Mean of Grouped Data: Direct, Assumed Mean and Step-deviation Methods

Mean is the calculated central value of a group of numbers; the average of the provided values. Simply put, it is the average of a group of variables. Apart from the mode and median, the mean is one of the Measures of Central Tendency in statistics. However, the central value of a set of data or observations is defined by all three metrics (mean, median, and mode).

Mean = (Sum of all the observations/Total number of observations)

Keyterms: Mean, Central Tendency, median, mode, numbers, data, variables, metrics, statistics, probability

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What is Mean?

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The average of a group of values is known as the mean. It refers to a data set's values being distributed evenly. The statistical metric that recognises a single value as typical of the whole distribution is known as central tendency. It makes an effort to offer a complete explanation of the data. It is the one-of-a-kind value that reflects the collected data. The three most popular measures of central tendency are the mean, median, and mode. The mean of a discrete probability distribution of a random variable X is the total of all possible values weighted by the likelihood of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x) and then adding all of these products together.

- Mean Symbol

The symbol ‘’ is commonly used to represent the mean. The mean of x number of values is shown by the bar above the letter x.

= (Sum of values ÷ Number of values)

= (x₁ + x₂ + x₃ +….+x_n)/n

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Mathematical Mean

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The average value of the provided numbers or data is known as the mean in mathematics. To find the mean, add all of the total values in a datasheet and divide the total by the total number of values. Assume that the pricing values of ten different garment fabrics are listed in a data table. If we need to calculate the average price, add the prices of each garment material and divide the total by 10. The outcome will be an average value. Another example is determining the average age of students in a class by adding the ages of individual students in the class and then dividing the result by the total number of students in the class.

In statistics, the mean is a regularly used approach. We taught this concept in primary school using the phrase "average." When we are exposed to the concept mean in upper grades, it relates to a more complex form of sequence or series of a number. When dealing with large amounts of data in the real world, we employ statistics. We also learn about median and mode in addition to mean. When all the values are organised in ascending order, the median is the middle value of the data. The number in the list that is repeated the most times is known as mode.

Formula for Mean: The basic formula for calculating the mean is determined by the data collection. When calculating the mean, each term in the data set is taken into account. The ratio of the sum of all terms to the total number of terms is the standard formula for mean. As a result, we may say; 

Mean = Sum of the Given Data/Total number of Data

To find the arithmetic mean of a set of data, add (sum) all of the data values (x), then divide the result by the number of values (n). We get the following formula for the mean (x): \(\bar{X}\) =∑ x/n

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How to Calculate Mean?

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We can apply the average to calculate the mean of any given data collection. The example below will demonstrate how to calculate the mean of a set of data. 

For example, in a class there are 20 students and they have secured a percentage of: 88,82,88,85,84,80,81,82,83,85,84,74,75,76,89,90,89,80,82,83. Calculate the average of the class's percentages.

Solution: Average = Total of percentage obtained by 20 students in class/Total number of students

Avg = [88+82+88+85+84+80+81+82+83+85+84+74+75+76+89+90+89+80+82+83]/20

Avg.=1660/20 = 83

As a result, each student's average percentage in class is 83 percent. In statistics, we find the mean of a given data collection in the same way.

Mean of Negative Numbers

Until now, we've only seen examples of determining the mean of positive numbers. But what if any of the integers in the observation list are negative? Let's have a look at an example.

Example: Find the mean of 9, 6, -3, 2, -7, 1.

Add all the numbers first:

Total: 9+6+(-3)+2+(-7)+1 = 9+6-3+2-7+1 = 8

Now divide the total from 6, to get the mean.

Mean = 8/6 = 1.33

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Types of Mean

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Arithmetic Mean

Arithmetic Mean is the result of adding all the values and dividing by the number of values. To calculate, add all of the supplied numbers together and divide by the number of numbers supplied.

Example: What is the mean of 3, 5, 9, 5, 7, 2?

Now add up all the given numbers:

3 + 5 + 9 + 5 + 7 + 2 = 31

Now divide by how many numbers provided in the sequence:

316= 5.16 is the answer

Geometric Mean

y is the geometric mean of the two integers x and y. The geometric mean of three integers, x, y, and z, is 3xyz. 

Example: Find the geometric mean of 4 and 3?

Geometric Mean =sqrt{4 \times 3} = 2 \sqrt{3} = 3.46

Harmonic Mean

To average ratios, the harmonic mean is utilised. The harmonic mean of two integers x and y is 2xy(x+y). The harmonic mean of three integers x, y, and z is 3xyz(xy+xz+yz).

Root Mean Square (Quadratic)

Many technical and statistical applications employ the root mean square, especially when there are data points that might be negative.

Contra-harmonic Mean

The contra-harmonic mean of x and y is (x² + y²)/(x + y). For n values,