Arithmetic Mean Formula

Arithmetic Mean in Statistics

We are able to create a statistical summary of the massive organised data thanks to the measures of central tendency. The arithmetic mean is one such statistical measure of central tendency. Measures of central tendency are used to condense enormous amounts of data into a single number.

You noticed the daily temperature reports while reading the newspaper in the morning, for instance. Even though the temperature changes throughout the day, how can a single reading represent the day’s conditions? Or, when you receive your exam scorecard, your performance is evaluated based on the aggregate percentage rather than the percentage in each subject.

You are able to create a statistical summary of the massive organised data thanks to the measures of central tendency. The arithmetic mean is one such statistical measure of central tendency. Measures of central tendency are used to condense enormous amounts of data into a single number.

Have you ever noticed the daily temperature reports while reading the newspaper in the morning, for instance. Even though the temperature changes throughout the day, how can a single reading represent the day’s conditions? Or, when you receive your exam scorecard, your performance is evaluated based on the aggregate percentage rather than the percentage in each subject.

What is Arithmetic Mean in Statistics?

The arithmetic mean is the most widely used indicator of central tendency. The mean of data, to put it simply, denotes the average of the specified collection of data.  It is determined by dividing the total number of values by the sum of all the values in the data set.

The mean of the data is given as for n values in a set of data, specifically as x1, x2, x3,… xn:

It may also said to mean:

x1, x2, x3,… xn are the recorded observations, and f1, f2, f3,… fn are the corresponding frequencies of the observations when the frequency of the observations is supplied, for the purpose of calculating the mean;

When the data is ungrouped in nature, the arithmetic mean is determined using the procedure described above. We determine the class mark before calculating the mean of the grouped data.  For this, the class interval midpoints are determined as follows:

The mean is computed following the calculation of the class grade, as was previously stated. The direct technique is a name for this approach of figuring out the mean.

Mean Definition in Statistics

Given what we now know about the arithmetic mean, let’s examine what the statistical term “mean” actually means.

The only thing a data set’s mean is is the average of its values.

Mean is calculated as the product of all the values and their sum.

The average of the sample is typically used to define the mean, whereas the average is just the sum of all the values divided by the total number of values. But rationally, the mean and average are equivalent.

Find the mean of the following values, for instance: 2, 3, 4, 5, 6,

Mean = (2+3+4+5+6+6)/6 = 26/6 = 13/3

Question 1

There are 8 students who took a class test and obtained marks as follows: 15, 27, 14, 31, 24, 23, 16, and 18. We need to calculate the arithmetic mean of the marks obtained by the students.

Solution:

The arithmetic mean can be calculated using the formula:

Arithmetic mean = {Sum of Observation} ÷ {Total numbers of Observations}

Arithmetic mean = (15 + 27 + 14 + 31 + 24 + 23 + 16 + 18) ÷ 8

= 21.125

Question 2

The heights of five students are 143 cm, 151 cm, 162 cm, 169 cm, and 175 cm respectively. We need to find the mean height of the students.

Solution:

The mean height can be calculated using the formula:

Mean height = {Sum of Observation} / {Total numbers of Observations}

Mean height = (143 + 151 + 162 + 169 + 175) / 5

= 800 / 5

= 160 cm

Question 3

The mean monthly salary of 5 workers in a group is $1260. One more worker whose monthly salary is $1450 has joined the group. We need to find the arithmetic mean of the monthly salary of 6 workers in the group.

Solution:

Given n = 5 and x̄ = 1260. Using the formula for arithmetic mean, we get:

x̄ = ∑xi / n

∴∑xi = x̄ × n

∑xi = 1260 × 5 = 6300

The total salary of 5 workers in the group is $6300. After the new worker joined, the total salary of 6 workers in the group becomes $6300 + $1450 = $7750.

Hence, the arithmetic mean of the monthly salary of 6 workers in the group is:

Arithmetic mean = Total salary of 6 workers / Number of workers

Arithmetic mean = $7750 / 6

= $1291.67 (rounded off to two decimal places)