Basic Statistics
KNES 510
Research Methods in Kinesiology
1
How Software is Used
in Statistics

Types of software for statistics
Minitab
 Statistical Analysis system (SAS)
 Statistical Package for the Social Sciences (SPSS)


Predictive Analytics Software (PASW)
Why not Microsoft Excel?
 http://www.youtube.com/

2
Why We Need Statistics


Statistics is an objective way of interpreting a
collection of observations
Types of statistics
1.
Descriptive
Central tendency
 Variability

2.
3.
Correlational
Inferential

Differences within or between groups
3
Ways to Select a Sample
1.
2.
Random sampling: tables of random numbers
Stratified random sampling

3.
Systematic sampling

4.

Pick a start and sample every nth number.
Random assignment
Justifying post hoc explanations


Strata=small groups. Sample from each strata
Convenience sample?
How good does the sample have to be?

Good enough for our purposes!
4
Descriptive Statistics


Descriptive statistics are used to summarize or
condense a group of scores
They include measures of central tendency
and measures of variability
Humans
Mean=100
SD=15
5
Central Tendency


Measures of central tendency describe the
average or common score of a group of scores
Common measures of central tendency include
the mean, median, and mode
6
Mean



The mean is the arithmetic average of the scores
The calculation of the mean considers both the
number of scores and their value
The formula for the mean of the variable X is:
X
M
n
7
Mean



Six men with high serum cholesterol participated
in a study to examine the effects of diet on
cholesterol
At the beginning of the study, their serum
cholesterol levels (mg/dL) were:
366, 327, 274, 292, 274, 230
Determine the mean
8
Mean
366 327 274 292 274 230
M
6
1,763
M
6
M  293.83
9
Calculating the Mean Using SPSS

Analyze -> Descriptive Statistics -> Frequencies
command may be used to determine the mean (you will
need to select the “Statistics…” button to choose the
“Mean”
Statistics
Cholesterol
N
Valid
Missing
Mean
6
0
293.83
10
Median


The median is the middle point in an ordered
distribution at which an equal number of scores
lie on each side of it
It is also known as the 50th percentile (P50), or
2nd quartile (Q2)
11
Median

The position of the median (Mdn) can be
calculated as follows:
n 1
Mdn 
2
12
Median

Example: Calculate the median for the following
measurements for height:
71”, 73”, 74”, 75”, 72”
13
Median


Step One: Place the scores in order from lowest
to highest.
71”, 72”, 73”, 74”, 75”
Step Two: Calculate the position of the median
using the following formula:
n 1
Mdn 
2
5 1
Mdn 
 3rd score
2
14
Median

Step Three: Determine the value of the median
by counting from either the highest or the
lowest score until the desired score is reached (in
this case the 3rd score)
15
Median


Suppose that in our previous distribution we had
a sixth score as follows:
71”, 72”, 73”, 74”, 74”, 75”
What are the position and value of the median?
16
Median



Consider the following example: Nine people
each perform 40 sit-ups, and one does 1,000
The median score for the group is 40, and the
mean (arithmetic average) is 136
The median would still be 40 even if the highest
score were 2,000 instead of 40
17
The Median is Unaffected by
Extreme Scores
Statistics
Sit-Ups
N
Valid
Missing
Mean
Median
10
0
136.00
40.00
Statistics
Sit-Ups
N
Mean
Median
Valid
Missing
10
0
236.00
40.00
18
Mode




The mode is the most frequently occurring score
Which of the following scores is the mode?
3, 7, 3, 9, 9, 3, 5, 1, 8, 5
Similarly, for another data set (2, 4, 9, 6, 4, 6, 6,
2, 8, 2), there are two modes; What are they?
What is the mode for 7, 7, 6, 6, 5, 5, 4 and 4
19
Mode


A distribution with a single mode is said to be
unimodal
A distribution with more than one mode is said
to be bimodal, trimodal, etc., or in general,
multimodal
20
Calculating the Mode Using SPSS


Analyze -> Descriptive Statistics -> Frequencies
command may be used to calculate the mode
(you will need to select the “Statistics…” button
to choose the mode, etc
Note differences in the SPSS output when the
distribution is unimodal, multimodal, or when
there is no mode
21
SPSS Output - Unimodal
Statistics
Scores
N
Valid
Missing
Mode
10
0
3
22
SPSS Output - Bimodal
Statistics
Scores
N
Valid
Missing
Mode
10
0
2a
a. Multiple modes exist. The smallest value is shown
23
SPSS Output – No Mode
Statistics
Scores
N
Valid
Missing
Mode
8
0
4a
a. Multiple modes exist. The smallest value is shown
24
Variability


Measures of variability describe the extent of
similarity or difference in a set of scores
These measures include the range, standard
deviation, and variance
25
Standard Deviation (SD)


