Methods to find Mean

Mean is the average, which is a basic principle of statistics and mathematics. The term refers to a measure of central tendency; it represents a value or a mean that has been taken among the numbers of a specified set. The significance of the mean calculation in areas like economics, finance, and social sciences is very high. The following article will discuss the techniques to find mean. Also, these techniques include the formula of the mean, grouped data, ungrouped data, the different ways to calculate the mean, etc.

Mean can be understood as a statistical measure that is used for the analysis of the datasets and shows the average value of all the numbers that are given. Simply put, it is the sum of all the values divided by the total number of values. The sample mean reflects the widely accepted measure of central tendency for collecting data about a group. It is a very popular tool for studying data from a number of points of view and for generalization. For instance, by the mean score in a set of exam scores, it can be understood how well the students did in the exam.

The formula for calculating the mean is straightforward:

Mean = Sum of all values/Total number of values

This formula is used to calculate the mean for both grouped and ungrouped data.

Grouped data, as such, is a type of data formed of ranges or intervals and values are grouped into them. For example, we can group a set of scores into ranges of 0–50, 51–75, and 86–100. For calculating the mean for grouped data, you should know the frequency of each group and the midpoint of each group. The formula to calculate the mean for grouped data is:

Mean = ∑?_× · ? / ∑?

Where ? is the frequency of each group and ? is the midpoint of each group.

Example: Calculating Mean for Grouped Data

Suppose we have a set of exam scores grouped into ranges of 0-50, 51-75, and 76-100. The frequency of each group is as follows:

The midpoint of each group is:

To calculate the mean, we first multiply the frequency of each group by the midpoint of each group:

Next, we add up these products:

125 + 504 + 616 = 1245

Finally, we divide the sum by the total frequency:

1245 ÷ 20 = 62.25

Ungrouped data is a type of data where each value is unique and not grouped into ranges. There are three methods used for the calculation of the mean for ungrouped data which are:

• Direct Method
• Assumed Mean Method
• Step-Deviation Method

Let’s understand each one by one

Direct Method

The direct method is an easy-and-simple technique which is used to discover the mean for ungrouped data. The aggregate approach in it is that you add up all the individual values of the data set and divide it by the total number of values. The formula for the mean using the direct method is:

Mean = ∑?/?

Where ? represents each individual value in the data set and ? is the total number of values.

Example: Direct Method

Let’s consider a set of numbers: 10, 15, 20, 25, 30. To find the mean using the direct method, we sum up all the values:

10 + 15 + 20 + 25 + 30 =1 00

Since there are 5 values in the data set, we divide the sum by 5:

Mean = 100/5 = 20

Therefore, the mean of the given data set using the direct method is 20.

Assumed Mean Method

The assumed mean method is another approach to finding the mean for ungrouped data. In this method, you assume a value as the mean and calculate deviations from this assumed mean. The formula for the mean using the assumed mean method is:

Mean = Assumed Mean + ∑?_?/?

Where ?_? represents the product of frequency and deviation from the assumed mean, and ? is the total number of values.

Example: Assumed Mean Method

Consider a set of numbers: 12, 15, 18, 21, 24. Let’s assume the mean to be 20. The deviations from the assumed mean are -8, -5, -2, 1, 4 with corresponding frequencies of 2, 3, 4, 2, 1. Using the assumed mean method formula:

Mean = 20 + [(−8×2)+(−5×3)+(−2×4)+(1×2)+(4×1)/5]

Calculating the sum of the products of frequency and deviation:

(−16) + (−15) + (−8) + 2+ 4 = −33

Substitute back into the formula:

Mean = 20 + [−33/5] = 20 − 6.6 = 13.4

Therefore, the mean of the given data set using the assumed mean method is 13.4.

Step-Deviation Method

Step-deviation method is a variation of the assumed mean method that simplifies calculations by using deviations from a convenient value. The formula for the mean using the step-deviation method is:

Mean = Assumed Mean+∑?_? / ? · ℎ

Where ?_? represents the product of frequency and step deviation, ? is the total number of values, and ℎ is the common difference between values.

Example: Step-Deviation Method

Consider a set of numbers: 5, 10, 15, 20, 25. Let’s assume the mean to be 15 and the common difference to be 5. The step deviations are -2, -1, 0, 1, 2 with corresponding frequencies of 3, 4, 2, 1, 2. Using the step-deviation method formula:

Mean = 15 + [(−2×3)+(−1×4)+(0×2)+(1×1)+(2×2)/5×5]

Calculating the sum of the products of frequency and step deviation:

(−6) + (−4) + 0 + 1 + 4 = −5

Substitute back into the formula:

Mean = 15 + [−5/25] = 15 − 0.2 = 14.8

Therefore, the mean of the given data set using the step-deviation method is 14.8.

In mathematics, there are several types of means that are commonly used to represent the central tendency of a set of numbers. These include the Arithmetic Mean, Geometric Mean, Harmonic Mean, and Weighted Mean.

