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Psychology 217 Statistical Methods

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Lesson U1-3: Summarizing Data Tables and graphs* –Tables –Pie charts –Histograms and Polygons –Scatterplots/Scattergrams *Note: The APA Publication Manual (6 th ed.) dedicates 43 pages (pp. 125-167) to Tables and Figures Central Tendency –Mean –Standard Error of the Mean (SE M ) –Median –Mode

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Graphics A picture is worth 1,000 words If it takes 1,000 words to explain a picture, then the point of diminishing returns has been violated!

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Tables Tables are most useful when trying to convey a large amount of data in a small amount of space. Tables should not be overused. –The 1 st APA Guideline for Table use is: Is the table necessary? Tables are not stand-alone. –They must be discussed in the narrative as well. For example,…

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Additionally, preference grouping significantly varied in relation to PPRS ranking (χ2(3)=87.19, p<.001) (See Table 6). Christian non-Catholic participants (LDS and Christian remainder) significantly underrepresented in the low PPRS ranking and overrepresented in the high PPRS ranking. Conversely, those without preference significantly overrepresented in the low PPRS ranking and underrepresented in the high PPRS ranking. Table 6. PPRS Ranking by Religious Preference Grouping Low PPRSHigh PPRS LDS20 (14%) 64 (47%) Catholic11 (8%) 21 (15%) Christian remainder29 (21%) 44 (32%) None79 (57%) 8 (6%) signifies high cell percentage (p<.05) determined by cell standard deviate calculation signifies low cell percentage (p<.05) determined by cell standard deviate calculation

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Pie Charts Pie charts are almost never worth the space –Simple visual depiction of proportions APA guidelines: –The number of items compared should be kept to five or fewer. Order the segments from large to small, beginning the largest segment at 12 oclock […] making the smallest segment [shaded] the darkest.

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Shapes of Distributions Frequency Distribution – a table of counts –2 5 3 6 5 8 2 3 4 1 0 6 8 9 2 3 7 2 9 0 –2 zeros, 1 one, 4 twos, 3 threes, 1 four, 2 fives, 2 sixes, 1 seven, 2 eights, and 2 nines

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Cumulative Frequency Distribution and Histogram Xfcf 9220 (N) 8218 7116 6215 5213 4111 3310 2 4 7 11 3 02 2

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Frequency Polygon and Histogram

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Normal, Negative Skew, Positive Skew, Bi-modal

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Scatterplots/Scattergrams The plotting of paired values –For example,… Age Toys –Bob5 6 –Anne6 12 –Sue3 15 –Bill5 7 –Tim4 15

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Central Tendency The location of data Mean –For Interval and Ratio data and normal distributions Median –For Ordinal data or skewed distributions Mode –For nominal data or multi-modal distributions

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The Mean (aka, the Arithmetic Mean, Average) The arithmetic center of the distribution Symbolized as –M in academic literature –X among statisticians Operationalized as (X)/N –Sigma () means sum of –Meaning, add up all the X scores and divide by the number of X scores

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The Mean Data set of X values –2 5 3 6 5 8 2 3 4 1 0 6 8 9 2 3 7 2 9 0 X = 85 N = 20 M = (X)/N = 85/20 = 4.25

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Standard Error of the Mean (SE M ) When we collect statistics from a sample, –our data do not represent the population parameters with 100% accuracy –we must adjust for this error in our data The Standard Error of the Mean (SE M ) –is an estimate of this margin of error –is a necessary calculation in inferential statistics The sample mean is a point estimate (location) SE M is an interval estimate (spread)

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Sampling Distribution of Means The Normal Distribution is a distribution of all the individual raw scores The Sampling Distribution of Means is a distribution of all the means of the samples that can be drawn from a population –The Central Limit Theorem is that this distribution is normal and centers around the population mean –By definition, SE M is the standard deviation of the Sampling Distribution of the Means

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Calculating SE M The larger the N, the smaller the SE M The lesser the variability, the smaller the SE M MS Within SE M = N ……………………………….Stay tuned!

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Median The physical center of the distribution Symbolized as Mdn Operationalized as the X value at the 50 th % ile –Half the data are below, half the data are above Data set of X values –2 5 3 6 5 8 2 3 4 1 0 6 8 9 2 3 7 2 9 0 –Ascending order –0 0 1 2 2 2 2 3 3 3 4 5 5 6 6 7 8 8 9 9 Median is (3 + 4) / 2 = 3½

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The Mode The most frequently occurring value Symbolized as Mode Operationalized as the X value(s) with the largest frequency(ies) on the frequency distribution –(aka, the tallest bar(s) on the histogram)

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Mode = 2 Xf 92 82 71 62 52 413 2 41 02

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Positive Skew (Mode = 2) < (Mdn = 3.5) < (M = 4.25)

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Central Tendencys Interpretive Power (the need for Dispersion) XY ZXY Z 3511 34 1 33 1 32 1 31 1 X = 15 Y = 15 Z = 15 M X = 3 M Y = 3 M Z = 3

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