PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 3
Central Tendency & Variability
PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Q1. A researcher examined the effect of amount of relaxation
training on insomnia. Four treatment groups were used.
Subjects received relaxation training for 2, 4, or 8
sessions. A control group received no training (0
sessions). Following training, the researcher measured
how long it took the subjects to fall asleep. The average
time for each group is presented in the following table:
PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Q1.
a.
Identify the IV and DV for this
study.
b. What is scale of the measurement
was used for the IV and the DV?
c. If the researcher used a graph to show
the obtained relationship between the
IV and the DV, what kind of graph would
be appropriate? Sketch the graph
showing the results of this experiment?
Training
sessions
Average time
(in minutes)
0
72
2
58
4
33
8
14
PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Q2. For the following set of the scores
4, 6, 9, 5, 3, 8, 9, 4, 2, 5, 10, 7, 4 ,9, 8, 3
a. Construct a frequency distribution table
b. Sketch a polygon showing the distribution
c. Describe the shape of the distribution
d. What is the percentile rank for X=6?
e. What is the 70th percentile?
Central tendency
is a statistical measure that identifies a
single score as representative of an entire
distribution.
MEAN
MEDIAN
MODE
The Mean
The mean for a distribution is the sum of the scores
divided by the number scores.
Quiz score
(x)
f
10
9
8
1
2
4
7
6
0
1
Charactistics of the Mean
1. Changing a score or introducing a new
score.
2. Adding or subtracting a constant from
each score.
3. Multiplying or dividing each score by a
constant.
The Median
is the score that divides a distribution
exactly in half.
Three types of data?
1.
When N is odd number
2.
When N is even number
3.
When there are several scores with the same
value in the middle of the distribution.
The Median
1.
2.
3.
EX:
3, 5, 8, 10, 11
3, 3, 4, 5, 7, 8
1, 2, 2, 3, 4, 4, 4, 4, 4, 5
EX: Find the median for this data
3, 4, 3, 2, 1, 3, 2, 4
The Mode
In a frequency distribution,
the mode is the score or
category that has greatest
frequency.
Quiz score
(x)
f
5
2
4
6
3
4
2
2
1
1
How do you decide which measure of
central tendency to use?
When
to
use
mode?
It can be used with any scale
of measurement.
Academic major
f
Biology
2
Psyhics
6
Sociology
2
Mathematic
1
How do you decide which measure of
central tendency to use?
When to use median?
1.
2.
3.
4.
There are a few extreme scores in the distribution
Some scores have undetermined values
There are open ended distribution
The data measured on an ordinal scale
How do you decide which measure of
central tendency to use?
When to use median?
1.
There are a few extreme
scores
in
the
distribution
Errors committed
before reaching
learning criterion (x)
f
10
1
11
4
12
3
13
1
100
1
How do you decide which measure of
central tendency to use?
When to use median?
1.
Some
scores
have
undetermined values
Person (x)
Time (amount of time
to complete puzzle)
1
8
2
11
3
12
4
13
5
17
6
Never finished
How do you decide which measure of
central tendency to use?
When to use median?
1.
There are open ended
distribution
Number of
children (x)
f
5 or
more
3
4
2
3
2
2
3
1
6
0
4
Central tendency and the shape of the
distribution
1. Symmetrical distribution
2. Skewed distributions
Central tendency
EX:
Find the mode, median
and mean
X
f
5
10
4
6
3
2
2
1
1
1
Variability
provides a quantitative measure of the
degree to which scores in a distribution are
spread out or clustered together.
A. RANGE – B. SEMI-INTERQUARTILE
RANGE C. STANDARD DEVIATION
PURPOSE: Are the scores all clustered
together, or are they scattered over a wide
range of values?
Range
The range is the distance between the
largest score and the smallest score in the
distribution.
range = URL Xmax – LRL Xmin
Ex. 3, 7, 12, 8, 5, 10
The Interquartile range and semiinterquartile range
The interquartile range is the distance
between the first quartile and the third
quartile.
interquartile range = Q3 – Q1
semi-interquartile range= ½ * (Q3 – Q1)
Ex. 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10, 11
Standard deviation and variance for a
population (a measure of distance from the
mean)
Step 1. The first step in finding the
standard distance from the mean is to
determine the deviation for each individual
score.
Deviation is the distance from the mean
Deviation score = X - µ
Standard deviation and variance for a
population
Step 2. The next step is to calculate the
mean of the deviation scores.
Step 3. Use mean
(Variance)
Population Variance
deviation = SS/N
(SS = Σ (X - µ)2 )
squared
=
Mean
deviation
squared
Standard deviation and variance for a
population
Step 4. Simply make a correction for
having squared all the distances. EX:
standard deviation = √variance
σ = √SS/N
σ2 = SS/N
x
1
0
6
1
Standard deviation and variance for a
population
Use the following population of
scores to calculate SS, variance,
and standard deviation
EX:
Scores: 1, 9, 5, 8, 7
Standard deviation and variance for a
sample
Notations: Notice that these sample
formula use n-1 instead of n.
Population
Sample
Mean
µ
X
variance
σ2 = SS/N
s2=SS/n-1
Standard deviation
σ = √SS/N
s = √SS/n-1
Standard deviation and variance for a
sample
Why do we use n-1 for the
sample?
Sample variability tends to underestimate
variability unless some correction is made.
population
Dividing by a smaller value produces a larger result and
makes sample variability an accurate, or unbiased, estimator
of population variability.
Standard deviation and variance for a
sample
Degrees of freedom: n-1
df = n-1
s2=SS/df
s = √SS/df
Properties of SD
1. descriptive measure: distance from the mean
2. a measure of how big the error will be.
Properties of SD
1. Adding a constant to each score will not change the SD.
2. Multiplying each score by a constant causes the SD to be
multiplied by the same constant.
SD or Range? When?
1. Extreme scores.
2. Sample size.
3. Stability under sampling
4. Open-ended distributions
SD or Range? When?
Range
Semiinterquartile
range
Variance
SD
Extreme scores
most
least
Be careful
Be careful
Sample size
Directly
related
Relatively
unaffected
Relatively
unaffected
Relatively
unaffected
Stability under
sampling
unstable
Reasonably
stable
stable
stable
Open-ended
distributions
N/A
Only available
one
N/A
N/A