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Central Tendency

Central Tendency. Mon. Feb 16 th , 2004. Central Tendency. One number that summarizes an entire distribution. 3 measures of central tendency: Mean – arithmetic average of scores Median – score that divides distribution exactly in half (50% percentile) Mode – most frequent score. Mean.

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Central Tendency

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  1. Central Tendency Mon. Feb 16th, 2004

  2. Central Tendency • One number that summarizes an entire distribution. • 3 measures of central tendency: • Mean – arithmetic average of scores • Median – score that divides distribution exactly in half (50% percentile) • Mode – most frequent score

  3. Mean • Book notation: Ybar = Y / N • Where  (sigma) indicates ‘sum’ • Labs often use “X bar” rather than “Y”, they are interchangeable • Properties of the mean: • It is highly influenced by extreme scores (outliers) (3, 5, 9, 11, 15): Y bar = 43 / 5 = 8.6 (3, 5, 9, 11, 54): Y bar = 82 / 5 = 16.4 May not be very representative when outliers

  4. Properties of Mean (cont.) • Add/subtract a constant from each score, the mean changes by adding/subtracting that constant. (3, 5, 9, 11, 15): y bar = 8.6 Add 2 to each..(5, 7, 11, 13, 17): y bar = 53 / 5 = 10.6 • Multiply/divide each score by constant, mean changes by mult/div that constant: (3, 5, 9, 11, 15) Y bar = 8.6 Multiply each by 3 (9, 15, 27, 33, 45): y bar = 129 / 5 = 25.8 Check (8.6 x 3) =25.8

  5. Median • Divides distribution in half, 50% scores above & 50% scores below • If Odd # scores (N): find median position with (N+1) / 2 (3, 4, 4, 5, 5, 5, 6, 6, 7), N=9, so (9+1)/2 = score in 5th position  5

  6. Median (cont.) • If Even N: find median position with (N+1) / 2  (will give a ½ answer, so take average of scores in those 2 adjacent positions) N = 10: (8, 10, 12, 15, 15, 18, 18, 19, 25, 60), med position is (10+1) / 2 = 5.5, so take average of scores in 5th and 6th positions 15 + 18 / 2 = 16.5 is median score

  7. (cont.) • Median is unaffected by outliers (unlike mean), so a better choice for distributions w/outliers • Note that in SPSS, easy way to find median in freq distribution tables…look for score/category that includes the 50% percentile (c%)

  8. Mode • Most frequent score • May have bimodal distribution (2 modes) • Not too informative, but best used when categorical/nominal data (median & mean don’t make sense there)

  9. CT & Shape of Distribution • Can determine relative order of mean, median, mode based on skewness of distribution • In pos skew distribution, from L  R will find mode, median, mean (see example) • In neg skew distribution, from LR will find mean, median, mode • If symmetrical distribution, mean, median, mode will all be equal

  10. Lab 9 • 3 parts: • 1. SPSS - Use students.sav data (used last week)…Do Q1 by hand • 2. Use applet (click on button, opens window, then choose “Mean & Median”) • 3. SPSS – use study.sav data, get histograms & examine for skewness, then predict position of mean, md, mode

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