Statistics Key Points
1.1.1
Discrete data can only take on certain values.
Continuous data can take on any value, possibly within a limited
range.
1.1.2
For ungrouped data, the mean is
𝑥=
𝛴𝑥
𝑛
𝑥=
𝛴𝑥𝑓
𝛴𝑓
For grouped data,
1.1.3
For ungrouped coded data, the mean is
1 𝛴(𝑎𝑥 − 𝑏)
𝑥= (
+ 𝑏.
𝑎
𝑛
For grouped coded data,
1 𝛴(𝑎𝑥 − 𝑏)𝑓
𝑥= (
+ 𝑏.
𝑎
𝛴𝑓
These formulae can be summarised by writing
1
𝑥 = ⋅ [mean(𝑎𝑥 − 𝑏) + 𝑏]
𝑎
1.1.4
Interquartile range = upper quartile – lower quartile
or
𝐼𝑄𝑅 = 𝑄! − 𝑄"
1.1.5
Standard deviation = √Variance
For ungrouped data,
𝛴 (𝑥 − 𝑥 ) #
𝛴𝑥 #
𝛴𝑥
#
𝜎=E
=E
− 𝑥 where 𝑥 =
𝑛
𝑛
𝑛
For grouped data,
𝛴(𝑥 − 𝑥 )# 𝑓
𝛴𝑥 # 𝑓
𝛴𝑥𝑓
#
𝜎=E
=E
− 𝑥 where 𝑥 =
𝛴𝑓
𝛴𝑓
𝛴𝑓
1.2.1
A probability distribution shows all the possible values of a
variable and the sum of the probabilities is
𝛴𝑝 = 1
1.2.2
The expectation of a discrete random variable (DRV) is
𝐸 (𝑋) = 𝛴𝑥𝑝 = 𝛴 [𝑥 ⋅ 𝑃(𝑋 = 𝑥)]
1.3.1
If 𝑿~𝑩(𝒏, 𝒑) then the probability of 𝑟 successes is
𝑛
𝑝$ = S T 𝑝$ (1 − 𝑝)%&$
𝑟
1.3.2
The mean and variance of 𝑋~𝐵(𝑛, 𝑝) are given, respectively, by
𝜇 = 𝑛𝑝 and 𝜎 # = 𝑛𝑝(1 − 𝑝) = 𝑛𝑝𝑞
1.4.1
A random variable 𝑋 that has a geometric distribution is denoted
by 𝑿~𝑮𝒆𝒐(𝒑), and the probability that the first success occurs on
the 𝑟th trial is
𝑝$ = 𝑝(1 − 𝑝)$&" for 𝑟 = 1,2,3, …
1.4.2
When 𝑋~𝐺𝑒𝑜(𝑝) and 𝑞 = 1 − 𝑝, then
• 𝑃 (𝑋 ≤ 𝑟) = 1 − 𝑞 $
• 𝑃 (𝑋 > 𝑟) = 𝑞 $
1.4.3
The mode of all geometric distributions is 1.
1.5.1
𝑋~𝑁 (𝜇, 𝜎 # ) describes a normally distributed random variable.
We read this as “𝑋 has a normal distribution with mean 𝜇 and
variance 𝜎 # ”
1.5.2
The standard normal variable is 𝑍~𝑁 (0,1)
1.5.3
When 𝑋~𝑁(𝜇, 𝜎 # ) then
𝑋−𝜇
𝜎
has a normal distribution. [Refer to 2.2.3]
A standardised value
𝑥−𝜇
𝑧=
𝜎
tells us how many standard deviations 𝑥 is from the mean.
𝑍=
1.5.4
𝑋~𝐵(𝑛, 𝑝) can be approximated by 𝑁(𝜇, 𝜎 # ), where 𝜇 = 𝑛𝑝 and
𝜎 # = 𝑛𝑝𝑞, provided that 𝑛 is large enough to ensure that 𝑛𝑝 > 5
and 𝑛𝑞 > 5
1.5.5
Continuity corrections must be made when a discrete distribution is
approximated by a continuous distribution.
2.1.1
A Poisson distribution can be used to model a discrete probability
distribution in which the events occur singly, at random and
independently, in a given interval of space or time. The mean and
variance of a Poisson distribution are equal; hence, a Poisson
distribution has only one parameter.
2.1.2
When modelling data using Poisson distribution:
• Work out the mean and variance and check if they are
approximately equal.
• If mean and variance are not approximately equal, the
Poisson distribution is not a suitable model to use with the
data.
