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Geometric Mean G M Notes Study Business Mathematics and Logical Reasoning & Statistics CA Foundation

It tells nothing about the degree to which the returns vary over time. To understand this, let us go back to the previous example. If you look at the table, you will see that Nifty has generated a mean annual return of 17.53% over the past 23 years. However, if you closely observe, you will notice that the returns each year have fluctuated substantially from as low as -52% to as high as +76%.

formula of geometric mean in statistics

The geometric mean, often referred to as the geometric average, is a so-called specialized average and is defined as the n-th root of the product of n numbers of the same sign. Any time you have several factors contributing to a product, and you want to calculate the “average” of the factors, the answer is the geometric mean. The geometric mean differs from the arithmetic average, or arithmetic mean, in how it is calculated because it takes into account the compounding that occurs from period to period.

There are same cases when adjustments are justified and the first one is similar to the negative numbers case above. If the data is percentage increases, you can transform them into normal percentage values in the way described for negative numbers. Zeros then become 100% or 1 and the calculation proceeds as normal.

Solved Examples:

We will apply the above result on three numbers 3a, 4b and (72 – 3a – 4b). We have selected these numbers as there sum will be a constant number and their product will have the terms required in the question. Three numbers a, b and c are said to be in Geometric progression if i.e. if the ratio of the terms is same. It is used to find the annual return on investment portfolios. Now, we want to show that the length of MN is the square root of a times b, i.e., the geometric mean of a and b.

In Mathematics, we come across statistics to measure central tendency and dispersion. The standard measures of central tendencies are mean, median, and mode. The dispersion consists of variance and standard deviation. In this section, we will discuss central tendency and learn what it means in math. The different types of means in statistics are Arithmetic Mean , Geometric Mean and Harmonic Mean . In this particular article, we will be focusing on GM.

Indicates the central tendency of a set of data by using the product of their values rather than their sum. The Median is the middle number in a sorted, ascending, or descending set. On the other hand, the mode is the most frequent score in our data set. Ans.1 The Geometric Mean or GM is the average value or mean which implies the central tendency of the set of numbers by using the root of the product of the values. If the geometric mean replaces each observation in the given data set, then the product of observations does nor change. Mathematics and statistics use the measures of central tendency to express the summary of all the values in a data collection.

Finally, observe the formula bar to understand how the geometric mean is calculated in cell H11. So, the geometric mean or the compounded annual returns of Nifty for the past 23 years is 13.17%. The geometric mean is also used in finance to find the average growth rates known as the compounded annual growth rate.

formula of geometric mean in statistics

The ratio of the observations of the geometric mean in two series is equivalent to the ratio of their geometric means. As a result, investors consider the geometric mean to be a more accurate indicator of returns than the arithmetic mean. The product of the values equals the geometric mean raised to the nth power.

Among these, the data set’s mean provides an overall picture of the data. Quartiles are those values of the variate which divide the total frequency into four equal parts. Obviously there will be three such points Q1, Q2 and Q3 such that Q1≤ Q2 ≤ Q3 termedas the three quartiles. Q1is known as the lower or first quartile and is the value which has 25% of the items of the distribution below it and consequently 75 percent of the items are greater than it. Q3is known as the upper or third quartile and has 75percent of the observations below it and consequently 25 percent of the observations above it.


Thus the mode is an important measure in case of qualitative data. In the above equation, 250 represents the approximate number of trading sessions in a year. Download data of historical prices for the period for which you are looking to calculate variance and standard deviation . Here we will learn all the Geometric Mean Formula With Example. The Geometric Mean is nth root of the product of n quantities of the series. It is observed by multiplying the values of items together and extracting the root of the product corresponding to the number of items.

So, inputting 0.05% in the above equation, we get annualized return as 13.31%. We can calculate the geometric mean of more than two numbers also. For calculating the geometric mean we have to multiply all the numbers and then take the nth root of that number.i.e. formula of geometric mean in statistics If we are multiplying two numbers, we are taking the square root, as we had taken in the above example. A certain store made profits of Rs 5,000, Rs 10,000 and Rs 80,000 in 1965, 1966 and 1967 respectively. Determine the average rate of growth of its profits.

The strongest drawback of arithmetic mean is that it is very much affected by extreme observations. Solve example 1 with short-cut and step-deviation method. It should not be unduly affected by extreme observations. Σ can be compared across securities, such as stocks, sectors, indices etc. The statistics are as mentioned in each of the tables below, with the table on the top representing the statistics for 2019 while that at the bottom representing the statistics for 2020. And m1, m2, …, mn are the mid points of class intervals.

Geometric Mean (G.M.) Notes | Study Business Mathematics and Logical Reasoning & Statistics – CA Foundation

For example, replacing 30 with 100 would yield an arithmetic mean of 25.80, but a geometric mean of just 9.17, which is very desirable in certain situations. However, before settling on using the geometric mean, you should consider if it is the right statistic to use to answer your particular question. The geometric mean is applied in stock indexes as many of the value line indexes handled by the financial departments uses G.M. The products of the similar elements of the geometric mean in two series are equivalent to the product of their GM.

  • Use this online calculator to easily calculate the Geometric mean for a set of numbers or percentages.
  • In this case, 1000 is the outlier and the average is 39.5.
  • The rate usually indicates the relation between two different types of measuring units that can be expressed reciprocally.

Geometric mean is also defined as the \(\) root of the product of \(n\) values. Suppose, if you have two values, take the square root; if you have three, take the cube root; if you have four, then take the \(\) root, and so on. In other cases, zeros mean non-responses and in some cases they can just be deleted before calculation.

Previous Year Questions with Solutions

The geometric mean formula applied only on the positive set of numbers. The geometric mean is less than the arithmetic mean for any set of positive numbers but when all of a series’ values are equal, however, G.M. In this article, we will discuss the geometric mean, geometric mean definitions, and formula, the geometric mean formula for grouped data, properties of geometric mean, etc. is.

Objective type Questions

The arithmetic mean between two numbers is a number, which, when placed between them, forms with them an arithmetic progression. The geometric mean between two numbers is a number, which, when placed between them, forms with them a geometric progression. The geometric mean is based on all the items of the series. Geometric mean is used in stock indexes because many of values line indexes which are used by financial departments. As you can see, the geometric mean is significantly more robust to outliers / extreme values.

See above that the geometric mean is less than the arithmetic mean. In fact, whenever there is dispersion in data , the geometric mean will always be less than the arithmetic mean. Furthermore, the larger the dispersion, the larger will be the difference between arithmetic and geometric mean . For instance, in the above case of Nifty, if you observe each of the annual returns, you will see there is quite a bit of difference over the years. As such, the arithmetic return is notably higher than the geometric return. The only instance when the two will be equal is when there is no dispersion in the data set.

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