Download Basic Biostatistics formula sheet and more Cheat Sheet Biostatistics in PDF only on Docsity!
Basic Biostatistics
Formulas
Jane Pham & B. Burt Gerstman
C:\data\biostat-text\formulas1.doc Last printed 12/20/2007 4:39:00 PM
Page 1 of 3
Exploratory and Summary Statistics (Chapters 3 & 4) Statistic
Parameter
Point Estimate
Formula
InterprƩtation
Notes
Sum of squares
df
2 Ć
Ļ^
SS
ā=
n i
i^
x
x
SS
1
(^
No easy interpretation.
Mean
Ī¼^
x^
ā=
n i
xi
n
x^
1
A measure of centrallocation; balancing pt.
Variance
2
2 s
2
=^
SSn
s^
A measure of spreadexpressed in units squared
StandardDeviation
Ļ^
s^
2 s
s^
=^
or
SSn
A measure of spreadexpressed in data units.More appropriate fordescriptive purposes.
-^
Mean and standard deviation are bestsuited to symmetrical distributions.
-^
When distribution is Normal, 68% of datapoints lie within +
Ļ^
of Ī¼, 95% within
Ļ^
of Ī¼, and 99.7% lie within +
Ļ^
of Ī¼
-^
For other distributions, use Chebychevāsrule (e.g., at least 75% of data lie within+
Ļ^
of Ī¼).
Statistic
Formula
Interpretation
5-point Summary
Notes of boxplot
Median
Median has depth of
n^
A measure of centrallocation
InterquartileRange
(
)
IQR
Q
Q
IQR
=^
A measure of spread,aka āhinge-spreadā
Lower Fence^ (^
) l
F^
(^
)
IQR
Q
Fl
Helps determine:Lower inside valueLower outside value(s)
Upper Fence^ (^
) u
F^
(^
)
IQR
Q
Fu
Helps determine:Upper inside valueUpper outside value(s)
Q0 ā MinimumQ1 ā First QuartileQ2 ā MedianQ3 ā Third quartileQ4 ā Maximum
-^
Provide information about locations, spread, andshape. The box contains middle 50% of data. Lineinside the box is the median.
-^
Anything above the upper fence or below the lowerfence is āoutside.ā (Fences are
not
drawn.)
Plot
outside values as separate points.
-^
The lower whisker is drawn from Q1 to the lowerinside value. The upper whisker is drawn from Q3to the upper inside value.
Basic Biostatistics
Formulas
Jane Pham & B. Burt Gerstman
C:\data\biostat-text\formulas1.doc Last printed 12/20/2007 4:39:00 PM
Page 2 of 3
Probability (Chapters 5ā7)
Ā^
Probability
relative frequency in the population; expected proportion after a very long run of trials; can be used to quantify subjective statements.
Ā^
Properties of probabilities Basic: (1) 0
Pr(A)
1; (2) Pr(S) = 1; (3) Pr(
āPr(A); and (4) Pr(A or B) = Pr(A) + Pr(B) for disjoint events.
Advanced: (5) If A and B are independent, Pr(A and B) = Pr(A) Ā· Pr(B) (6) Pr(A or B) = Pr(A) + Pr(B)
Pr(A and B) (7) Pr(B|A) = Pr(A and B) / Pr(A) (8) Pr(A
and B) = Pr(A) Ā· Pr(B|A) (9) Pr(B) = [Pr(B and A)] + Pr(B and
) (10) Bayesā Theorem (p. 111)
Ā^
Binomial variables
:^ X
~ b(
n ,^
p ),
x n x x n^
q p C x X^
ā
=^
Pr(
where
!^ x n x
n
C^ xn
=^
and
q
p
Ā^
Cumulative probability:
Pr(
X^
ā¤^
x ) = sum all probabilities up to and including Pr(
X^
=^
x ); corresponds to AUC in the left tail of the
pmf
or
pdf.
Ā^
Normal variables
:^ X
~ N(
Ī¼,
Ļ
). To determine Pr(
X^
ā¤^
x ), standardize
=^
x z^
and look up cumulative probability in
Z
table. Use the fact that the AUC sums to 1
to determine probabilities for various ranges.To find a value that corresponds to a given probability, look up closest
z^ p
in the Z table and then unstandardize according to
x^
=^
Ī¼^
+^
z^ p
Ā·Ļ.
Introduction to Inference (Chapters 8ā11)
Ā^
The
sampling distribution of the mean (SDM)
is governed by the central limit theorem, law of large numbers, and square root law. When
n^
is large,
~^
x
N x
Ī¼^
where
x
Ļ^
is the standard error (
SE
) and is equal to
Ļ^ n
. The standard estimate is estimated by
s^ n
when the population standard deviation is
not known. Ā^
Ī±)100% confidence interval for
Ī¼
.^ Use
x SE z x^
Ā±^
Ī±^ ā 12
when
Ļ
is known. Use
x
n^
SE
t x^
ā
Ā±^
ā ā^ 12 , 1
Ī±^
when relying on
s.
Ā^
Hypothesis testing basics.
Know all the steps, not just the conclusion and keep in mind that hypothesis tests require certain conditions (e.g., Normality,
independence, data quality) to be valid. The steps are:A.
H
and 0
H
[For one-sample test of a mean, 1
H
0: Ī¼ = Ī¼
where Ī¼ 0
is the mean specified by the null hypothesis.] 0
B. Test statistic [For one-sample test of a mean, use either
x x SE
z^
0
stat
=^
or
with 0
stat
=^
n df
x SE
t
Ī¼^ x
.]
C.
P
-value. Convert the test statistic to a
P
-value. Small
P
strong evidence against
H
D. Significance level. It is unwise to draw too firm a line. However, you can use the conventions regarding marginal significance, significance, and highsignificance when first learning. Ā^
Power and sample size basics.
Approach from estimation, testing, or āpowerā perspective. Sample size requirement for limiting margin of error
m
is given by
2
1
2
=^
ā^
m z n
The power of testing a mean is
ā^ āāā
ā^
ā
Ī±
n
z^
2 1
.^ The sample size requirement of a one-sample
z^
or
t^
test:
(^
2
2 1
1 2
2
=^
ā
ā
Ī±
Ī²
Ļ^
z
z
n^
. It is OK to use
s^
as a substitute for
Ļ
in power and sample size formulas, when necessary.
