Statistics-Mean

Introduction

In statistics, the mean, median, and mode are the three most often used measures of central tendency. Any data set’s central location may be determined by describing the data set. In statistical terms, this is called the central tendency. Mean refers to the average of the values, which we can also understand as the sum of all values divided by the total number of values in a particular set. It is one of the most commonly used measures of central tendency employed to make a statistical summary of enormous data. A simple example of mean is how the report card focuses on the aggregate marks to simplify the interpretation instead of stating the different marks one has scored in multiple subjects throughout the year. This central tendency is, thus, beneficial in interpreting large value sets to come to valuable conclusions. 

What is Mean?

The sum of all observations divided by the number of observations is the mean of a particular dataset. For example, a cricketer’s five one-day international scores are as follows: 

12,34,45,50,24

With the mean formula, we can compute the mean of all of the data and arrive at his average match score:

The average is calculated as the sum of all observations and the total count of observations. The average is 165/5 = 33, x is the mean of a set of values (pronounced as x bar).

Different kinds of data

It’s possible to have raw data or tabular data. To compare the two, we’ll use the mean.

• Raw data

There are x₁, x₂, x₃, x₄…xₙ observations in this case. The mean formula may be used to calculate the mean.

Mean, x̄ = x₁+x₂+…xₙ/xₙ

Example: Five persons, each measuring 142 cm in height, 156 cm, 150 cm, 153 cm, 149cm. Identify the average height.

The average person’s height, x = (142 + 150 + 149 + 156 + 153)/5= 750/5 = 150. The average height is 150 cm, hence x=150 cm.

Mean Properties

Let us now look at some mean properties to understand the concept better.

1. If all the numbers in a given set have the same value, k, then the mean would also be k. For example: The mean of the five numbers 12, 12, 12, 12, and 12 will be (12+12+12+12+12)/5 = 12. 
2. The algebraic sum of the deviations of a given set from their mean is always zero. It can be stated as (x1−x̄)+(x2−x̄)+(x3−x̄)+…+(xn−x̄) = 0. For ungrouped data, it can be written as ∑(xi−x̄) = 0, and for grouped data, it can be written as ∑fi(xi−x̄) = 0.
3. If each number in a set decreases or increases by the same value, the mean would also decrease or increase by a similar value. Suppose the mean of a set is x1, x2, x3 ……xn is X̄, then x1+k, x2+k, x3 +k ……xn+k will also be X̄+k. 
4. If each number in a set gets multiplied or divided by the same value, then the mean would also be multiplied or divided by a similar value. If the mean of a set  x1, x2, x3 ……xnis X̄, then x1/k, x2/k, x3/k ……xn/k will also be X̄/k. As for division, the fixed value must be a non-zero number since division by 0 does not give a defined number. 

Advantages of mean 

The mean is useful in statistics, mathematics, economics, experimental science, sociology, and other similar disciplines. Here are some benefits of mean: 

1. The formula for finding out the mean is rigid and does not change based on the position of the value in any given set. Unlike median, mean is a more stable and rigid central tendency. 
2. Mean is constituted by considering every value present in any given set. 
3. The formula for calculating the mean is simple. Any person with basic addition and division skills can find out the mean. 
4. Mean provides valuable results irrespective of the size of the data set. It helps in the interpretation of a large value set with ease. 
5. Mean can be used for further mathematical operations, unlike other algebraic expressions like mode and median. 
6. Mean also has applicability in geometry. For instance, the coordinates of the centroid of a triangle are also the mean of the vertex coordinates. 

Disadvantages of mean 

Along with advantages, there are also some disadvantages of mean, such as:

1. One of the main disadvantages of mean is that it gets affected by large values in the data set. For example, if the marks scored by different students in a particular exam are 10, 20, 30, 20, 30, and 90, the mean is (10+20+30+20+30+90)/6 = 33.33 that is majorly affected by 90, an extreme value in the set. 
2. Mean value can solely be used for quantitative data and not qualitative data such as honesty, hard work, etc. 
3. Mean value cannot be calculated even if a single value is unknown since every value impacts the average. 
4. There is no means to locate the mean, either graphically or through inspection. 
5. mean cannot be found out in the case of open-ended classes without making a rough assumption of the class size. 

Conclusion :

There are three central tendency measures: the mean, the median, and the mode. The mean of a collection of data is its arithmetic average. This may be calculated by multiplying the total number of observations in a data collection by the sum of the observations. Mean is an easy and valuable concept helpful in multiple disciplines such as mathematics, statistics, economics, and geometry. It is a useful mathematical operation used in everyday life to find the average and interpret data effectively. Ranging from weather statistics to the average marks secured by the students in particular subjects, all require using the arithmetic mean.