Central Tendency

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
  (N1) / 2
• (3, 4, 4, 5, 5, 5, 6, 6, 7), N9, so (91)/2
  score in 5th position ? 5

6
Median (cont.)

• If Even N find median position with (N1) / 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 (101) / 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 tableslook 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 dont
  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