Central Tendency. Ch 3. Essentials of Statistics for the Behavior Science Ch.3

Transcription

1 Central Tendency Ch 3

2 Ch. 3 Central Tendency 3.1 Introduction 3.2 Mean 3.3 Median 3.4 Mode 3.5 Selecting a Measure of Central Tendency 3.6 Central Tendency & Shape of the Distribution Summary

3 3.1 Introduction Central Tendency To describe a group of individuals with a single measurement. A statistical measure that identifies a single score as representative for an entire distribution. To find out the average or typical individual Average value provides a simple description of the entire population/sample Useful for making comparisons between groups of individuals Find center: Figure 3.1 (p.54) Three methods for measuring central tendency Mean, Median, Mode

4 3.2 Mean Mean: the arithmetic average By adding all the scores in the distribution and dividing by the number of scores Mean for population (Greek letter mu) : μ = X/N Mean for sample (M): Balance point for a distribution Data: 2, 2, 6, 10 M X n

5 Weighted Mean Weighted Mean Combine two sets of scores and then find the overall mean for the combined group. Eg.) Calculate weighted mean for Test score 6.0 for 12 students in Group A and Test score 7.0 for 8 students in Group B. Weighted mean of group A & B:? Find two values: Total number of individuals for groups (n), Overall sum of scores for the combined group ( X) Weighted mean is not obtained by simply averaging the means Different sample size: larger sample will carry more weight in the combined group 6.5 vs. 6.4 (closer to 6)

6 Characteristics of the Mean Changing a score will change the mean Every single score in the distribution contributes to the value of mean Adding or subtracting a constant from each score If a constant value is added to every score in a distribution, the same constant will be added to the mean, vice versa. Multiplying or dividing each score by a constant If every score in a distribution is multiplied (or divided) by a constant value, the mean will be changed in the same way

7 3.3 Median Median: the score that divides a distribution exactly in half (no symbol or notation). Exactly 50% of the individuals in a distribution have scores at or below the median. To determine the precise midpoint of a distribution Different types of data set 1. When N is an odd number: middle score in the list 3, 5, 8, 10, 11 (Figure 3.4, p.63) 2. When N is an even number: halfway between the middle two scores 3, 3, 4, 5, 7, 8 (Figure 3.5, p.64)

8 3.4 Mode Mode: Popular style? The most common observation among a group of scores. The score or category that has the greatest frequency No symbols, no notation Useful used to determine the typical or average value for any scale of measurement, including nominal scale when the data should not be calculated a mean or median One mean, One median, but multiple Modes are possible (bimodal, multimodal) Casually refer to scores with relatively high frequencies Major mode & minor mode (Figure 3.7 p.66 )

9 3.5 Selecting a Measure of Central Tendency Again, the goal of central tendency is to find the single value that best represents the entire population. Mean: Most often the preferred measure of central tendency Closely related to variance and standard deviation Common measure of variability (Ch.4 will discuss this in great detail)

10 When to use the Median Alternative to the mean 1. There are a few extreme scores in the distribution 2. Some scores have undetermined values 3. An open-ended distribution 4. Data are measured on an ordinal scale When to use the Mode Easy to compute & used with any scale of measurement Supplementary measure along with the mean or median Gives an indication of the shape of the distribution Mean of 72, Median of 75 and Mode of 80

11 Median 1. Extremely scores or skewed distribution One or two extreme values can have a large influence and cause the mean to be displaced Eg.: Salary distribution, Number of errors (Figure 3.8, p.68): Mean 20.3 vs. Median Undetermined values When an individual has an unknown or undetermined score (Table 3.5, p.69): Impossible to compute the Mean! 3. Open-ended distribution When there is no upper or lower limit for one of the categories It is impossible to compute a mean for this case E.g.) Number of children cases (p.69) 4. Ordinal scale Not appropriate to use the mean to describe central tendency for ordinal data

12 3.6 Central Tendency & Shape of Distribution The relationship among the mean, median, and mode are determined by the shape of the distribution. Symmetrical distribution Overall average will be exactly at the middle The mean and median will be the same (Figure 3.11 (a), p.73) Skewed distribution Positively skewed distribution (Figure 3.12 (a), p.74) Position: Mode, Median, Mean (from the left) Value size: Mean > Median > Mode

13 Assignment Ch.3 Problems (p.78) Question # 2, 4, 7, 9, 23

14 Summary 中庸 in English? Q & A