Mean Statistics Formula: Definition, Mean for Grouped and Ungrouped data, Calculation, and Types

Mean Statistics Formula Overview

Mean statistics formulas are fundamental to the study of mathematics and data science. They can tell us a lot about how certain variables interact with each other, and they are essential to understanding both basic and complex relationships between numbers or patterns. In statistics, in addition to the mode and median, the mean is one of the measures of central tendency. Simply put, the mean is the average of the values in the given collection. It indicates that values in a certain data collection are distributed equally. The three most often employed measures of central tendency are the mean, median, and mode. The total values provided in a datasheet must be added, and the sum must be divided by the total number of values in order to get the mean. When all of the values are organized in ascending order, the median is the median value of the provided data. While the number on the list that is repeated the most times is known as the mode. 

Mean Statistics Formula: Definition and Symbol of Mean 

The mean is the average of the given numbers and is computed by dividing the total number of numbers by the sum of the given numbers.

Mean is equal to (Sum of All Observations/Total Observations).

Usually, the letter "x" is used to represent the mean. The bar above the letter x represents the average of X values.

What are Mean Statistics?

Mean statistics is the study of the mathematical properties of numerical data sets. It is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data. In applying statistical methods to a data set, one can often make inferences about the population that the data set represents.

Also Read More About- Mean, Median and Mode

How to calculate mean statistics Formula

In order to calculate mean statistics, you will need to first determine the sum of all the data points. Once you have determined the sum, you will then divide this by the total number of data points. This will give you the mean statistic.

Mean Statistics Formula: What is the mean formula?

The basic formula for calculating the mean is determined by the data collection. When calculating the mean, every term in the data set is taken into account. The standard mean formula is the total number of phrases divided by the sum of all terms. For example, if you roll the dice ten times and get an average roll of 4, your mean score would be 4. If you rolled three times and received a 3, 9, and 12, your mean would be 6 (12 - 3).

The mean is equal to the product of the given data and the total amount of data. We must first add up (sum) all of the data values (x) in order to get the arithmetic mean of a set of data, and then divide the result by the total number of values (n). Since the symbol for summarizing values is (see Sigma Notation), we arrive at the formula shown below for the mean (x):

x̄=∑ x/n

Mean Statistics Formula: How to Locate Mean

The mean is the sum of all the values divided by the number of values. It's also called the average or expected value.

For example, if you roll the dice ten times and get an average roll of 4, your mean score would be 4. If you rolled three times and received a 3, 9, and 12, your mean would be 6 (12 - 3).

Mean statistics formula

In statistics, the mean is the sum of all the values in a data set, divided by the number of values in the data set. The mean can be calculated for numerical data sets and for categorical data sets.

To calculate the mean for a numerical data set, add up all the values in the data set and divide by the number of values in the data set. For example, if you have a data set with five values: 1, 2, 3, 4, and 5, the mean would be (1+2+3+4+5)/5 = 15/5 = 3.

To calculate the mean for a categorical data set, count how many times each category occurs and divide by the total number of values in the data set. For example, if you have a data set with two categories: Category A occurs three times, and Category B occurs seven times, then (3+7)/2 = 10/2 = 5.

Mean Statistics Formula: Mean for Ungrouped Data

The example given below will help you understand how to find the mean of ungrouped data.

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, and 83.

Find the mean percentage obtained by the class.

Solution:

Mean = Total of percentage obtained by 20 students in class/Total number of students

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

= 1660/20

= 83

Hence, the mean percentage of each student in the class is 83%.

Mean Statistics Formula: Mean for Grouped Data

For grouped data, we can find the mean using either of the following formulas.

Direct method:

Mean,x―=∑i=1nfixi∑i=1nfi

Assumed mean method:

Mean,(x―)=a+∑fidi∑fi

Step-deviation method:

Mean,(x―)=a+h∑fiui∑fi

Go through the example given below to understand how to calculate the mean for grouped data.

Example:

Find the mean for the following distribution.

Solution:

For the given data, we can find the mean using the direct method.

