quartile Range IQR: Alternative measure of variability IQR = 3rd quartile - 1st quartile Middle 50% of the scores Works well for skewed distributions Inter-Quartile Range

Inter-Quartile Range 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% 25% 25% IQR Inter-Quartile Range

Standard Deviation & Variance Most utilized & important measure of variability Standard deviation measures how far off all of the scores are from a standard Standard deviation average of deviations ~ μ Typically mean is used as standard. Standard Deviation & Variance

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

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

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

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

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

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

Computing standard deviation (population) Step 3: Compute Variance. To get mean squared deviation, divide SS by number of scores Our population 2, 4, 6, 8 So N = 4 Variance = σ2 = SS/N (Population) MEMORIZE! = 20/4 = 5.0 Computing standard deviation (population)

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

Computing standard deviation (population) A good idea: check your answer, does it appear to be in the right ballpark? Our population 2, 4, 6, 8 μ +2.24 -2.24 1 2 3 4 5 6 7 8 9 10 X - μ = deviation scores 2 - 5 = -3 6 - 5 = +1 4 - 5 = -1 8 - 5 = +3 Computing standard deviation (population)

Computing standard deviation (population) 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)

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

Computing standard deviation (sample) 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)

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

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

Computing standard deviation (sample) 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 = s2 Notational differences Computing standard deviation (sample)

Computing standard deviation (sample) 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)

Characteristics of a standard deviation 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

Characteristics of a standard deviation 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. X old Characteristics of a standard deviation

Characteristics of a standard deviation 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. X old Characteristics of a standard deviation

Characteristics of a standard deviation 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. X old Characteristics of a standard deviation

Characteristics of a standard deviation 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. X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X new Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old Characteristics of a standard deviation

Characteristics of a standard deviation 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 X old X new Characteristics of a standard deviation

Characteristics of a standard deviation 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 2, 4, 6, 8 Original mean 5 2 - 5 = -3 4 - 5 = -1 6 - 5 = +1 8 - 5 = +3 Original SS 20 New sample 1, 3, 5, 7 New mean 4 1 - 4 = -3 5 - 4 = +1 Original SS 20 3 - 4 = -1 7 - 4 = +3 Characteristics of a standard deviation

Characteristics of a standard deviation 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 X 21 - 22 = -1 (-1)2 23 - 22 = +1 (+1)2 s = Characteristics of a standard deviation

Characteristics of a standard deviation 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. 42 - 44 = -2 (-2)2 40 42 44 46 48 X 46 - 44 = +2 (+2)2 s = Sold=1.41 Characteristics of a standard deviation

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

Today’s lab: compute several measures of variability both by hand and using SPSS Questions? Wrap up