Calculate the arithmetic mean of 5.7, 6.6, 7.2, 9.3, 6.2

In daily lives, people encounter various situations where dealing with numerical facts or information is required. These facts can pertain to anything, be it a teacher grading students’ papers, data about the ages of people in one’s locality, monthly consumption of groceries by a family for one whole year, etc. It is only natural that a teacher would like to know the class average or the total percentage of students who cleared the test. A person collecting data on age groups might be interested in knowing how many people around him are of the same age group as him. A family which prepares such a chart showing monthly consumption might be interested in comparing it with the previous year or to cut down certain expenses.

All those activities discussed are termed as ‘analysis’. It is imperative to note that they collected all such facts first, organized them in a meaningful pattern, then analyzed it to form an interpretation, and then take necessary action. Hence, one cannot do such an interpretation without collecting numerical facts first. Also, it is naturally not possible to collect the data and expect it to yield some meaningful conclusion. One needs to use some tools or procedures in order to arrive at a conclusion. This is when the concept of statistics comes into the picture.

Statistics

In simple words, statistics implies the process of gathering, sorting, examine, interpret and then present the data in an understandable manner so as to enable one to form an opinion of it and take necessary action, if necessary. Examples:

A teacher collecting students’ marks, organizing them in ascending or descending manner, and calculating the average class marks, or finding the number of students who failed, informing them so that they start working hard.
Government officials collecting data for the census, and comparing it with the previous record to see whether population growth is in control or not.
Analyzing the number of followers of a particular religion of a country.

Statistical Tools

The most popular tools of statistics are as follows,

Arithmetic Mean Also known as average, arithmetic mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations.
Median Such a value as separates the higher and lower values of a given set of statistical data is called the median.
Mode Such value that occurs most frequently in a given series of statistical data is called the mode.
Standard Deviation Such a value as indicates the extent to which certain values of a statistical series tend to vary or disperse from its mean or median is called standard deviation.
Range Such a value depicts the difference between the highest and lowest values in a series.
Correlation Such a statistical tool as helps study the relationship between two variables is called correlation.

Arithmetic Mean

Arithmetic mean also known as average, arithmetic mean for a given set of data is calculated by adding up the numbers in the data and dividing the sum so obtained with the number of observations. It is the most popular method of central tendency.

Formula

The arithmetic mean is calculated using the following formula,

Sum of observation/ Number of observations

Mean of the series = x̄ = Σx/ N.

Properties of Arithmetic Mean

Deviations from the arithmetic mean of all items in a statistical series would always add up to zero, i.e. ∑(x – X) = 0.
The squared deviations from the arithmetic mean is always minimum, i.e., less than the sum of such square deviations from other values like the median, mode, or another tool.
Replacing all the items in a statistical series with its arithmetic mean has no effect on the sum of the said items.

Example

Arithmetic mean of the series: 0, 1, 2, 3, 4, 5.

Here, Σx = 0 + 1 + 2 + 3 + 4 + 5 = 15

n = Number of terms = 6

Hence, A.M. of the given series = Σx/ n = 15/ 6 = 2.5