Download Basic Statistical Concepts - Illinois State University Department of

Basic Statistical Concepts
www.phdcomics.com
So, you have collected your data …
Now what?
We use statistical analysis to


test our hypotheses
make claims about the population
This type of analyses are called
inferential statistics
But, first we must …
Organize, simplify, and describe our
body of data (distribution).
These statistical techniques are called
descriptive statistics
Distributions
Recall a variable is a characteristic that
can take different values
A distribution of a variable is a
summary of all the different values of a
variable

Both type (each value) and token (each instance)
Distribution
How excited are you about learning statistical concepts?
1
2
3
4
5
Comatose
1
2
6
7
Hyperventilating
2
3
7 Types: 1,2,3,4,5,6,7
4
4
5
6
9 Tokens: 1,2,2,3,4,4,5,6,7
7
Distribution
2
1
1
2
3
4
N=9
5
6
7
Properties of a Distribution
Shape


symmetric vs. skewed
unimodal vs. multimodal
Central Tendency


where most of the data are
mean, median, and mode
Variability (spread)


how similar the scores are
range, variance, and standard deviation
Representing a Distribution
Often it is helpful to visually represent
distributions in various ways
Graphs


continuous variables (histogram, line graph)
categorical variables (pie chart, bar chart)
Tables

frequency distribution table
Distribution
What if we collected 200 observations
instead of only 9?
…
Distribution
N = 200
50
40
30
20
10
1
2
3
4
5
6
7
Frequency
Continuous Variables
20
18
16
14
12
10
8
6
4
2
0
18
17
12
11
10
8
7
5
3
1
5054
5559
6064
6569
7074
7579
8084
Exam scores
8589
9094
95100
Categorical Variables
Cu tt in g
Doe
Missing
Sm it h
Frequency Distribution Table
VAR00003
Val id
Cu m u l at ive
Pe r cen t
7.7
1.00
Fr e q u e n cy
2
Pe r cen t
7.7
2.00
3.00
4.00
3
3
5
11.5
11.5
19.2
11.5
11.5
19.2
19.2
30.8
50.0
5.00
6.00
7.00
8.00
4
2
4
2
15.4
7.7
15.4
7.7
15.4
7.7
15.4
7.7
65.4
73.1
88.5
96.2
1
26
3.8
100.0
3.8
100.0
100.0
9.00
To t al
Val id Per ce n t
7.7
Shape of a Distribution
Symmetrical (normal)

scores are evenly distributed about the
central tendency (i.e., mean)
Shape of a Distribution
Skewed

extreme high or low scores can skew the
distribution in either direction
Negative skew
Positive skew
Shape of a Distribution
Unimodal
Multimodal
Minor Mode
Major Mode
Distribution
So, ordering our data and
understanding the shape of the
distribution organizes our data
Now, we must simplify and describe
the distribution
What value best represents our
distribution? (central tendency)
Central Tendency
Mode: the most frequent score


good for nominal scales (eye color)
a must for multimodal distributions
Median: the middle score


separates the bottom 50% and the top
50% of the distribution
good for skewed distributions (net worth)
Central Tendency
Mean: the arithmetic average


add all of the scores and divide by total
number of scores
This the preferred measure of central
tendency (takes all of the scores into account)
X

N
population
X
X
n
sample
Computing a Mean
10 scores: 8, 4, 5, 2, 9, 13, 3, 7, 8, 5
ξΧ = 64
ξΧ/n = 6.4
Central Tendency
Is the mean always the best measure
of central tendency?
No, skew pulls the mean in the
direction of the skew
Central Tendency and Skew
Mode
Median
Mean
Central Tendency and Skew
Mode
Median
Mean
Distribution
So, central tendency simplifies and
describes our distribution by providing a
representative score
What about the difference between the
individual scores and the mean?
(variability)
Variability
Range: maximum value – minimum value


only takes two scores from the distribution into
account
easily influenced by extreme high or low scores
Standard Deviation/Variance



the average deviation of scores from the mean of
the distribution
takes all scores into account
less influenced by extreme values
Standard Deviation
most popular and important measure
of variability
a measure of how far all of the
individual scores in the distribution are
from a standard (mean)

Standard Deviation
mean
mean
low variability
high variability
small SD
large SD
Computing a Standard Deviation
10 scores: 8, 4, 5, 2, 9, 13, 3, 7, 8, 5
8 – 6.4 =
1.6
2.56
4 – 6.4 =
- 2.4
5.76
5 – 6.4 =
- 1.4
1.96
2 – 6.4 =
- 4.4
19.36
9 – 6.4 =
2.6
6.76
13 – 6.4 =
6.6
43.56
3 – 6.4 =
- 3.4
11.56
7 – 6.4 =
0.6
0.36
8 – 6.4 =
1.6
2.56
5 – 6.4 =
- 1.4
1.96
ξΧ/n = 6.4
SS = 96.4
variance = 2 = SS/N
10.71
X  
2
standard deviation =  =  
2
3.27

N
Standard Deviation
In a perfectly symmetrical (i.e. normal)
distribution 2/3 of the scores will fall
within +/- 1 standard deviation
-1
3.13
+1
6.4
9.67
Variance vs. SD
So, SD simplifies and describes the
distribution by providing a measure of
the variability of scores
If we only ever report SD, then why
would variance be considered a
separate measure of variability?
Variance will be an important value in
many calculations in inferential statistics
Review
Descriptive statistics organize, simplify, and describe
the important aspects of a distribution
This is the first step toward testing hypotheses with
inferential statistics
Distributions can be described in terms of shape,
central tendency, and variability
There are small differences in computation for
populations vs. samples
It is often useful to graphically represent a
distribution