Transcript Statistical Tools in Evaluation Chapter 2

Statistical Tools in Evaluation
Part II
Percentiles and the Normal Curve
Comparing Different Groups
Consider two different distributions (groups of scores)
with different Xs and SDs. The same score in both
distributions will have different meanings.
Class 1
X = 32
SD = 2
Class 2
X = 28
SD = 4
Same score of 34 in both groups is different within each
group.
Class 1: 34 is 1 SD above X; percentile rank of 84.13%
(50 + 34.13)
Class 2: 34 is 1.5 SD above the X; percentile rank is
90.925%
Distributions
Question:
• Can scores from different kinds of tests be
added to make a total score?
Example
How do you combine scores on the 1-mile run test
and sit-up test into an overall fitness score?
Standard Scores
• Conversion of raw scores into standard scores
– Allows for the combining and comparison of
information that is in different measurement units
– Examples: time + distance + repetitions, etc.
Types of Standard Scores
• Z scores – represent the number of standard
deviations a raw score is from the mean
• Z scale has a X of 0 and a SD of 1 (see normal
curve)
• Calculation:
Z=X-X
SD
Z=X–X
SD
*high to low
*low to high
Examples and Practice Problems
Class 1
X = 32
SD = 2
score = 30
Z = -1
Class 2
X = 28
SD = 4
score = 30
Z = + .5
• Z scores can be decimals and neg. values
• Z scores represent position on the normal
curve
Other Types of Standard Scores?
Types of Standard Scores
• T scores – represent Z scores that have been
converted for ease of use
• T scale has a X of 50 and a SD of 10
( No decimals or negative numbers! )
• T scores represent position on the normal curve
T score calculations:
T = 10 ( X – X ) + 50
SD
*high to low
T = 10 ( X – X ) + 50
SD
*low to high
Put another way…..
T = 10 Z + 50
(Use to convert Z score to T score)
Examples and Practice Problems
X = 80
X = 65
SD = 5
T = 10(65-80) +50
5
T = 20
X = 25
X=8
SD = 10
T = 10(8 – 25) +50
10
T = 33
T-score and the Normal Curve
Practical Use of T scores
• Add values from unrelated measurements
• Example: Fitness Test Battery
–
–
–
–
Curl ups
Pull ups
Flexibility
Mile run
• Compare to 50th percentile: T50 x 4 = 200
Example
•
•
•
•
Curl-ups
Pull-ups
Flexibility
Mile Run
• Fitness
Score = 21
Score = 13
Score = 8 cm
Score = 6:05
T = 30
T = 70
T = 42
T = 58
Score = ???
T total = 200
= Average
Questions?
Relationships
• Similarities
• Connections
• What one factor has to do with another
Determining Relationships Between
Scores
Graphing Technique
Scattergram /Scatter plot
Graph used to demonstrate the relationship between
two groups of variables or scores.
Determining Relationships Between
Scores
Scattergram
Correlations
X
• A correlation is a statistical technique used to express the
relationship between two sets of scores or variables
• Correlation coefficient (r) - a number that is used to
represent the strength of the relationship
Key points:
– r values range from - 1.00 to + 1.00
– The larger the r, the stronger the relationship.
– 1.00 is a perfect relationship (either pos. or neg.)
– A correlation coefficient near zero indicates no
relationship; the variables are not related in any way
Key points:
(+) correlation means a positive relationship; as one
variable increases, so does the other one
High
Low
Low
High
Key points:
(-)
correlation means an inverse relationship; as one
variable goes up, the other goes down
High
Low
Low
High
Key points:
(0) Low correlation means no relationship; as one
goes up the other goes up or down – not connected
High
Low
Low
High
Significance of the Correlation Coefficient:
r = .80 - 1.00: very strong relationship
r = .60 - .79: strong
r = .40 - .59: moderate
r=
< .40: weak
• Correlation coefficient meaning should be
considered within the context of the study.
• Not cause and effect – just strength of
connection.
Coefficient of determination (r2)
The amount of variability in one score that can be
accounted for (explained) by the variability in
another score.
Example 1:
leg power and vertical jump height
r = .90
r2 = .81
81% of the variability in VJH can be explained
by the variability in LP
Coefficient of determination (r2)
The amount of variability in one score that can be
accounted for (explained) by the variability in
another score.
Example 2:
shoe size and type of car you drive
r = .30
r2 = .09
9% of the variability in the type of car you
drive can be explained by the variability in
shoe size
Graphing Technique
Scattergram
Graph used to demonstrate the relationship between
two groups of variables or scores
Line of best fit (regression line) - A line representing
the trend in the data (the plotted points)
Example
r = .79
Example
r = - .68
Cause and Effect?
• Correlations only show strength of
relationship!!!
• They don’t show cause and effect!!!!
Prediction-Regression Analysis
• When two variables are correlated, it
becomes possible to predict one variable
from the measure on the other
Prediction-Regression Analysis
• Simple prediction – use one variable to
predict another
Y’ = bX + c
also the equation for a line
Y’ = [r(sy/sx)](X - X) + Y
measured
Prediction-Regression Analysis
measured
predicted
Prediction-Regression Analysis
• Multiple Prediction – Use several
variables to predict one variable.
 R – multiple correlation coefficient
 R2 – percentage of variance in one score
explained by the prediction equation.
Prediction-Regression Analysis
Example – prediction equation for the one
mile walk test:
VO2max = 132.853 – (.0769 x W) – (.3877 x A) + (6.315 x G)
– (3.2649 x T) – (.1565 x HR)
Questions?
