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Stats 1 Formula
Measure of location: 𝑥̅ vs 
Sample
𝑥̅ =
𝑥̅ =
𝑥̅ =
Population
∑ 𝑥𝑖
𝑛
𝜇=
∑ 𝑥𝑖 𝑓𝑖
∑ 𝑓𝑖
𝜇=
∑(𝑥𝑖 − 𝑎)
+𝑎
𝑛
𝜇=
∑ 𝑥𝑖
𝑛
∑ 𝑥𝑖 𝑓𝑖
∑ 𝑓𝑖
∑(𝑥𝑖 − 𝑎)
+𝑎
𝑛
Measure of spread:
Sample
𝑠2 =
Population
1
∑(𝑥 − 𝑥̅ )2
𝑛−1
𝜎2 =
(∑ 𝑥)2
1
2
𝑠 =
{∑ 𝑥 −
}
𝑛−1
𝑛
2
𝑠2 =
𝜎2 =
(∑ 𝑓𝑥)2
1
{∑ 𝑓𝑥 2 −
}
𝑛−1
𝑛
SD    var
1
∑(𝑥 − 𝑥̅ )2
𝑛
1
(∑ 𝑥 2 ) − 𝜇 2
𝑛
(∑ 𝑓𝑥)
1
𝜎 = (∑ 𝑓𝑥 2 ) − {
}
𝑛
𝑛
2
2

var   2
1 standard deviation includes 2/3 of data
2 standard deviations include 95% of data
3 standard deviations include ‘almost all’ data
“a quantity expressing by how much the members of a group differ from the
mean value for the group”
Probability
Mutually Exclusive
Non-Mutually Exclusive
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴 | 𝐵) × 𝑃(𝐵)

𝑃(𝐴 ∩ 𝐵)
𝑃(𝐴 | 𝐵) =
𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵 ∩ 𝐶) = 𝑃(𝐴) × 𝑃(𝐵) × 𝑃(𝐶)
Exclusive – no overlap of possible events
Exhaustive – all possible events are accounted for
If any of the following are true then they are all true & events are independent:
𝑃(𝐴 | 𝐵) = 𝑃(𝐴 | 𝐵′)
𝑃(𝐵| 𝐴) = 𝑃(𝐵 | 𝐴′)
𝑃(𝐴 | 𝐵) = 𝑃(𝐴)
𝑃(𝐵 | 𝐴) = 𝑃(𝐵)
𝑃(𝐴 | 𝐵′) = 𝑃(𝐴)
𝑃(𝐵 | 𝐴′) = 𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
Binomial distribution (a discrete distribution):
X ~ B ( n, p )
P( X  x) n C x p x q n  x
Expectation & Variance of Binomial distribution:
X ~ B ( n, p )
E ( x)    np
Var( x)   2  npq
Normal distribution (a continuous distribution):
𝑍~𝑁(𝜇, 𝜎 2 )
The standard normal distribution: 𝑍~𝑁(0 , 1)
Conversion: 𝑍 =
𝑋−𝜇
𝜎
To find  and/or  using given probabilities:
1. Draw a sketch or curve
2. Use the probabilities table to find corresponding z value
3. Use the z value together with the conversion to find  and/or . Finding both
 and  will involve solving simultaneous equations.
Estimation
Sample mean = 𝑥̅
Sample variance = 𝑠 2
Unbiased estimator of
population mean = 𝑋̅
Unbiased estimator of
population variance = 𝑆̅
(ie the distribution of many sample means)
(ie the distribution of many sample variances)
𝑋̅~𝑁 (𝜇,
𝜎2
)
𝑛
𝑍̅ =
𝑋̅ − 𝜇
𝜎 ~𝑁(0,1)
√𝑛
Confidence Interval
90%
95%
98%
99%
99.8%
z
1.6449
1.9600
2.3263
2.5758
3.0902
𝑥̅ ± "z"
𝜎
√𝑛
Product Moment Correlation Coefficient:
r
S xy
S xx S yy
S xy   xi y i 
S xx   xi 
2
1
 xi 2
n
1
 xi  y i
n
S yy   y i 
2
1
 yi 2
n
Regression:
y on x
x on y
y  a  bx
x  a 'b' y
a  y  bx
a '  x  b' y
S xy
b' 
S yy
b
S xy
S xx