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Economics 250 Formula Sheet
Descriptive Statistics
Population Sample
Mean μ = (^) N^1
∑N
i=1 xi^ x^ =^
1 n
∑n i=1 xi
Variance σ^2 = (^) N^1
∑N
i=1(xi^ −^ μ)
(^2) s (^2) = 1 n− 1
∑n i=1(xi^ −^ x)
2
CV σμ × (^100) xs × 100
Covariance σxy = (^) N^1
∑N
i=1(xi^ −^ μx)(yi^ −^ μy^ )^ sxy^ =^
1 n− 1
∑n i=1(xi^ −^ x)(yi^ −^ y)
Correlation ρxy = σxy σxσy rxy^ =^
sxy sxsy
Grouped Data
With K classes, with midpoints mi and counts ci, the sample mean is x = (^) n^1
∑K
i=1 cimi,^ and the sample variance is s^2 = (^) n^1 − 1
∑K
i=1 ci(mi^ −^ x) (^2) , where n = ∑K i=1 ci.
68–95–99.7 Rule
For a normal distribution 68% of the observations are in μ ± 1 σ, 95% are in μ ± 2 σ, and almost all (99.7%) are in μ ± 3 σ.
Normal Distribution
For −∞ < x < ∞
f (x) = (2πσ^2 )−^1 /^2 exp
[ (^) −(x − μ)^2 2 σ^2
]
with mean μx and standard deviation σx, then
x ∼ N (μx, σx)
z =
x − μx σx
∼ N (0, 1)
Warning: Some people record the normal distribution as N (μx, σ x^2 ) i.e. the second number in brackets is the variance rather than the standard deviation.
Uniform Distribution
For a ≤ x ≤ b
f (x) =
b − a
E(x) =
a + b 2
V ar(x) =
(b − a)^2 12
Random Variables
Let x be a discrete random variable, then:
μx = E(x) =
x
xP (x)
σ^2 x =
x
(x − μ)^2 P (x)
The covariance of x and y is:
cov(x, y) = σxy = E(x − μx)(y − μy ) =
x
y
(x − μx)(y − μy )P (x, y)
The correlation between x and y is:
ρxy =
σxy σxσy
For a continuous rv replace the sums by integrals.
Functions of Random Variables
If y = a + bx then: E(y) = μy = a + bμx,
and σ^2 y = b^2 σ x^2.
If w = cx + dy then: E(w) = μw = cμx + dμy ,
and σ^2 w = c^2 σ^2 x + d^2 σ^2 y + 2cdσxy.
Sampling Distribution of the Sample Mean
For large samples,
x ∼ N
μ,
σ √ n
Probability Theory
P (A) = 1 − P (A) (complement rule)
P (A ∪ B) = P (A) + P (B) − P (A ∩ B) (addition rule)
P (A ∩ B) = P (A|B)P (B) (multiplication rule)
A and B are independent if P (A ∩ B) = P (A)P (B)
Marginal probabilities add entries in a joint probability table. If Bi are mutually exclusive and exhaustive events then:
P (A) =
∑^ n
i=
P (A ∩ Bi)
Bayes’s Rule:
P (B|A) =
P (A|B)P (B)
P (A)
- Differences in Means: Independent Samples
(variances unknown and not assumed equal):
sx 1 −x 2 =
s^21 n 1
s^22 n 2
can be used with tν,α/ 2 and
ν =
[(s^21 /n 1 ) + (s^22 /n 2 )]^2 (s^21 /n 1 )^2 /(n 1 − 1) + (s^22 /n 2 )^2 /(n 2 − 1)
or often degrees of freedom ν approximated by the smaller of n 1 − 1 and n 2 − 1.
Inference for Proportions
- For large samples, a 100(1 − α)% CI for p is:
pˆ ± zα/ 2
pˆ(1 − pˆ) n
(Or replace ˆp = x/n by ˜p = (x + 2)/(n + 4) when α = 1%, 5%, or 10%.)
- Differences in Proportions
pˆ 1 − pˆ 2
has standard deviation: (^) √ p ˆ 1 (1 − pˆ 1 ) n 1
pˆ 2 (1 − pˆ 2 ) n 2
which can be used with zα/ 2 to form a confidence interval. (Or add 1 success and 1 failure to each sample when α = 1%, 5%, or 10%.)
- Testing the Hypothesis of Equal Proportions
For this test, use the pooled estimate of the common value of p 1 and p 2 :
pˆpool =
X 1 + X 2
n 1 + n 2
to form
SEDp =
p ˆpool(1 − pˆpool)
n 1
n 2
