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 10s 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 XiXiXiXi 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.