Measures of Central Tendency Psych 101 with Professor

Four “Averages” § Mode: The Mode of Variable X is the value of X

“Averages” claims to fame: § The Mode is most often correct § The Median

Example: Suppose we have n = 5 numbers, as follows: X 1 = 10

Find the Midpoint X 1 = 10 X 2 = 0 X 3 =

Summary of Example Case, i Xi 1 10 2 0 3 10 4 8

Next Idea: Variance § If we must guess, and if we are penalized by

Slides: 10

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Measures of Central Tendency Psych 101 with Professor Michael Birnbaum

Four “Averages” § Mode: The Mode of Variable X is the value of X that is most frequent. § Median: The Median of X is the value of X such that an equal number are above and below X. § Midpoint: Value of X that is half-way between the smallest and largest value. § Mean: The Arithmetic Average (Sum of X, divided by the number).

“Averages” claims to fame: § The Mode is most often correct § The Median minimizes the sum of absolute deviations. § The Mean minimizes the sum of squared deviations (sum of squared errors). § The Midpoint minimizes the biggest deviation.

Example: Suppose we have n = 5 numbers, as follows: X 1 = 10 X 2 = 0 X 3 = 10 X 4 = 8 X 5 = 9

Find the Mode: X 1 = 10 X 2 = 0 X 3 = 10 X 4 = 8 X 5 = 9 The Mode is the value of X that is most frequent. There are more 10 s than any other value. So, the Mode is 10.

Find the Median X 1 = 10 X 2 = 0 X 3 = 10 X 4 = 8 X 5 = 9 The Median has an equal number above and below it. Here, 9 has 2 numbers above and 2 below, so the Median is 9.

Find the Midpoint X 1 = 10 X 2 = 0 X 3 = 10 X 4 = 8 X 5 = 9 The Midpoint is half-way Between the smallest Value of X and the largest.

Find the Mean X 1 = 10 X 2 = 0 X 3 = 10 X 4 = 8 X 5 = 9

Summary of Example Case, i Xi 1 10 2 0 3 10 4 8 5 9 Mode= 10 Median = 9 Midpoint = 5 Mean = 7. 4 All are “averages”

Next Idea: Variance § If we must guess, and if we are penalized by the sum of squared errors, the Mean is our best guess. § If we always guess the Mean, the cost will be minimal. § The average cost per guess is the Variance, or Mean Squared Error. § Square root of that Mean Square is the standard deviation.