Transcript Module 7: Statistical Reasoning

Statistical Reasoning
“He told me I was average. I told him he was mean.”
Descriptive Statistics
• Used to organize and summarize data in a
meaningful way.
Frequency distributions – Where are the majority
of the scores?
• Used to organize raw scores, or data, so that
information makes sense at a glance.
• They take scores and arrange them in order of
magnitude and the number of times each score
occurs.
Histograms & Frequency Polygons
(showing you data a glance)
• Ways of showing your frequency distribution data.
1. Histogram – graphically represents a frequency
distribution by making a bar chart using vertical bars
that touch
•
•
When you have a continuous scale (for example, scores on a test go from 0-100,
continuously getting larger.) the bars touch, because you have to have a class for each score
to fall into, and you can’t have any “gaps.”
Different than a simple Bar Graph which is used when you have non-continuous classes
(example, which candidate do you support, Obama or McCain? You’d have a bar for each,
with gaps in between, because you can’t fall between two candidates, you have to pick
one.)
Histogram
Uses a Bar Graph to show data
Frequency Polygon
Uses a line graph to show data
2. Frequency Polygon – graphically represents a frequency distribution
by marking each score category along a graph’s horizontal axis,
and connecting them with straight lines (line graph)
Measures of Central Tendency
•
A single number that gives us information about
the “center” of a frequency distribution. Measures
of central tendency – 3 types
4, 4, 3, 4, 5
1. Mode=most common=4
(Reports what there is more of – Used in data with no
connection. Can’t average men & women.)
2. Mean=arithmetic average=20/5=4
(has most statistical value but is susceptible to the effects of extreme scores )
3. Median=middle score=4
(1/2 the scores are higher, half are lower. Used when there are extreme scores)
Central Tendency
An extremely high or low price/score can skew the mean. Sometimes the
median is better at showing you the central tendency.
1968
TOPPS
Baseball
Cards
Nolan Ryan
$1500
Elston Howard
Billy Williams
Luis Aparicio
Harmon Killebrew
Orlando Cepeda
Maury Wills
Jim Bunning
Tony Conigliaro
Tony Oliva
Lou Pinella
Mickey Lolich
$8
$5
$5
$3.50
$3.50
$3
$3
$3
$3
$2.50
Jim Bouton
Rocky Colavito
Boog Powell
Luis Tiant
Tim McCarver
Tug McGraw
Joe Torre
Rusty Staub
Curt Flood
With Ryan:
Median=$2.50
Mean=$74.14
$2.25
$2
$2
$2
$2
$1.75
$1.75
$1.5
$1.25
$1
Without Ryan:
Median=$2.38
Mean=$2.85
Does the mean accurately portray the central
tendency of incomes?
NO!
What measure of central tendency would more accurately
show income distribution?
Median – the majority of the incomes surround that number.
Measures of Variability
•
•
Gives us a single number that presents us with
information about how spread out scores are in a
frequency distribution. (See example of why this
is important).
Range – Difference b/w a high & low score
–
•
Take the highest score and subtract the lowest
score from it. (can be skewed by an extreme score)
Standard Deviation – How spread out is your data?
–
–
The larger this number is, the more spread out
scores are from the mean.
The smaller this number is, the more
consistent the scores are to the mean
Calculating Standard Deviation
How spread out (consistent) is your data?
1. Calculate the mean.
2. Take each score and subtract the mean from it.
3. Square the new scores to make them positive.
4. Mean (average) the new scores
5. Take the square root of the mean to get back to your original
measurement.
6. The smaller the number the more closely packed the data. The
larger the number the more spread out it is.
