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Learn about measures of central tendency (mode, median, mean) and their importance in summarizing distributions of variables. Explore how to compute these statistics and their relevance in data analysis.
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Variables have distributions • A variable is something that changes or has different values (e.g., anger). • A distribution is a collection of measures, usually across people. • Distributions of numbers can be summarized with numbers (called statistics or parameters).
Middle of the Distribution (Common Statistics) • Mode • Most common score • Median • Top from bottom 50 percent • Mean • Arithmetic mean or average
p Mode • The most frequently occurring score. Can have bimodal and multimodal distributions. Modal psychology student is female. Modal number of pubs from grad school is zero.
Median • Score that separates top 50% from bottom 50% • Even number of scores, median is half way between two middle scores. • 1 2 3 4 | 5 6 7 8 – Median is 4.5 • Odd number of scores, median is the middle number • 1 2 3 4 5 6 7 – Median is 4
Mean • Sum of scores divided by the number of people. Population mean is (mu) and sample mean is (X-bar). • We calculate the sample mean by: • We calculate the population mean by: Raw score is X. N is number of people. Sigma (Greek symbol like big E) is summation sign. Add up scores and divide by the number of people.
Deviations from the Mean • Deviation defined. • Deviations sum to zero. • Raw scores: • Deviation scores:
Comparison of stats (1) • Mode • Good for nominal variables • Good if you need to know most frequent observation • Quick and easy
Comparison of stats (2) Median • Good for “bad” (skewed) distributions • Good for distributions with arbitrary ceiling or floor • Often used with distributions of money
Comparison of stats (3) • Mean • Used for inference as well as description; best estimator of the parameter • Based on all data in the distribution • Generally preferred except for “bad” distribution. • Most commonly used statistic for central tendency.
Review • What is central tendency? • Mode • Median • Mean
Computation • Consider the following scores: 1, 2, 2, 3, 3, 3, 4, 5 • For the above set of scores, what is N? • Cannot be determined • 2 • 3 • 8
Computation • Consider the following scores: 1, 2, 2, 3, 3, 3, 4, 5 • For the above set of scores, what is the percentage (relative frequency) of 2s? • 2 • 10 • 20 • 25
Computation • Consider the following scores: 1, 2, 2, 3, 3, 3, 4, 5 • For the above set of scores, what is the mode? • 2 • 3 • 4 • 5
Discussion Questions • Name a variable where it would be better to find the median than the mean. • Why is it misleading to say that the average person has 1.2 brothers? Why might it be useful or helpful to say it anyway?