Ibrahim Altubasi, PT, PhD The University of Jordan Central Tendency Ibrahim Altubasi, PT, PhD The University of Jordan

Descriptive Statistics Descriptive Statistics: statistical procedures used to summarize, organize, and simplify data. Shape of Distribution Central Tendency Variability Mean, Median, Mode A statistical measure that identifies a single score (usually a central value) to serve as a representative for the entire group.

Central Tendency Mean is the value obtained when the sum of the scores is divided by the number of scores: Central Tendency Mean (average) Median Mode

Central Tendency Properties related to mean: • Change the value (including adding or deleting a new observation score) in the data may change the mean; • Mean tends to be affected a lot by the extreme observation(s); • Add or subtract a constant from each score in the data …; • Multiplying or dividing each score by a constant in the data… Mean (average) Median Mode

Central Tendency Add or subtract a constant from each score in the data …; = 4.33 = ?? Amount of food (in grams) consumed before and after diet drug injections (from Table 3.2, p.81 in G&W).

Central Tendency Multiplying or dividing each score by a constant in the data… = 10 = ?? Measurement of five pieces of wood (Table 3.3, p.81 in G&W)

Central Tendency Averaging Means: Suppose a teaching assistant gives his instructor the mean final exam scores for each of her two statistics classes. Suppose further that the instructor wishes to obtain a measure of the average performance of all her students, regardless of class. The mean final examination score was 77 in class 1 and 83 in class 2. What is the mean score of all her students in the two classes? Case 1: n1 =20, n2 = 20 Case 2: n1 =10, n2 = 30

Central Tendency Unweighted mean is the average of two or more means. 1 + 2 + … + k _________________ Unweighted = k Weighted mean is the average of two or more means, calculated so that each mean is weighted by the number of scores it represents . ΣX1 + ΣX2 +…+ ΣXk n1 1 +n2 2+…+ nk k __________________ _______________________ weighted = n1+ n2+…+ nk n1+ n2+…+ nk Question: Can unweighted mean be equal to weighted mean? Under what conditions?

Central Tendency Central Tendency The score that divides a distribution exactly in half-the 50th percentile Mean (average) Median Mode Ways to identify the median: Order the raw score (from lowest to highest); If n is an odd number, the median is the middle score. If n is an even number, the median is the point halfway between the middle two scores.

Central Tendency The score that has the greatest frequency. Mean (average) Median Mode

Central Tendency Measures of central tendency for three symmetrical distributions: normal, bimodal, and rectangular

Central Tendency Measures of central tendency for skewed distributions

Central Tendency Central tendency of variables on different scales of measurement: Interval or ratio: mean, median, mode Ordinal: median, mode Nominal: mode With symmetric or nearly symmetric distributions, the mean or median is preferred to the mode The median is often used when the distribution is asymmetric with a few extreme scores Mode is the least stable measure of central tendency