Education

# Introduction To Geometric And Harmonic Mean

Geometric and Harmonic mean! The inference of the data set given by using a single value that reflects the center of the distribution of data is called central tendency. The different measures of central tendency involve mean median and mode. Mean can be called the most used measure of central tendency. It can be applied to both discrete and continuous data sets. The ratio of the sum of the observations by the number of observations is defined as the value of x {bar} (mean).

x {bar} (mean) = [x1 + x2 + …… + xn] / n

x {bar} = ∑x / n

The above formula is for the sample mean. The population means is denoted by “mu”. Its formula is given by

𝛍 = ∑x / n

The value of the arithmetic mean represents a model of the given data set. One of the important properties of the mean is that it reduces the error of prediction of any value in the set of data. Every data point is given the same importance while calculating the value of the mean. The sum of the deviations of each value from the mean is 0. The value of the mean is affected by the outliers. The mean is inappropriate for the distributions that are skewed. It can’t be calculated for nominal data. The different types of mean include the weighted mean, geometric mean (GM), and harmonic mean (HM).

Weighted mean: It is used when certain values of the dataset are considered to be more important when compared to others. A weight is associated with each value of the data set which symbolizes its importance.

Weighted mean =  ∑wx / ∑w

= summation

w = the weights

x = the data value

Weighted means are often used in the study of populations.

Geometric mean: It is the nth root of the product of the given observations. The geometric mean is a suitable measure when the data value grows exponentially and if in case of a skewed distribution, a log transformation can be used to convert into a symmetrical distribution.

Geometric mean = (x1 . x2 ….. xn)1/n

log (geometric mean) = ∑ (log x) / n

It finds its application in the fields of microbiology and serology. It cannot be found if one of the values is 0 or negative in nature. The geometric mean is used as a filter for noise in the case of image processing. In order to choose a compromise aspect ratio for film and videos, the concept of geometric means comes into the picture as it provides a compromise between them.

Harmonic mean: It is defined as the reciprocal of the arithmetic mean of the given data values. Harmonic mean can also be explained as the reciprocal of the arithmetic mean of the individual observations. It is mostly used when the reciprocals of values are more desired.

Harmonic mean = 1 / [∑ (1 / x) / n] = n / ∑ (1 / x)

It is commonly used to find the average sample size involving a number of groups, out of which each is of a different sample size.

If the values of the data set are the same, then the 3 means will be the same. As the data gets more variable, the difference among the means also increases.

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