Arithmetic mean(introduction, example)

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 What is Arithmetic mean

The term "arithmetic mean" is also used to refer to the mean or arithmetic average. It is determined by adding up each number in a set of data, then dividing that total by the number of elements in the set. The middle number is the arithmetic mean (AM) for equally dispersed numbers. Additionally, various approaches are used to calculate the AM, which depends on the volume and distribution of the information. The symbol for it is x̄.

The simple average, or sum of a number sequence divided by the count of that sequence of numbers, is what is referred to as the arithmetic mean.

In the area of finance, the mean is typically not a good way to determine an average, particularly when one outlier can significantly skew the mean.

The geometric and harmonic means are two further averages with more financial use.

Example

Let's talk about a situation where arithmetic means is used. Since 15 is the result of 15 divided by 3 (there are three numbers), the mean of the number system 5, 3, and 7 is 5.

Formula:

Mean (x̄) = Sum of total observations / Number of observations

Different formats can be used to present data. For instance, when we receive raw data, such as a student's marks across six topics, we add the six marks and divide the result by six because there are six topics altogether.



Arithmetic Mean Characteristic


Let's examine some of the key characteristics of the arithmetic mean. If x is the arithmetic mean of the n samples defined by the letters x1, x2, x3,..., xn, then:

  • If the provided data set's observations all have a value, let's say "m," then the data set's arithmetic mean will also be "m."
  • A set of observations' total algebraic deviations above the arithmetic mean is equal to zero. (x1x) + (x2xx) + (x3xx) +... + (xnxx) = 0. The discrete data formula is (xix) = 0. f(xix) = 0 for grouped frequency and percentage.
  • The mean will change by the very same amount whether all data values grow or decrease by a constant amount. If x1, x2, x3,......xn have a mean of X, then x1+k, x2+k,......xn+k will have a mean of X+k.
  • The mean is multiplied or divided by the identical value if each value in the data is multiplied or divided by a constant number. If x1, x2, x3,......xn has a mean of X, then kx1, kx2,......xn+k has a mean of kX. Similar to this, Xk will be the mean of x1/k, x2/k, x3/k,... xn/k.
  • The total amount of the items' departures from their arithmetic mean is always zero, or (x - X) = 0.
  • The minimum sum of squared deviations of the elements from the arithmetic mean (A.M.) is less than the maximum sum of squared deviations of the items of any other values.
  • The sum of these changes will be identical to the sum of the individual items if the mean is used to replace each element in the arithmetic series.

Making Arithmetic Mean Calculations for Ungrouped Data

Mean x̄ = Sum of observations / Number of total observations

Calculate the first four even natural integers' arithmetic mean.

2 4 6 8

2+4+6+8/4= 20/4= 5

Arithmetic mean=5

Reasons to Use Arithmetic Mean

The arithmetic mean has applications outside statistics and mathematics, including experimental research, finance, psychology, and other interdisciplinary fields of study. Some of the arithmetic mean's main benefits are enumerated below.

  • The arithmetic mean calculation formula is rigorous, hence the outcome is constant. It is unaffected by the location of the value within the data set in contrast to the median.
  • Every value in the data set is taken into account.
  • Determining arithmetic mean is fairly easy to do; even a common man with limited financial and mathematical knowledge may do it.
  • Due to its propensity to produce useful results even with vast sets of numbers, it is also a valuable measure of central tendency.
  • In contrast to mode and median, it can be further subjected to a variety of algebraic operations. For instance, the mean of each data series can be used to determine the average of two or more series.
  • Geometry also makes extensive use of the arithmetic mean. For instance, the arithmetic mean of the values of the vertices serves as the "centroid" coordinates of triangles (or any other shape limited by line segments).
  • The arithmetic mean is straightforward to comprehend and simple to compute.
  • It is affected by how much each component in the series is worth.
  • A.M. is firmly established.
  • It can be subjected to more algebraic processing.
  • It is a sample mean rather than one based on a series' location.

Limitations of Arithmetic mean


Let's now examine a few drawbacks or shortcomings of applying the arithmetic mean.
  • The arithmetic mean's greatest flaw is that it is impacted by the high values in the data.
  • The number of means cannot be calculated in distribution with open-end classes without making assumptions about the class's size.
  • Extreme goods, such as extremely small and very huge items, alter it.
  • Rarely can it be determined through a visual inspection.
  • A.M. doesn't always show the original item. As an illustration, a hospital typically admits 10.7 patients every day.
  • Extremely asymmetrical distributions do not lend themselves to the arithmetic mean.







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