Standard Deviation – a measure of the
variability, or spread, of a set of scores around
the mean
Intuitively, the sum of the differences between
each score and the mean (known as deviation
scores) appears to be a good approach for
measuring variability around the mean
26
SD

Symbolically, we can write this as
 X  M 

Let’s use the scores 1, 2, 6, 6, and 15, where
M 6
27
SD

Now let’s calculate the sum of the deviation scores:
 X  M 
= (1-6) + (2-6) + (6-6) + (6-6) + (15-6)
= (-5) + (-4) + (0) + (0) + (9)
= = -9 + 9 = 0
28
SD


We can avoid this problem (deviation scores
sum to 0) by squaring each deviation score
before summing them
This would be written symbolically as
 X  M 
2
29
SD

Substituting our X scores again,
= (1-6)2 + (2-6)2 + (6-6)2 + (6-6)2 + (15-6)2
= (-5)2 + (-4)2 + (0)2 + (0)2 + (9)2
= 25 + 16 + 0 + 0 + 81
= 122
30
SD


We then divide this value by n-1 to arrive at the
mean squared deviation
122/4 = 30.5
We then take the square root of this value to
bring the units back to the raw score units
30.5  5.52
31
Example calculation of variance and standard deviation on strength scores.
Subj
Score (x)
Deviation
(x)2
1
216
22.7
515.29
2
144
-49.3
2430.49
X 1740

X=

 193.3
n
9
2

(
x

X
)
11774.01
2
s 

 147175
.
n 1
8
3
183
-10.3
106.09
4
138
-55.3
3058.09
5
212
18.7
349.69
6
180
-13.3
176.89
7
200
6.7
44.89
8
264
70.7
4998.49
9
203
9.7
94.09
=1740
=0
=11774.01
2
(
x

X
)

s
 38.4
n 1
Calculating the SD Using SPSS

Analyze -> Descriptive Statistics -> Frequencies
command may be used to determine the standard
deviation (you will need to select the “Statistics…”
button to choose the “Std. deviation”
Statistics
Scores
N
Std. Deviation
Valid
Missing
5
0
5.523
33
Variance


The variance is the square of the standard
deviation
It is used most commonly with more advanced
statistical procedures such as regression analysis,
analysis of variance (ANOVA), and the
determination of the reliability of a test
34
Variance

The variance is also known as the mean square
(MS)
 X  M 
s 
n 1
2
2
35
Range


The range is equal to the high score minus the
low score in a distribution
It is considered an unstable measure of
variability, and can change drastically if extreme
scores are introduced to the distribution
36
Range


As a result of gas analysis in a respirometer, an
investigator obtains the following four readings
of oxygen percentages:
14.9, 10.8, 12.3, and 23.3
What is the range?
37
Calculating the Range Using SPSS

Analyze -> Descriptive Statistics -> Frequencies
command may be used to calculate the range (you will
need to select the “Statistics…” button to choose
“Minimum,” “Maximum,” and “Range”
Statistics
Oxygen_Content
N
Valid
Missing
Range
Minimum
Maximum
4
0
12.5
10.8
23.3
38
Example of Descriptive Statistics
39
Confidence Intervals


Provide an expected upper and lower limit for a
statistic at a specified probability level (usually
95% or 99%)
CI is dependent upon the sample size,
homogeneity of values within the sample and
the level of confidence selected by the
researcher
40
Confidence Interval, cont’d
For example, a sample mean is an estimate of
the population mean
 A confidence interval provides a band within
which the population mean is likely to fall
CI = mean ± (standard error × confidence level)
 The standard error (sM) is the variability of the
sampling distribution of the statistic

sM  s / n
41
Calculating a CI



Example: n = 30, M = 40, s = 8
CI = 40 ± (1.46 × 2.045)
CI = 40 ± 2.99 = 37.01 to 42.99
The value “1.46” came from the following
formula:
sM  8 / 30
The value “2.045” came from table A.5 (next
slide)
42
43
Correlation

Correlation “indicates the extent to which two
variables are related or associated

The extent to which the direction and size of
deviations from the mean in one variable are
related to the direction and size of deviations
from the mean in another variable”
Z

r
X
ZY 
N
44
Example of Correlations
45
Categories of Statistical Tests

Parametric
Normal distribution
 Equal variances
 Independent observations


Nonparametric (distribution free)


Distribution is not normal
Normal curve
Skewness
 Kurtosis

46
Normal Curve
47
Skewness
48
Kurtosis
49
Statistics

What statistical techniques tell us
Reliability (significance) of effect
 Strength of the relationship (meaningfulness)


Types of statistical techniques
Relationships among variability
 Differences among groups


Cause and effect

Correlation is no proof of causation
50
Next Class


Chapter 7 and 8
Full Lit Review
51