Central tendency • A statistical measure • A single number to define the center of a set of scores Purpose • Find the single number that is most typical or best represents the entire group Estimate a population • The central tendency (average) of a sample is sometimes used to estimate the entire population
Mode – most common value in the data Median – middle case (data point) in the data Mean – balance point of all the data Choose based on scale (level) of measurement and skewness
Used with interval or ratio data, except when thedistribution is skewed or indeterminate
The mean: sum of all the scores divided by the number of scores in the data Symbols: Greek μ for population, M for sample Some texts and publications use X but we will not use it in this course. Population Mean Sample Mean X X M N n
The balance point for a distribution of scores, equal weight on either side.
Compute Σf = n Compute f · X for each value in the data set Compute Value (X) f fX 10 1 10 ΣX = Σ(f · X) 9 2 18 M = ΣX / n 8 4 32 7 0 0 6 1 6 Total n = Σf = 8 ΣfX = 66 M = ΣX / n = 66/8 = 8.25
1. Determine the combined sum of all the scores ΣX1 + ΣX2 = ΣXcombined2. Determine the combined number of scores n1 + n2 = ncombined3. Compute new mean Mcombined = ΣXcombined /ncombined X1 X2 overall mean M n1 n2
Changing the value of one score always changes the mean. Introducing a new score or removing a score usually changes the mean (unless the score is exactly equal to the mean). Adding or subtracting a constant from each score changes the mean by the same constant. Multiplying or dividing each score by a constant multiplies or divides the mean by that constant.
Used with ordinal data.Used with skewed interval or ratio data
The median is the midpoint of the scores in a distribution when they are listed in order from smallest to largest. The median divides the scores into two groups of equal size. If the data have an even number of scores, median is the midpoint between two scores.
Mean is the balance point of a distribution • Defined by distance from center (“weight”) • Not necessarily the midpoint Median is the midpoint of a distribution • Defined by number of scores • Usually is not the balance point Both measure central tendency, using two different concepts of “middle”
Used with nominal dataOften reported as percentage, not category
The mode is the score or category that has the greatest frequency of any in the frequency distribution. • Can be used with any scale of measurement • Corresponds to an actual score in the data It is possible to have more than one mode
The extreme values affect the Mean more thanthey affect the Median
In positively skewed data (a) Mean is larger than Median In negatively skewed data (b) Mean is less than Median When you do not have a graph of a distribution, comparing the Mean and the Median tells you if there is skew present.
Mean, influenced by extreme scores, is pulled toward the long tail a lot (positive or negative) Median, in order to divide scores in half is less affected by the extreme scores Mode is not affected by extreme scores If Mean – Median > O, the distribution is positively skewed. If Mean – Median < O, the distribution is negatively skewed
Measure of Appropriate to choose when … Should not be used when…Central TendencyMean •No situation precludes it •Extreme scores •First choice measure of •Skewed distribution central tendency •Ordinal scale •Nominal scaleMedian •Extreme scores •Nominal scale •Skewed distribution •Ordinal scaleMode •Nominal scales •Interval or ratio data, except •Discrete variables to accompany mean or median •Describing shape of distribution
Core concept for rest of statistics.The mean will reappear all semester.