Arithmetic Mean

Arithmetic Mean is the most common type of mean, calculated by summing all the numbers in a dataset and dividing by the total number of values. It is widely used to find the average value of a set of numbers.

Formula: The formula for the Arithmetic Mean is:

Arithmetic Mean=∑?/?

Example: Consider a dataset with values 10, 20, 30, 40, and 50. To find the Arithmetic Mean:

Arithmetic Mean = 10+20+30+40+50/5 = 150/5 = 30

Therefore, the Arithmetic Mean of the dataset is 30.

Weighted Mean

Weighted Mean is a customized mean that considers the relative importance or weight of each value in a dataset. It is used when some values contribute more to the mean than others.

Formula: The formula for the Weighted Mean is:

Weighted Mean = ∑? · ?/∑?​

Where ? represents the weight of each value and ? is the value itself.

Example: Suppose we have values 5, 10, and 15 with corresponding weights 2, 3, and 4. To find the Weighted Mean:

Weighted Mean=(2×5) + (3×10) + (4×15)/2+3+4 = 10+30+60/9 = 100/9 ≈ 11.11

Therefore, the Weighted Mean of the dataset is approximately 11.11.

Geometric Mean

Geometric Mean is the nth root of the product of all values in a dataset. It is commonly used when dealing with growth rates or returns.

Formula: The formula for the Geometric Mean is:

Geometric Mean = (∏?)^1/?

Example: Consider a dataset with values 2, 4, and 8. To find the Geometric Mean:

Geometric Mean = (2×4×8)^1/3 = (64)^1/3 = 4

Therefore, the Geometric Mean of the dataset is 4.

Harmonic Mean

Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals of the dataset. It is suitable for situations where rates or ratios are of interest.

Formula: The formula for the Harmonic Mean is:

Harmonic Mean = ?/∑1/?​

Example: Consider a dataset with values 2, 4, and 8. To find the Harmonic Mean:

Harmonic Mean=3/(1/2+1/4+1/8) = 3/(4+2+1/8) = 3/(7/8) = 24/7 ≈ 3.43

Therefore, the Harmonic Mean of the dataset is approximately 3.43.

The mean is important in real life because it provides a summary of the central tendency of a set of data. Here are some ways the mean is used:

• Economic Analysis: The mean is applied to economic trends, and it helps in understanding the average cost of living.

• Finance: The mean measures up the average return on investment and helps us get a better picture of stocks’ performance.

• Social Sciences: The mean is used to analyze social trends and understand the average income of a population.

Basically mean is a simple concept in both statistics and mathematics. It is one of the most used measures of central tendency from the set of numbers a real value that shows the average. The ability to calculate the mean is a must in many fields, such as economics, finance, and social sciences. Through its use, you can get key insights into the central tendency of a given set of data and this knowledge comes in handy in every conceivable decision-making.

Example 1: Find the mean of following data set: 10, 20, 36, 12, 35, 40, 36, 30, 36, 40

Solution:

Given:

?? = 10,20,36,12,35,40,36,30,36,40

? = 10

Mean = ∑??/? = 10+20+36+12+35+40+36+30+36+40/10 = 295/10 = 29.5

Therefore, the mean of the given data set is 29.5.

Example 2: Find the mean of following data set.

Solution:

Calculating the sum:

∑???? = 500+43+152+84+495+672+812+120=2878

∑?? = 100

Mean = ∑????/∑?? = 2878/100 = 28.78

Therefore, the mean of the given distribution is 28.78.

Example 3: Find the mean of following data set.

Solution:

Assumed mean: 34

Calculating the sum of products:

∑???? = −17

Mean = 34 + (4×(−17)/20) = 34−3.4 = 30.6

Therefore, the mean age of the data using the step-deviation method is 30.6.

Question 1. Data Set: 15, 25, 30, 35, 40, 45. Calculate the mean using the Direct Method.

Question 2. The marks obtained by 8 students in a test are: 75, 80, 85, 90, 95, 100, 105, 110. Calculate the mean using the Assumed Mean Method with an assumed mean of 90.

Question 3. The following data represents the weights (in pounds) of 50 computer towers:

Calculate the mean weight using the Step-deviation Method with an assumed mean of 15 and a common difference of 5.

Related Articles:

• Mean
• Median
• Mode

What is the mean in statistics?

Mean is a measure of central tendency, often referred to as the average, calculated by summing up all values in a dataset and dividing by the number of values.

How do you find the mean?

To find the mean, add up all the values in the dataset and then divide the sum by the total number of values.

When is the mean useful?

Mean is useful when you want to understand the typical value or central tendency of a dataset, especially when the values are relatively evenly distributed.

What are some limitations of using the mean?

Mean can be sensitive to extreme values, known as outliers, which can skew its value, making it less representative of the dataset’s typical value.

Are there alternative methods to find the mean?

Yes, there are alternative measures of central tendency such as the median and mode, which can be used instead of or in conjunction with the mean, depending on the characteristics of the dataset and the research question.