Geometric Mean FormulaGeometric Mean FormulaGeometric Mean Definition : Geometric mean is a kind of average of a set of numbers that is
different from the arithmetic average. The geometric mean is well defined only for sets of
positive real numbers. This is calculated by multiplying al the numbers (cal the number of
numbers n), and taking the nth root of the total. A common example of where the geometric
mean is the correct choice is when averaging growth rates.
A geometric mean is often used when comparing different items- finding a single "figure of
merit" for these items- when each item has multiple properties that have different numeric
ranges.
For example, the geometric mean can give a meaningful "average" to compare two
companies which are each rated at 0 to 5 for their environmental sustainability, and are rated
at 0 to 100 for their financial viability.
If an arithmetic mean was used instead of a geometric mean, the financial viability is given
more weight because its numeric range is larger- so a small percentage change in the
financial rating (e.g. going from 80 to 90) makes a much larger difference in the arithmetic
mean than a large percentage change in environmental sustainability (e.g. going from 2 to 5).
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The use of a geometric mean "normalizes" the ranges being averaged, so that no range
dominates the weighting, and a given percentage change in any of the properties has the
same effect on the geometric mean. So, a 20% change in environmental sustainability from 4
to 4.8 has the same effect on the geometric mean as a 20% change in financial viability from
60 to 72.
The geometric mean is similar to the arithmetic mean, except that the numbers are multiplied
and then the nth root (where n is the count of numbers in the set) of the resulting product is
taken. For instance, the geometric mean of two numbers, say 2 and 8, is just the square root
of their product; that is 22 x 8 = 4. As another example, the geometric mean of the three
numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is 34 x 1 x
1/32 = 1/2 . More general y, if the numbers are , the geometric mean satisfies
The latter expression states that the log of the geometric mean is the arithmetic mean of the
logs of the numbers. The geometric mean can also be understood in terms of geometry. The
geometric mean of two numbers, a and b, is the length of one side of a square whose area is
equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of
three numbers, a, b, and c, is the length of one side of a cube whose volume is the same as
that of a cuboid with sides whose lengths are equal to the three given numbers.
The geometric mean applies only to positive numbers.[2] It is also often used for a set of
numbers whose values are meant to be multiplied together or are exponential in nature, such
as data on the growth of the human population or interest rates of a financial investment.
The geometric mean is also one of the three classical Pythagorean means, together with the
aforementioned arithmetic mean and the harmonic mean. For al positive data sets containing
at least one pair of unequal values, the harmonic mean is always the least of the three means,
while the arithmetic mean is always the greatest of the three and the geometric mean is
always in between (see Inequality of arithmetic and geometric means.)
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What is Central TendencyWhat is Central TendencyThe normal distribution is a continuous probability distribution used to model continuous
random variables - these variables that can take any value and are typical y natural y
occurring (e.g. height). As a continuous distribution the probability of, for example, a random
person being 180cm tal is zero (i.e. P(X = 180) = 0). As the variable is continuous it is ranges
you are concerned with such as 'What is the probability of a random person being smal er
than 180cm?'
Notation :- If a continuous random variable X with a mean of and variance (where is the
standard deviation) is normal y distributed it is notated like so:
Example :- The heights of men in a town can be model ed by the normal distribution with a
mean of 182cm and a standard deviation of 10cm. This can be notated:
Standard normal distribution :- When calculating the probability of a normal distribution it
must be transformed from the standard normal distribution. The continuous random variable Z
is used to denote the standard normal distribution. The standard normal distribution has a
mean of 0 and a variance of 1:
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In order to transform from the normal distribution variable X to the standard normal distribution
variable Z the following equation is used:
Where is the mean, is the standard deviation (square root of the variance), is a particular
value of the random variable X and is the corresponding value of random variable Z.
Calculating probabilityThe probability is calculated using the fol owing function: This function gives the area under
the curve (the probability) to the LEFT of the value z. That is to say it is cumulative, the value
of is equal to the probability of the random variable Z being less than z. See examples for
more information.
Values for this function for multiple values of z are found in the tables of values in your formula
book.
Worked examples :- The weight of adults in the UK is normally distributed with a mean of 15
Stone and a standard deviation of 3 stone. Find the probability that a randomly selected adult
is:
Less than 18 stone
Less than 12 stone
Over 17 stone
Between 12 and 18 stone
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