• Use the mean to calculate probabilities and expected
frequencies.
• Compare expected frequencies with observed frequencies.
2.1.3
If the random variable 𝑋 has a Poisson distribution with parameter
𝜆, where 𝜆 > 0, we write 𝑋~𝑃𝑜(𝜆) and:
'!
• 𝑃 (𝑋 = 𝑟) = 𝑒 &' ⋅
where 𝑟 = 0,1,2, …
• 𝐸 (𝑋) = 𝜆
• 𝑉𝑎𝑟(𝑋) = 𝜆
$!
2.1.4
In a Poisson distribution, events occur at a constant rate; the mean
average number of events in a given interval is proportional to that
interval.
2.2.1
For a random variable 𝑋 and constants 𝑎 and 𝑏:
𝐸 (𝑎𝑋 ± 𝑏 ) = 𝑎𝐸 (𝑋) ± 𝑏 and 𝑉𝑎𝑟(𝑎𝑋 + 𝑏) = 𝑎# 𝑉𝑎𝑟(𝑋)
2.2.2
For two independent random variables 𝑋 and 𝑌 and constants 𝑎
and 𝑏:
𝐸 (𝑎𝑋 ± 𝑏𝑌 ) = 𝑎𝐸 (𝑋) ± 𝑏𝐸 (𝑌)
and
𝑉𝑎𝑟(𝑎𝑋 ± 𝑏𝑌) = 𝑎# 𝑉𝑎𝑟 (𝑋) + 𝑏 # 𝑉𝑎𝑟(𝑌)
These results can be extended to any number of independent
random variables.
2.2.3
If a continuous random variable 𝑋 has a normal distribution, then
𝑎𝑋 + 𝑏, where 𝑎 and 𝑏 are constants, also has a normal
distribution.
If continuous random variables 𝑋 and 𝑌 have independent normal
distributions, then 𝑎𝑋 + 𝑏𝑌, where 𝑎 and 𝑏 are constants, has a
normal distribution.
2.3.1
A graph, f(x), representing a continuous random variable is the
probability density function (PDF).
The PDF has following properties:
• It cannot be negative since you cannot have a negative
probability; 𝑓 (𝑥 ) ≥ 0.
• Total probability of all outcomes = 1; hence,
)
n 𝑓(𝑥 ) 𝑑𝑥 = 1
&)
In many situations, the data are defined across a specified interval
or across specified intervals, outside of which 𝑓(𝑥) = 0.
2.3.2
• With continuous random variables, each individual value has
zero probability of occurring.
For a continuous random variable with PDF 𝑓 (𝑥 ),
𝑃 (𝑋 = 𝑎 ) = 0.
• Because we cannot find the probability of an exact value,
when finding the probability in a given interval it does not
matter whether you use < or ≤.
𝑃 (𝑎 < 𝑥 < 𝑏 ) = 𝑃 (𝑎 ≤ 𝑥 < 𝑏) = 𝑃(𝑎 < 𝑥 ≤ 𝑏) = 𝑃 (𝑎 ≤ 𝑥 ≤ 𝑏 )
Note that this does not imply that 𝑋 cannot take the value 𝑎, it just
means the probability of the exact value 𝑎 is zero.
2.3.3
The probability of 𝑋 lying in the interval (𝑎, 𝑏) is given by the area
under the graph between 𝑎 and 𝑏.
That is:
*
𝑃(𝑎 < 𝑋 < 𝑏 ) = n 𝑓 (𝑥 ) 𝑑𝑥
+
2.3.4
The median, 𝑚, of a continuous random variable is that value for
which
,
1
𝑃(𝑋 < 𝑚) = n 𝑓 (𝑥 ) 𝑑𝑥 =
2
&)
2.3.5
The 𝑟th percentile, 𝑘, of a continuous random variable is that value
for which
𝑟
𝑃 (𝑋 < 𝑘 ) = n 𝑓(𝑥) 𝑑𝑥 =
where 0 < 𝑟 < 100
100
&)
2.3.6
For continuous random variables with PDF 𝑓(𝑥):
)
𝐸 (𝑋 ) = n 𝑥𝑓 (𝑥 ) 𝑑𝑥
&)
and
)
)
&)
&)
𝑉𝑎𝑟(𝑥 ) = n 𝑥 # 𝑓(𝑥) 𝑑𝑥 − sn 𝑥𝑓 (𝑥 ) 𝑑𝑥t
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