Normal distribution, Mean & Standard deviation

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Normal distribution, Mean & Standard deviation
Mean
most commonly used measure of central tendency
influenced by every value in a sample

X
N
µ is population mean
X is sample mean
Standard deviation
measure of variability

(X  )
2
N
if µ is unknown, use X
correct for smaller SS bias by dividing by n-1

Normal distribution
(X  X )
2
n 1
bell shaped curve
reasonably accurate description of many
distributions
properties : unimodal
symmetrical
points of inflection at µ ± σ
tails approach x-axis
completely defined by mean and SD
Sampling & population inference
population
sample
entire collection of units of interest
collection of observations from a well defined
population
random sample each unit in a population has an equal chance of
being sampled and each unit is independent of each
other
population inference
to form a conclusion about a population from a
sample
central limit theorem
distribution of random samples of mean tends
towards a normal distribution, even if parent
population isn’t normal
normal distribution is model for distribution of sample stats
approximation to normal distribution improves as n increases
standard error of mean
standard deviation of sampling error of
different samples of a given sample size
how great is sampling error of ( X - µ)
as n increases, variability decreases
X 

n
z, t & F distributions
hypothesis testing: often want to know the likelihood that a given
sample has come from a population with known characteristic(s)
1. define H0
2. test likelihood of H0
z
X 
X
normal distribution with mean 0, standard deviation 1 (cf central limit
theorem)
e.g.
X = 104.0
H0 : µ = 100
X= 3
z = (104 – 100) / 3 = 1.33
α = 0.05
therefore retain H0
t
X 
sX
for a given mean and sd, normal distribution is completely defined
there are a family of t curves, depending on degrees of freedom
n – 1 degrees of freedom associated with deviations from a single
mean
with infinite degrees of freedom, t = z
H0 : µ = 100
X = 120
n = 25
sx = 35.5
sX 
t
sx
n

35.5
25
 7.1
X   120  100

 2.82
sX
7.1
df = 24
α = 0.05
tcrit = 2.06
therefore reject H0
t values may be converted to z values via p values
Analysis of variance
SStotal = SSwithin + SSbetween
dfwithin = ntotal – k
dfbetween = k – 1
Within-groups variance estimate:
SS w
df w
s w2 
(‘mean square within’)
estimates inherent variance
Between-groups variance estimate:
2
sbet

SS bet
df bet
(‘mean square between’)
estimates inherent variance + treatment effect
F
2
sbet
s w2
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