Mean

How To Find Mean Value

Before heading to mean, let’s get to know where it arises from and why it is so important to define mean in math. You might already be familiar that it is super complicated to decide and deem from anything unprocessed until the data is arranged properly. For a data arranged properly into a frequency distribution, the details contained in the data could be understood effortlessly. Now the point is where does the mean come from? Mean is one type of average that represents all the values of the distribution in some specific way.


Types of Averages

Most frequently used average are:-

  1. Mean

  2. Median

  3. Mode

  4. Range

When you have a huge set of data there are all sorts of easy ways to mathematically express the data. "Average" is used profusely with data sets. Mean, median, and mode, together with range are all different kinds of averages which help simplify the data for easy description.

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Define Mean In Math

When people say "average" they are generally talking about the mean. And it is commonly the arithmetic mean. Mean, typically in arithmetic math is the "average" number; established by summing up all data points and dividing by the number of data points.


Significance Of Mean In Mathematics And Statistics

Whether you are a math student, researcher, econometrician, statistician, you would invariably need to compute the average of multiple numbers from now and then.

A crucial characteristic of the mean is that it includes every value in your data series as a component of the calculation. More so, the mean is the only measure of central tendency where the total value of the deviations of each observation from the mean is always zero. The statistical mean also provides an effective scheme about simplifying the statistical data.


Calculating The Mean

You can calculate the arithmetic mean by totaling the value of all the numbers in the data and then dividing by the number of data points. To say, mean=sum of data# of data points

Formula to calculate mean written in mathematical expression: Mean = \[\frac{\sum_{x_{i}}^{n}}{n\sum x_{i}}\]​​

Mean = Sum of all Data Points/Number of Data Points

Mean = Assumed Mean + (Sum of all Derivations/Number of Data Points)


Solved Example 

Example Problem Finding Mean

Find the mean of the following data set:

5, 11, 15, 17, 22

Finding the mean:

First add up all the numbers: 5 + 11 + 15 + 17 + 22 = 70

Then divide 70 by the total number of data points i.e. 5, and you get 14. Thus, the mean is 14.


Specific “Means” Commonly Used In Statistics

Statistical mean is a computation of central tendency and provides you a proposition about where the data appears to assemble around. Most common types of mean applied in stats for a number of different experimentation include;-

  • Sample Mean: The average value in a set of distribution sample

  • Population Mean: The average value in a population


Example Problem Of Using Statistical Mean

The mean marks secured by a student Jessica in an examination are needed to accurately measure the performance of a student in that test. If the student obtains a low percentage, but is far ahead of the mean, then it states the exam paper is difficult and thus Jessica’s performance is satisfactory, something that simply a percentage will be unable to deduce.


Mean and its Types in Statistics

There are various other types of mean in statistics that can be used in various branches of math. Most have very narrow applications as well as meanings to domains like physics, finance.

Most common types of mean you’ll come across are as follows:-

  • Weighted mean

  • Geometric mean.

  • Arithmetic-Geometric mean

  • Root-Mean Square mean

  • Harmonic Mean

  • Heronian mean


  1. Weighted Mean

Fairly frequent in statistical application, weighted mean are particularly used when studying populations. Rather of each data point bestowing equally to the final average, some data points impart more than others. If all the weights are equal, then this will make the arithmetic mean equivalent.


  1. Geometric Mean

With ultra limited, and specific uses in finance, social sciences, stock analysis and technology., geometric mean can be used to determine the average rate of return. For example, you own mutual funds that earn 7% the first year, 25% the second year, and 15% the third year. If you seek to find out the average rate of return, you can’t use the arithmetic average. Wondering Why? It is due to the reason that when you are identifying rates of return you are multiplying, not adding.


  1. Arithmetic-Geometric Mean

This is widely used in calculus and in machine computation or computer calculations. It has a relation to the perimeter of an ellipse as because when it was first invented by Gauss, it was used to compute planetary orbits. With arithmetic-geometric, you can find simple and clear explanations of complicated math concepts.


  1. Root-Mean Square

With wide and specific application, this type of mean works quite fruitfully in fields that study electrical engineering like electromagnetism, sine waves. This specific type is also known as the quadratic average.


  1. Harmonic Mean

Fairly used in the field of finance, the harmonic mean is a type of average used to identify the average for financial numerals such as price-to-earnings (P/E) ratio. To figure out the harmonic mean, use the harmonic formula which is dividing the number of observations in the data series by the sum of each value of reciprocals (1/x_i) in the data series.


  1. Heronian Mean

This mean type is used in geometry to determine volume of a pyramidal frustrum. A pyramidal frustrum is a pyramid made by chopping off the top of the pyramid. 


Fun Facts

  • There are some situations where weighted mean can provide inaccurate information, as illustrated by Simpson’s Paradox

  • Harmonic mean provides better average than the arithmetic mean in certain circumstances involving rates and ratios

  • “Median” contains just the same number of letters as “Middle”.