Central Tendency Lecture Notes PDF

CENTRAL TENDENCY Descriptive Statistics Overview  Central Tendency  Mean  Weighted Mean  Median  Mode  What Is Central Tendency? 3  Score that best captures the entire group of scores  Mean  Median  Mode  M...

CENTRAL TENDENCY Descriptive Statistics Overview  Central Tendency  Mean  Weighted Mean  Median  Mode  What Is Central Tendency? 3  Score that best captures the entire group of scores  Mean  Median  Mode  Mean 4  Arithmetic average  Add scores & divide by # of scores X  More Terms X   Population mean m (“mu”) Sample mean M or X n  Population N  Sample n  When writing in APA style N total sample, n for portion of sample  Mean: an exercise 5 Here is a data set: 2 3 3 5 6 7 9 X What is the mean? X n  Answer 6 Here is a data set: 2 3 3 5 6 7 9 X What is the mean? X n 2+3+3+5+6+7+9 35 ------------------------------- = ------ = 5 7 7 Weighted Mean  Overall mean of 2 or more separate groups X 1  X 2   WeightedX  n1  n2 X  Sample 1: M = 5, n = 4 X  n Sample 2: M = 3, n = 6  Σx1=  Σx2 = Weighted Mean: Answers  Overall mean of 2 or more separate groups X 1  X 2   WeightedX  n1  n2 X  Sample 1: M = 5, n = 4 X  n Sample 2: M = 3, n = 6  Σx1= 5 x 4 = 20 X 20  18  38 3.8  Σx2 =3 x 6 = 18 46 10 Weighted Mean Example  Hypothetical evaluations:  Instructor effectiveness 1-5 (high scores more effective)  Intro to Bio M = 3.25, n = 11  Intro to Bio M = 4.12, n = 25  What would the weighted mean be? Weighted Mean Example  Hypothetical evaluations:  Instructor effectiveness 1-5 (high scores more effective)  Intro to Bio M = 3.25, n = 11  Intro to Bio M = 4.12, n = 25  What would the weighted mean be?  [(3.25 x 11) + (4.12 x 25)]/(11 + 25)  (35.75 + 103)/36  138.75/36  3.85  Median 11  Score at the 50th percentile (Mdn)  List scores in order from low to high  When n is odd Ex: 2, 2, 4, 5, 6  Median is middle Mdn = 4 number Ex: 2, 2, 3, 3, 5, 5,  When n is even 6, 8  Median is halfway Middle scores: 3 between middle 2 &5 numbers Halfway: (3 + 5)/2 Mdn = 4  Mode 12  Most frequently occurring score  Any scale of measurement  Used for nominal scales  Problems:  Could be many modes  Does not take into consideration all of the data Mode  What is the mode of these data:  2 3 5 5 6 7 9  What is the mode of these data:  2 3 5 5 5 6 7 7 7 9 Mode  What is the mode of these data:  2 3 5 5 6 7 9 5  What is the mode of these data:  2 3 5 5 5 6 7 7 7 9 5 and 7 (these data are bimodal)  Distributions 15 Unimodal Bimodal Measures of Central Tendency  Measures of Central Tendency 17  Symmetrical (normal curve): M = Mdn = Mode  Positively skewed: M > Mdn  Mode  Negatively skewed: M < Mdn  Mode  Which measure to use? 18 5 Score f f*score 4 3 5 1 f 2 4 2 1 3 4 0 1 2 3 4 5 2 2 1 1 What is n? What is Σx? What is M? What is Mdn? What is  Answers: Which measure to use? 19 5 Score f f*score 4 3 5 1 5 f 2 4 2 8 1 3 4 12 0 1 2 3 4 5 2 2 1 1 4 1 What is n? (1+2+4+2+1) What is Σx? = 10 Mean usually used for What is M? Sum of f*score interval or ratio data What is = 1, 302, 2, 3, 3, 3, 3, 4, when distribution is Mdn? 4, 5 symmetrical & 30/10 =3 What is 3 unimodal.  Which measure to use? 20 5 Score f f*score 4 55 1 55 3 f 2 4 2 8 1 3 4 12 0 1 2 3 4 55 2 2 1 1 4 1 What is n? What is Σx? What is M? What is Mdn? What is  Which measure to use? 21 5 Score f f*score 4 55 1 55 3 f 2 4 2 8 1 3 4 12 0 1 2 3 4 55 2 2 1 1 4 1 What is n? (1+2+4+2+1) What is Σx? = 10 Median is preferred if What is M? Sum of f*score there are a few What is = 80 extreme scores or the Mdn? 80/10 = 8 distribution is very What is 3 skewed. Which measure to use?  Also use median if:  There are undetermined or unknown values  Open ended distributions 1, 2, 3, 4 or more  Ordinal data Which measure to use?  Mode is most appropriate for:  Nominal scales  Discrete variables  As an add on, gives a sense of distribution shape Example  Construct a frequency table and a graph.  Add columns for relative frequencies, cumulative frequencies, & cumulative percents.  Determine SX, n, M, Mdn, Mode.  Determine the shape of the distribution.  If you could only report one measure of central tendency then which would you choose and why. (interval data) 14 14 13 15 10 9 15 13 10 12 13 14 13 12 8 15 17 14 9 15 16 Answers Scores f f*score rf cf C% s 17 1 17.05 21 100% 16 1 16.05 20 95% 15 4 60.19 19 90% 14 4 56.19 15 71% 13 4 52.19 11 52% 12 2 24.10 7 33% 11 0 0.00 5 24% 10 2 20.10 5 24% 9 2 18.10 3 14% 8 1 8.05 1 5%  Frequency of Scores 5 4 3 f 2 1 0 8 9 10 11 12 13 14 15 16 17 Scores Answers  n = 21  SX = 271  M = 12.90  Mdn = 13  Mo = 13, 14, 15  There are three modes and there is a slight negative skew. It is not quite a normal distribution, but because the mean, median, and mode are very similar it suggests the skew is not dramatic.  I would report mean, because the distribution is not skewed dramatically.