Geometric Mean Statistics Homework Help

Geometric mean, as a mathematical average, has a lot of algebraic properties.

Some of the important properties are depicted are depicted here as under:

1. For any series of positive values, the geometric mean is smaller than the arithmetic mean. However, when all the values of a series are equal, G.M. equals to the arithmetic mean.

Proof. Let a be the largest and b the smallest positive values of a series.

Then      (a –b)2 > 0

So, a2 – 2ab + b2 > 0

So, a2 + 2ab+ b2 > 4ab     (adding 4ab to both the sides)

So,          (a + b)2 > (2√ab )2

So,          a + b > 2√ab,

So,  a+b/2 >√ab  , (Dividing each side by 2)

Now a+b/2 = Arithmetic mean and √ab = Geometric Mean

X > G and G < X

2.  If any value of a series is zero, the value of geometric mean is infinity and thus inappropriate.

Proof. Let there be 3 times viz. 15, 20 and 0. The geometric mean of these items would be

3√(15 x 20 x 0) = 3√0 = infinity, and hence inappropriate.

3. It can not be calculated, if the number of negative values is odd.This is because, the product of the values will become negative, and we can not find out the root of a negative product viz.

3√(-3 x -5 x -4 x 7) or 4√-420

4. Since, geometric mean or G = N√(X1. X2…..Xn),  the geometric mean raised to the nth power equals to the product of the values

i.e.          Gn =  X1.X2. X3…Xn.

5.  Any series of numbers having the same N, and the same product will have same geometric mean.

6. The product of the ratios of the values on one side of the geometric mean is equal to the product of the ratios of the geometric mean to the values of the other side of the geometric mean.

Proof. Let 4,5,20 and 25 be the values of a series. The G.M. of the series would be 4√(4 x 5 x 20 x 25 = 10. Here, the values 4 and 5 are on one side of the G.M. 10 and the values 20 and 25 are on the other side of the G.M. 10.

The product of the rations of the of the values 4 and 5 to G.M.

(4/10) x (5/10) = 1/5

And, the product of the rations of the G.M. to the values 20 and 25.

(10/20) x (10/25) = 1/5

Thus, the two products are equal.

The product of the reverse of the above rations will also be equal.

The product of the reverse of the above of rations will also be equal.

Thus, (10/4) x (10/5) = 5, (20/10) x (20/25) = 5

7. The products of a series of numbers remain the same even when each number of the series is replaced by its geometric mean.

Proof. Product of the series, 4,5, 20,25 = 10000

Geometric mean of the series = 4√(4 x 5 x 20 x 25) = 10

If each number of the series is replaced by G.M.10, the series becomes 10, 10, 0, 10.

 The product of this series 10 × 10 × 10 × 10 = 10000. Proved.

8. The sum of deviations of the logarithms of the original values above and below the logarithm of the G.M. is equal.

Here, sum of the deviations below G.M.

= .3979 + .3010 = .6989

And, sum of the deviations above the G.M.

= .3010 + .3979 = .6989

Note that the G.M. here =4√(4 x 5 x 20 x 25) = 10, and that its logarithm= 1.0000.

Geometric Mean is a type of mathematical average. It is defined as the nth root of the product of then items. The fundamental formula for its computation stands as under:

G = N√(X1X2X3……..Xn)

Where   G = Geometric mean

N = number of items

X1, X2, X3……XN = values of the 1st, 2nd, 3rd and so on items.

Thus, the geometric mean of 1,8 and 64 would be

3√(1x8x64 = 1x2x4= 8 )

Similarly, the geometric mean of 8, 27 and 64 would be

3√(8x27x64 = 2x3x4x = 24

The above fundamental formula of geometric mean will give us the value at an ease, if the values of the items are small, simple, and capable of being factorised readily. If the values of the variable are big or complicated ones, the geometric mean will have to be calculated by logarithmic operation through the aforesaid formula. In such a case, the said formula will be modified as under:

G = Antilog of  1/N  {log X1 + log X2 + log X3…. Log Xn}

= A.L. ∑ log x / N

Hence, under the direct method, the models of G.M. for three types of series will be as thus:

Where, ∑F log X = Sum of the products of the logarithms of the values of the X variable and their corresponding frequencies, and

N = total number of items, or .

Weighted Geometric Mean:

To calculate the weighted geometric mean of a series, the above formula will be modified by the substitution of ‘W’ in place of ‘F’ as follows:

G(w) = AL  ∑w log X / N

Where, G(w)   = weighted geometric mean

 ∑w log X = Sum of the products of the logarithms of the values of X and their corresponding weights and

∑w = Sum of the weights

Combined Geometric Mean:

To calculate the combined G.M. of a number of series, the relevant formula is :

G1.2. = AL  { N1 log G1 + N2 log G2+…../ N1+N2+…..}

From all the above cited formula, it must be noted that geometric mean is the antilogarithm of the arithmetic average of the logarithms of the values of a variable.

Steps for Computation

In the computation of the geometric mean through the logarithmic operation, the following steps are to be taken up in turn:

  1. Arrange the mid values, and their corresponding frequencies in a columnar manner, and see that these two columns remain adjacent to each other. However, and see that these two columns remain adjacent to each other. However, it is not required to arrange the series either in ascending or in descending order.
  2. Find the logarithms of the mid-values and put them in order in the third columns under the heading ‘log x’.
  3. Find the product of each pair of the logarithm and the respective frequency, or the weight as the case may be and put these products in order in the fourth column under the heading ‘F log X’ or ‘W log X’.

In case of simple series, however, this step is not necessary.

  1. Get the total of the Frequency/weight column as  ∑ F or ∑ W log X. and that of the product column as  ∑ F log X or,  ∑ W log X. In the case of a simple series, however, only the logarithm column will be totaled as  ∑ W log X.
  2. Divide the total of the products i.e. ∑ F log X, or ∑ W log X by the total of  ∑ F, or  ∑ W and ascertain its quotient. In the case of a simple series, however, total of the log X column i.e. ∑ log X will be divided by the number of items i.e. N.
  3. Find the antilongarithm of the quotient thus arrived at under the 5th step. This will be the required value of the geometric mean.
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