Measures of
Central
Tendency&
Variability
Di Fara
Gino’s
Di Fara
1
2
4
3
4
4
3
4
1
3
3
2
5
3
2
Σ 44/15
= 2.93
Gino’s
2
3
2
3
2
2
4
4
4
3
2
4
3
2
4
Σ 44/15
= 2.93
Measures of Central Tendency and Variability
So far, we have used very basic characterizations of distributions
•
•
Number of modes (unimodal, bimodal, multimodal)
Skew (positive or negative) & Symmetry
We need a way to characterize these same distributions quantitatively
(using numbers). This allows us to compare distributions.
We can describe distributions using two categories of measures:
Measures of Central Tendency
• mean, median, mode
Measures of Variability
• range, standard deviation, variance
Measures of Central Tendency
(where all the action is)
Mean- The average of all the scores. The sum of all the scores
divided by the number of scores.
Example: x : {1, 3, 4, 8 }
Σ x = (1 + 3 + 4 + 8) = 16 = 4
N
4
4
The mean is denoted differently depending on the
type of data from which it comes:
Population mean = μ (pronounced “myou”)
__
Sample mean = x (spoken as “x-bar”)
m
The median is the “middle score”
Median – The score that falls in the exact middle of the
distribution. (Half the scores are lower and half higher than the
median.
x = {5, 6, 2, 3, 1, 9, 8, 0, 2, 4, 5}
First, arrange the numbers in ascending order:
x = {0, 1, 2, 2, 3, 4, 5, 5, 6, 8, 9}
Find the number
that falls in the
middle.
For an even number of scores, average the two middle numbers.
x = {0, 1, 1, 2, 2, 3, 4, 5, 5, 6, 8, 9}
Mode – The score that occurs most frequently. The score with
the highest FREQUENCY.
Example: 1, 3, 1, 5, 2, 1, 1, 8, 2, 3, 1, 1, 1, 0, 1, 3, 2, 1, 1, 1
Mode – The score that occurs most frequently. The score with
the highest FREQUENCY.
Example: 1, 3, 1, 5, 2, 1, 1, 8, 2, 3, 1, 1, 1, 0, 1, 3, 2, 1, 1, 1
Relations between measures of central tendency
describe score distribution shape : Skewness
When the mean, median, and
mode agree, you have symmetry.
Pos Skew: Mean > Median
Pos Skew: Mean < Median
Review of Summation:
x: {1, 0, 3}
y: {2, 5, 1}
Sx = 1 + 0 + 3 = 4
Sx2 = 1 + 0 + 9 = 10
x
x
1
01
30
3
y
2
5
1
(Sx)2 = (1 + 0 + 3)2 = 42 = 16
S 3x = 3(1) + (3)0 + (3)3 = 3 + 0 + 9 = 12
S xy = 1(2) + (0)5 + (3)1 = 2 + 0 + 3 = 5
(Sx)(Sy) = (1+0+3)(2+5+1) = (4)(8) = 32
Measures of variability:
(how clustered or spread out the distribution is)
Relative
Frequency
The Normal Distribution
0.015
0.01
0.005
0
0
32
64
96
128
160
192
224
X
Range - The maximum difference in the data (Max-Min score)
Standard Deviation -The average amount that the scores
deviate from the mean.
Variance - Similar to the standard deviation but with special
properties.
# Canoli Eaten
4
5
6
6
7
8
8
9
10
10
10
10
11
11
11
12
12
14
14
14
16
16
21
The Range
Canolli Eating Contest
6
# C ontestants
Contestant
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
5
4
3
2
1
0
0
2
4
6
8
10
12
14
16
18
20
22
# Canolli Eaten
Minimum = 4
Maximum = 21
Range = Maximum - Minimum
= 21 - 4
= 17
Standard Deviation: example
Canolli Eating Contest
# C ontestants
6
5
How much does
each score in the
sample differ
from the average
score?
4
3
2
1
0
0
2
4
6
8
10
12
14
16
18
20
22
# Canolli Eaten
The amount by which each score differs from the mean
is called its deviation.
Standard Deviation (population)
Raw vs. Deviation Scores
How do you suppose we would go about finding the AVERAGE
amount by which each score DEVIATES from the mean?
x: { 1, 2, 3, 2}
x
1
2
3
2
μ
2
2
2
2
x–μ
-1
0
1
0
(x – μ)2
1
0
1
0
Σ(x-μ)2 ]- Sum of squares (SS)
√
Σ(x-μ)2
_______
=
N
√
SS =
N
√
2
4
= .7071
= .71
“deviation method”
s = .71
Standard Deviation (sample)
x: { 1, 2, 3, 2}
_
x
x
2
1
2
2
2
3
2
2
_
x–x
-1
0
1
0
_
(x – x)2
1
0
1
0
_
Σ(x-x)2 ]- Sum of squares (SS)
√
_
_______
Σ(x-x)2 =
N-1
√
SS =
N-1
√
2
3
= .8165
= .82
“deviation method”
s = .82
The “raw scores method” is an easier way to calculate
the Sum of Squares (SS)
Remember,
s=
√
SS
N
s=
√
SS
N-1
“raw scores method”
SS = Sx2
__
(Sx)2
N
Finding the standard deviation using the “raw scores
method” for finding the Sum of Squares (SS)
x2
x
_________
1
1
2
4
3
9
2
4
Sx = 8
Sx2 = 18
SS =
__
2
Sx
SS = 18
SS = 2
__
(Sx)2
N
(8)2
4
Finding the standard deviation using the “raw scores
Remember:
method” for finding the Sum of Squares (SS)
POPULATION:
x
x2
_________
SS
1
1
s4 =
2
N
3
9
2
4
Sx
=
8