Mean = ∑fixi/∑fi = 393/24 = 16.4

Mean Statistics Formula: Mean of Negative Numbers

We have seen examples of finding the mean of positive numbers until now. But what if the numbers on the observation list include negative numbers. Let us understand with an example:

Find the mean of 9, 6, -3, 2, -7, and 1.9+6+(-3)9+6+(-3)  +2+(-7)+1 = 9+6-3+2-7+1 = 8

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

Mean = 8/6 = 1.33

Mean Statistics Formuls: Types of Mean

The main types of mean, including:

1. Arithmetic Mean: This is the most common type of mean, also known as the average. It is calculated by adding all the values in a set and dividing by the total number of values. The arithmetic mean gives a general idea of what a typical value in a set might be. For example, if you have the set of numbers [3, 7, 11, 15], the arithmetic mean would be (3+7+11+15)/4 = 36/4 = 9.
2. Geometric Mean: This type of mean is used when working with sets of numbers that have different units or different scales, and it is calculated by finding the nth root of the product of all the values in the set, where n is the number of values in the set. The geometric mean is often used in finance and economics to calculate the average return on an investment. For example, if you have the set of numbers [2, 4, 8], the geometric mean would be (2 * 4 * 8)^(1/3) = 64^(1/3) = 4.
3. Harmonic Mean: The harmonic mean is used when working with sets of numbers that are reciprocals of each other, such as speed or frequency. It is calculated by dividing the total number of values by the sum of the reciprocals of each value in the set. The harmonic mean is often used to calculate the average speed or rate of something. For example, if you have the set of numbers [2, 4, 6], the harmonic mean would be 3 / (1/2 + 1/4 + 1/6) = 3 / (1.33) = 2.25.
4. Weighted Mean: This type of mean is used when some values in a set are more important or have more weight than others. It is calculated by multiplying each value in the set by its weight and adding up the results, then dividing by the total weight of all the values in the set. For example, if you have the set of numbers [3, 7, 11, 15] and the corresponding weights [1, 2, 3, 4], the weighted mean would be (31 + 72 + 113 + 154)/(1 + 2 + 3 + 4) = 98/10 = 9.8.
5. Root Mean Square (Quadratic): Root Mean Square (RMS), also known as the Quadratic Mean, is a statistical measure of the magnitude of a set of numbers. It is calculated by taking the square root of the average of the squares of the numbers in a dataset. The formula for RMS is given as: RMS = sqrt ((1/n) * ∑(x^2))
6. Contraharmonic Mean: The Contraharmonic Mean is a type of statistical average that is calculated by dividing the sum of the nth powers of the values in a dataset by the sum of the nth powers of the reciprocals of the values in the dataset. The formula for the Contraharmonic Mean is given as: C(n) = n / ((1/n) * ∑(1/x^n)) where n is the power to which the values and their reciprocals are raised, and x is the value of each data point in the dataset.

Mean Statistics Formula: Real-Life Applications of Mean

The average of a set of data can be calculated by adding up all the values in the data and dividing by the total number of values. For example, to find the average price of 10 clothing materials, the sum of all the prices is divided by 10. To find the average age of students in a class, the sum of the ages of all the students is divided by the total number of students in the class.

Mean Statistics Formula: Practice Problems

Q.1: Calculate the mean of the numbers 5, 10, 15, 20, 25.

Q.2: Determine the mean of the data set: 10, 20, 30, 40, 50, 60, 70, 80, 90.

Q.3: Determine the mean of the first 10 even numbers.

Q.4: Determine the mean of the first 10 odd numbers.

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Conclusion

Statistics formulas are essential tools for understanding data and making sense of it. A thorough understanding of mean statistical formulas is required to get the most out of statistical analysis. Thankfully, there are many resources available to help you gain an in-depth understanding of these formulas so that you can confidently use them when working with data sets. Whether you're interested in pursuing a career as a statistician or simply want to improve your skill set, learning more about statistical concepts such as the mean statistics formula is definitely worth your time.