Download Measures of Central Tendency

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:
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
Find the Mode:
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 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
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 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
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
The Midpoint is half-way
Between the smallest
Value of X and the largest.
X MIN  X MAX 0 10

5
2
2

Find the Mean
n
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
X
 Xi
i1
n
10  0 10  8  9

 7.4
5
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.