Sx2 = 18

s=
SS =
SAMPLE:
2
(Sx)
__
2
Sx
SS N
s=
N 1
SS = 18
__
4
SS = 2
√
s = .71
2
4
(8)2
s=
√
s = .82
2
3
Summary Slide for Standard Deviation
2
(
)
x

m

POPULATION:
SS
s=
N
SAMPLE:
SS
s=
N 1
 x)
(
x  N
2
2
(x  x)
2


 x)
(
x  N
2
2
A family of statistics to describe populations and samples
Central
Tendency
Variability
Population
Sample
- mean
m = (Sx)/N
- mean
_
x = (Sx)/N
- median
- mode
- same
- same
- range
- Std Dev.
s = √(SS/N)
- Variance
s2
- same
- Std Dev.
s = √(SS/N-1)
- Variance
s2
Revisiting Pizza…
s = 1.16
s = .88
The Normal Distribution and Z-scores
Are all unimodal, symmetrical distributions
normal? NO.
Kurtosis
The Normal Distribution and Z-scores
What did you get on your SATs?
• Prior to 2005, the highest possible score was 1600
• In 2005, an additional section was added to the SAT,
making the highest possible score a 2400
If my score (I took the SATs in 2002) was a 1400,
and my friend’s score (2006) was an 1800, did my
friend do better than I did or not?
We need to find a way to compare scores from different
distributions. We cannot compare the raw scores directly.
If we know that the particular variable on which our
score was measured is NORMALLY distributed:
• we can specify HOW MANY standard deviations our score is
above or below the mean.
• For example: We read on Princeton Review’s website that SAT scores
are normally distributed. Using the old scale of measurement, the
population mean SAT score was 1000, with a standard deviation of 150
points.
m = 1000
s = 150
How many standard
deviations away
from the
mean is a score of
1300?
.
.
.
.
.
600 800 1000 1200 1400
What about a score of 1325? How many standard deviations is it
from the mean?
1325 – 1000
150
= 325
150
1325
= 2.166
= 2.17
600 800 1000 1200 1400
The Z-score
z =
z =
x-m
s
x-x
s
for population
Measures how extreme or unusual
a score is within a population
for sample
*in units of standard deviation.
(this means it tells us exactly HOW
MANY standard deviations a score is
from the mean).
The Z-score
Example: MY SAT score (1400)
MY friend’s SAT score (1800)
Population of SAT scores (old grading
system):
Population of SAT scores (new
grading system):
m
s
1000 pts
150 pts
m
s
1500
200
z=
1400 – 1000
150
z=
1800 – 1500
200
=
400
= 150
z = 2.6666
= 2.67 standard deviations
above the mean
pts
pts
300
200
z = 1.5000
= 1.50 standard deviations
above the mean