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Pop mean =
μ
, sample mean = x
=
∑
x
i
/
n
Pop variance =
σ
2
, sample s
2
= ss/df
df – number of independent pieces of
info in a calculation (n1)
Var(x) =
∑
(x
i
x
)
2
/(n1), variance =
σ
2
/n
St. dev. =
σ
/
√
n
tstatistic = estimated – hypothesized
parameter value/SE(estimate)
SE(estimate) = s
p
√
(1/n
1
+1/n
2
)
CI: x±m*SE(x) or x±2*(
σ
/
√
n)
2SE(e) < e < 2SE(e)
Not skewed
e
≤
2SE(e)
Negative skew
e
≥
2SE(e)
Positive skew
2SE(u) < u < 2SE(u)
Not kurtotic
u
≤
2SE(u)
Negative kurtosis
u
≥
2SE(u)
Positive kurtosis
Quantitative – can be
meaningfully subtracted
 discrete – only whole numbers
 continuous – interval, fractions
Categorical – numbers are not
meaningful
 nominal  categorical and no
meaningful order
Bias
–
how far the average of many
measurements is from the true value
Skewness
– measure of asymmetry
IQR – measure of spread (Q3
Q1), more robust than var or
QN plot – if all points fall on or nearly
on diagonal line, Normal distribution
allows detection of nonnormality and
diagnosis of skewness and kurtosis
values too high in high&low range –
positive(right) skew
 high end points too high&low end
Probability mass function (pmf) –
discrete/cat.
Twolevel cat. variable, quantitative outcome >
independent samples ttest
or
ANOVA
tstatistic = difference between sample means divided by st. error of difference
pvalue = probability that any given experiment will produce value of chosen statistic
equal to observed value in experiment or something more extreme, when null
hypothesis is true and assumptions are met
reject null hypothesis when pvalue is less than acceptable type 1 error rate ( )
α
if null hypothesis is really true, approximate p=value will be less than .05 exactly
Measurement variability
– differences in
repeat measurement values (high precision)
Environment variability
– tradeoff – may
reduce external validity
Treatment application variability
–
difference in quality/quantity of treatment
among subjects assigned to same treatment
Internal validity – degree to which we can appropriately conclude that changes in X caused
changes in Y
random assignment of treatment – best way to assure that all potential confounding
variables are equal on average among groups
block randomization – randomization is performed separately for each level of the critical
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 Spring '09
 Variance

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