Standard Deviation
Punt
Deviation
Distance from Mean
36
38
41
45
36 - 40 = -4
38 – 40 = -2
41 – 40 = +1
45 – 40 = +5
Deviation
Squared
Numbers
multiplied by itself
& added together
16
4
1
25
Standard
Deviation:
variance=
11.5 = 3.4 yds
Mean:
160/4 = 40 yds
46
Variance:
46/4 = 11.5
Multiple Choice
13 A+ 40
12 A 39
38
11 A- 37
10 B+ 36
9 B 35
34
8 B- 33
7 C+ 32
6 C 31
30
5 C- 29
4 D+ 28
3 D 27
26
2 D- 25
1 F 24
<24
Composite
Essay
4 41%
11
11
6
4 31%
5
5
4
3 19%
3
2
4
1
8%
2%
A 12 23 52%
A- 11 10
B+ 10 15
B 9 5 41%
B- 8 6
C+ 7 2
C 6 1 5%
C- 5
D+ 4 2
D 3
3%
D- 2
F 1
0%
0
13
12
11
10
9
8
7
6
5
4
3
2
1
Mean=10.2
SD=2.0
1
Mean=34.3
SD=4.2
Are these scores consistent?
Is there a skew?
A+
A 11 39%
A- 14
B+ 9
B 12 45%
B- 8
C+ 2
C+ 2 11%
C- 3
D+
D 3 5%
DF
0%
Mean=9.3
SD=2.3
Z-Scores
A number expressed in Standard Deviation Units that shows
an Individual score’s deviation from the mean.
Basically, it shows how you did compared to everyone else.
+ Z-score means you are above the mean,
– Z-score means you are below the mean.
Z-Score = your score minus the average score divided by standard deviation.
Which class did you perform better in compared to your classmates?
Test Total Your
Score
Average
score
S.D.
Biology
200
168
160
4
Psych.
100
44
38
2
Z score in Biology: 168-160 = 8, 8 / 4 = +2 Z Score
Z score in Psych: 44-38 = 6, 6/2 = +3 Z Score
You performed better in Psych compared to your classmates.
9/14/2010
Photo courtesy of Judy Davidson, DNP, RN
14
Standard Normal Distribution Curve
Characteristics of the normal curve
•
Bell shaped curve where the mean, median and mode
are all the same and fall exactly in the middle
+ or - #
-3
-2
-1
0
+1
+2
Wechsler Intelligence Scores
+3
Skewed Curves
Skewed Distribution – when more scores pile up on one side of the
distribution than the other.
Positively skewed means more people have low scores. Negatively
skewed means more people have high scores.
•Positive & Negative refers to the direction of the “tail” of the curve,
they do not mean “good” or “bad.”
•Need more explanation? Try this website.
Inferential Statistics
•
•
•
•
Help us determine if our results are legit and can be
generalized to the public
Help to determine whether a study’s outcome is more
than just chance events.
Used to predict things about a population based on a
sample.
3 Principles of Inferential Statistics:
1. Non-biased sample - Representative Samples are
better than biased samples for generalizing data
2. Less-variability is better – the average is better
when it comes from scores of low variability
3. More cases are better than fewer – averages based
on many cases are more reliable.
Statistically Significant
• Possibility that the differences in results between the
experimental and control groups could have occurred by
chance is no more than 5 percent
• Must be at least 95% certain the differences between the
groups is due to the independent variable
Statistical Significance
p value = likelihood a result is caused by chance. In other
words, are they statistically significant? If the answer is yes,
then they can be generalized to a larger population
• Researchers want this number to be as small as possible to
show that any change in their experiment was caused by an
independent variable and not some outside force.
• Results are considered statistically significant if the
probability of obtaining it by chance alone is less than .05
or a P-Score of 5%.
p ≤ .05
• Researcher must be 95% certain their results are not caused
by chance.
• Replication of the experiment will prove the p value to be
true or not.
• Effect Size – Measure of the strength of a relationship
between variables (used with SS to report quality of results)
p Value
• Describes the percent of the population/area under
the curve (in the tail) that is beyond our statistic
• This means the percentage of chance that a
confounding variable may be responsible for our
results.
Check out P
Values
made
simple for
more help.