Probability
Probability of an event occurring is the number
of possible outcomes that satisfy the conditions
of the desired event divided by the total
number of possible outcomes (page 54).
1 coin
P(heads) = 1
2
2 coins
P(both heads) = 1
4
Probability
• What is the probability that a score is untrue
when many scores are gathered?
• What is the chance of making a mistake and
saying the score is true when it is not?
– 10 out of 100?
– 5 out of 100?
– 1 out of 100?
Probability
• How much risk of being wrong do you want
to take?
– For surgery
– For money
• 10 out of 100?
• 5 out of 100?
• 1 out of 100?
Probability
Probability
• The normal curve is used to make probability
statements.
– The normal curve represents 100 possible outcomes.
– Use tabled values to find probability – “alpha”
Probability
Probability
• The normal curve is used to make probability
statements.
– Alpha level is set ahead of time
• p < .10 - probability of being wrong 10 out of 100 times
• p < .05 - probability of being wrong 5 out of 100 times
• p < .01 - probability of being wrong 1 out of 100 times
Probability
• One Tailed Test
– Extreme or untrue score could only be high or low
• Two Tailed Test
– Extreme or untrue score could be either high or
low
Questions?
Determining the Difference Between
Group Scores
Null Hypothesis and Significant Difference
• Inferential statistics provides a method to make
an educated guess as to whether two groups are
really different (due to a treatment effect)
• or if a difference is just due to measurement
error or random chance
Determining the Difference Between
Group Scores
Null Hypothesis and Significant Difference
• Two sample Xs may have different numbers,
but when compared to each other in the “big
picture” of populations (are the populations
different?), they may, for all practical matters,
be the same.
• A meaningful comparison indicates whether the
Xs are statistically different or not.
• This information is generally inferred upon an
entire population
A statistical hypothesis is a prediction about the
difference between two or more variables.
 Null hypothesis - predicts that there is no
statistical difference between two Xs. Any
difference between the Xs is likely due to chance.
H0: X1 = X2 (diff = 0)
 Alternative hypothesis - predicts that there is a
difference between the Xs.
H1: X1  X2 (diff  0) (X1 > X2)
H2: X1  X2 (diff  0) (X1 < X2)
Level of Significance
 Level of Significance - This refers to the
probability of rejecting the null hypothesis
when it is true.
 Typical levels of significance are .05 and .01,
with .05 being the more commonly used
Examples: p < .05
p < .01
• If the null hypothesis is rejected, you are saying that
the means are truly different for some reason other
than measurement error or random chance.
• p < .05 means that the probability of being wrong is less
than 5 out of 100 times.
• Said another way, the difference between Xs is so great,
that out of 100 tries, this type of difference would occur
by measurement error or random chance less than five
times, which is very unlikely and worthy of that risk.
Degrees of Freedom
 df - the number of scores in a sample that are
free to vary and still maintain the same sample
mean
one group: df = n – 1
2 groups: df = n1 + n2 – 2
Testing Group Differences
• Is the difference found actually true?
– Is the difference the result of measurement
error?
– Is the difference because of chance?
– Is the difference found because the groups are
different?
t - Test: Independent and Dependent
Groups
• Independent Groups - This type of test
compares two sample Xs from two different
estimated populations ().
• two groups: df = n1 + n2 – 2
t - Test: Independent and Dependent
Groups
t ratio = X1 - X2
s
n
Example
• Q?:Is there a significant difference between
maximal strength between a group taking
creatine and a control group?
• A: A large t (ratio) indicates that the difference
between the two Xs is larger than the estimated
normal difference between two Xs.
Example
– If the calculated t > tabled t = statistically
significant diff. - groups are different
• Reject the null hypothesis
– If the calculated t < tabled t = no
statistically sig. diff. - groups are not
different
• Accept the null hypothesis
Dependent Groups
• Two different treatments are given to the same
group (sample) which comes from the same
population.
• Example: Pre and post testing, or using the
same subjects in both treatment and control
trials of a study
• 1 group: df = n - 1
Example
– If the calculated t > tabled t = statistically
significant diff. - groups are different
• Reject the null hypothesis
– If the calculated t < tabled t = no
statistically sig. diff. - groups are not
different
• Accept the null hypothesis
Situation:
• You compare a group on weight gain before
and after a training program. Group size is 15.
• You choose a probability level (level of sig.) of
.05 and your test is one tailed.
• You calculate t = 9.7
Questions About This Situation:
1.
2.
3.
4.
5.
6.
Independent or Dependent?
What are the degrees of freedom?
What is the tabled t value?
Is there a statistically sig. difference?
What do you do with the null hypothesis?
What do you do with the alternative
hypothesis?
7. What is the probability that you have made
a mistake by rejecting the null hypothesis?
Situation:
• You compare two groups on use of electrical
stimulator vs. weight training for strength
improvement. Each group has 60 subjects
• You choose a probability level (level of sig.) of
.01 and your test is one tailed.
• You calculate t = 1.683
Questions About This Situation:
1.
2.
3.
4.
5.
6.
Independent or Dependent?
What are the degrees of freedom?
What is the tabled t value?
Is there a statistically sig. difference?
What do you do with the null hypothesis?
What do you do with the alternative
hypothesis?
7. What is the probability that you have made
a mistake by accepting the null hypothesis?
Questions?
End Part II