Measures of Central Tendency and Variability: Chapter 3
Measures of Central Tendency for Quantitative Variables
- Mode most easily computed index of central tendency
- Score that occurs most frequently
- Problem of bimodal or multimodal distributions
- Median point in distribution that divides distribution into two equal parts
- 50% of scores occur above the median, 50% occur below
- Reflects the middle value when observations are ordered from least to most, or vice versa
- Mean arithmetic average of scores
- Sum all scores and divide by the total number of scores
- Equation: 
: Where 
represents the mean of a set of scores on variable X
Use of Mode, Median, and Mean
- When quantitative variable measured on a level that approximates interval characteristics, all three measures of central tendency are meaningful
- Concepts of median and mean are meaningless for qualitative variables because both involve ordering objects along dimension
Measures of Variability for Quantitative Variables
- Range highest score minus lowest score
- Interquartile Range (IQR)
- The difference between the highest and lowest scores after the top 25% of scores and bottom 25% of scores have been trimmed
- Represents the middle 50% of scores
- Sum of Squares
- Index of variability takes into account all of the scores in the data set
- How far do scores vary from "typical" score
- Represent typical score w/mean, concerned with how each score deviates from the mean
- Cant use the sum of the deviation scores, because those will equal zero
- Sum of Squares gets its name because it is the sum of squared deviations from the mean
- formula: 
where 
is the sum of the squared X scores and 
is the square of the summed X scores
- Variance
- Variance takes into account the number of cases in each set in computing an index of variability
- 
in the examples above,
- Standard Deviation The positive square root of the variance
- Represents the average deviation from the mean
Characteristics and Use of Sum of Squares, the Variance, and the Standard Deviation
- SS, s2, and s are always greater than or equal to 0
- Can never be negative because all based on squared deviation scores
- When SS = 0, variance and standard deviation will also equal 0
- 0 represents there is no variability in scores, all scores are the same
Skewness and Kurtosis
- Two other ways distributions of scores can differ
- Skewness tendency for scores to cluster on one side of mean
- Positively skewed "tail" is toward the positive (right) end of the abscissa
- Negatively skewed "tail" is toward the negative (left) end of the abscissa
- Kurtosis refers to the flatness or peakness of one distribution relative to another
- Platykurtic less peaked than another
- Leptokurtic more peaked than another
Sample vs. Population Notation
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Index |
Sample Notation "Statistic" |
Population Notation "Parameter" |
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Mean |
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Variance |
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Standard Deviation |
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- Greek notation makes it explicit that we are describing a population.
