Competitive Exams: Mean and its types
Describe arithmetic, geometric and harmonic means with suitable example. Explain merits and limitation of geometric mean.
The arithmetic mean is the “standard” average, often simply called the “mean” It is used for many purposes but also often abused by incorrectly using it to describe skewed distributions, with highly misleading results. The classic example is average income-using the arithmetic mean makes it appear to be much higher than is in fact the case. Consider the scores (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.16, but five out of six scores are below this!
The arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (The word set is used perhaps somewhat loosely; for example, the number 3.8 could occur more than once in such a “set” ). The arithmetic mean is what pupils are taught very early to call the “average.” If the set is a statistical population, then we speak of the population mean. If the set is a statistical sample, we call the resulting statistic a sample mean. The mean may be conceived of as an estimate of the median. When the mean is not an accurate estimate of the median, the set of numbers, or frequency distribution, is said to be skewed.
We denote the set of data by X = (x1, x2, … xn). The symbol μ (Greek: Mu) is used to denote the arithmetic mean of a population. We use the name of the variable, X, with a horizontal bar over it as the symbol ( “X bar” ) for a sample mean. Both are computed in the same way:
The arithmetic mean is greatly influenced by outliers. In certain situations, the arithmetic mean is the wrong concept of “average” altogether. For example, if a stock rose 10% in the first year, 30% in the second year and fell 10% in the third year, then it would be incorrect to report its “average” increase per year over this three year period as the arithmetic mean (10% + 30% + (-10%) )/3 = 10%; the correct average in this case is the geometric mean which yields an average increase per year of only 8.8%.
The geometric mean is an average which is useful for sets of numbers which are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth.
The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members. In a formula: The geometric mean of a1, a2, … an is, which is. The geometric mean is useful to determine “average factors” For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85) ⅓ = 1.0391… And we conclude that the stock rose on average 3.91 percent per year. The geometric mean of a data set is always smaller than or equal to the set's arithmetic mean (the two means are equal if and only if all members of the data set are equal). This allows the definition of the arithmetic-geometric mean, a mixture of the two which always lies in between. The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (an) and (hn) are defined:
and Then an and hn will converge to the geometric mean of x and y.
The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).
In mathematics, the harmonic mean is one of several methods of calculating an average.
The harmonic mean of the positive real numbers a1, … an is defined to be
The harmonic mean is never larger than the geometric mean or the arithmetic mean (see generalized mean). In certain situations, the harmonic mean provides the correct notion of “average” For instance, if for half the distance of a trip you travel at 40 miles per hour and for the other half of the distance you travel at 60 miles per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you traveled the entire trip at 48 miles per hour. Similarly, if in an electrical circuit you have two resistors connected in parallel, one with 40 ohms and the other with 60 ohms, then the average resistance of the two resistors is 48 ohms; that is, the total resistance of the circuit is the same as it would be if each of the two resistors were replaced by a 48-ohm resistor (Note: This is not to be confused with their equivalent resistance, 24 ohm, which is the resistance needed for a single resistor to replace the wo resistors at once.). Typically, the harmonic mean is appropriate for situations when the average of rates is desired.
Another formula for the harmonic mean of two numbers is to multiply the two numbers, and divide that quantity by the arithmetic mean of the two numbers. In mathematical terms:
Merits and limitation of geometric mean
- It is based on each and every item of the series.
- It is rigidly defined
- It is useful in averaging ratio and percentage in determining rates of increase or decrease.
- it gives less weight to large items and more to small items. Thus geometric mean of the geometric of values is always less than their arithmetic mean.
- It is capable of algebraic manipulation like computing the grand geometric mean of the geometric mean of different sets of values.
- It is relatively difficult to comprehend, compute and interpret.
- A GM with zero value cannot be compounded with similar other non-zero values with negative sign