Descriptive Statistics. Measures of Central Tendency

Measures of Central Tendency Descriptive Statistics Part 1 What is a typical score like?
There are three indices of this central tendency:  Mode  Median  Mean 2 Measures of Central Tendency Measure Definition Level of Disadvantage Measurement Mode Most frequent value Median Middle value ord., int./rat. Mean Arithmetic average nom., ord., int./rat. int./rat. Crude Only two points contribute Affected by skew 3 1 The Mode Mode: most common (frequent) number Peak of distribution
Always an observed score
With grouped data is merely the midpoint of the most frequent measurement class.
Thus, if a case were drawn at random from the distribution, that case is more likely to fall in the modal class than any other.

 Important interpretation 4 How to find the mode? One Mode Two Modes 5 Problems with the Mode
However, there are three disadvantages with the mode: 1. 2. 3. Multiple modes (especially with grouped data) The mode is very sensitive to the size and number of class intervals (different intervals = different modes) The mode of a sample undependable when estimated population 6 2 The Median
Median: middle of distribution (50th percentile)
Not applicable to nominal data; ordinal, interval & ratio data only
Is considerably less sensitive when grouping into class intervals.
More useful for making inferences (although not the best)
Not affected by outliers. 7 Median Median Balancing Point 8 Median
The median requires 1 or 2 pieces of information
If there are an odd number of scores the median is the center score
If there is an even number the median is midpoint between the two middle numbers 9 3 Median
Examples   Data Set 1: {7, 2, 9, 3, 4, 5, 8}
Ordered: ________________
Median = _______ Data Set 2: {11, 15, 10, 9, 5, 13, 12}
Ordered: _____________________
Median = _______ 10 Median Data set 1: {12, 9, 8, 5, 3, 2} Data set 2: {77, 50, 8, 5, 5, 5}
Example: In a six-score set of data the only scores that matter when calculating the median are the two middle scores
The rest of the data is ignored
Because of this the median is not sensitive to outlying scores 11 Median
The average absolute deviation is smallest when taken from the median. ∑i X i − Mdn A. A.D = n 12 4 The Mean
Only data. used for interval & ratio n ∑X Mean = M X = X =
Major i i =1 n advantages: sample value is a very good estimate of the population value.  Mean of the sample = X  Mean of the population = µ  The 13 The Mean
Example  Data Set: {3, 4, 6, 10}  Formula: X= X= ∑X n __ + __ + __ + __ = ____ __ 14 The Mean and Deviations
For the mean the following is always true. ∑( X ∑( X i −X)=0 2 i − X ) = minumum* *When compared to the other 2 measures of central tendency 15 5 Mean: Demonstration
Since X = 5.75 (X Score 3 4 6 10 Σ i −X) (X (3 - 5.75) = -2.75 (4 - 5.75) = -1.75 (6 - 5.75) = 0.25 (10 - 5.75) = 4.25 0 i −X) 2 7.563 3.063 0.063 18.06 28.75 16 The Mean Note: If the mean is guessed as the value of any case drawn at random from a distribution, on average the amount of signed error will be 0.
Mental  This is a most important interpretation of the mean and why it’s so most often!
Disadvantage  Highly affected by outliers The Influence of Outliers Ordered Data 6 6 8 9 9 9 9 10 10 13 Data 8 10 9 9 6 6 9 13 10 9 Mode = Median = Mean = Ordered Data 6 6 8 9 9 9 9 10 10 33 Data 8 10 9 9 6 6 9 33 10 9 Mode = Median = Mean = 18 6 Mode, Mean and Median Positively Skewed Distribution Mode Median Mean 19 Mode, Mean and Median Negatively Skewed Distribution Mean Median Mode 20 Mode = median = mean?
When the distribution is symmetric Mean Median Mode 21 7 The mean vs. the median
The degree of discrepancy between them indicates the skewness of the data
The closer the two values are the more symmetric the data 22 8