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# Arithmetic Mean – Definition, Formula, Properties

Arithmetic Mean, also known as the mean or average. It is a statistical measure of central tendency that is calculated by adding up all the values in a set of data and dividing by the total number of values. The arithmetic mean is a widely used statistical tool for summarizing a set of data by giving a typical value that represents the central tendency of the data.

## Arithmetic Mean Formula

The formula for calculating the arithmetic mean is:

Mean = (x1 + x2 + x3 + … + xn) / n

where x1, x2, x3, …, xn are the values in the data set, and n is the total number of values in the data set.

## Properties of Arithmetic Mean

If a value ‘c’ is added or subtracted from every value in a data set, the arithmetic mean is also added or subtracted by the same value.

Mathematically, the formula for additive property is:

Mean(x + c) = Mean(x) + c

Mean(x – c) = Mean(x) – c

where x is the original data set and c is a constant value.

1. ##### Distributive Property:

If a value ‘c’ is multiplied or divided by a constant, the arithmetic mean is also multiplied or divided by the same constant.

Mathematically, the formula for distributive property is:

Mean(cx) = c * Mean(x)

where x is the original data set and c is a constant value.

1. ##### Symmetry Property:

If the data set is symmetric around the mean, the mean will be equal to the median.

Mathematically, the formula for symmetry property is:

Mean(x) = Median(x)

where x is the original data set.

1. Always Exists:

The arithmetic mean always exists for any finite set of data. It is calculated by dividing the total sum of the values by the total number of values in the data set.

Mathematically, the formula for arithmetic mean is:

Mean(x) = (x1 + x2 + x3 + … + xn) / n

where x1, x2, x3, …, xn are the values in the data set, and n is the total number of values in the data set.

1. ##### Sensitive to Outliers:

The arithmetic mean is sensitive to outliers, meaning that extreme values in a data set can greatly affect the value of the mean. In such cases, the median or mode may be better measures of central tendency.

1. Can be Calculated for Continuous Data:

The arithmetic mean can be calculated for continuous data, which is data that is measured on a continuous scale, such as height or weight. It is calculated by dividing the total sum of the values by the total number of values in the data set.

Mathematically, the formula for arithmetic mean for continuous data is:

Mean(x) = ∫ x * f(x) dx / ∫ f(x) dx

where x is the continuous variable, f(x) is the probability density function of x, and the integrals are taken over the entire range of x.

1. ##### Useful in Comparing Data Sets:

The arithmetic mean is useful in comparing different data sets and identifying trends and patterns in the data. It provides a single value that represents the central tendency of the data set.

## Steps to Calculate Arithmetic Mean

To calculate the arithmetic mean of a set of data, follow these steps:

1. Add up all the values in the data set.
2. Count the total number of values in the data set.
3. Divide the sum of all the values by the total number of values.
4. The result is the arithmetic mean of the data set.

1. Simple to Calculate: The arithmetic mean is a simple and easy-to-understand method of calculating the average of a set of data.
2. Useful in Comparing Data Sets: The arithmetic mean is useful in comparing different data sets and identifying trends and patterns in the data.
3. Useful in Making Decisions: The arithmetic mean is useful in making decisions based on the data. For example, a company can use the arithmetic mean to determine the average sales revenue for a particular product.
4. Takes into Account All Values: The arithmetic mean takes into account all the values in the data set, giving an overall picture of the central tendency of the data.

1. Sensitive to Outliers: The arithmetic mean is sensitive to outliers, which are extreme values that are far from the other values in a set of data. Outliers can pull the arithmetic mean towards them, making it an inaccurate representation of the data.
2. Not Suitable for Skewed Data: The arithmetic mean may not be a suitable measure of central tendency for skewed data, where the data is not evenly distributed around the mean.
3. Not Appropriate for Categorical Data: The arithmetic mean is not appropriate for categorical data, where the data is divided into distinct categories rather than measured on a continuous scale.

## Arithmetic Mean Examples

Example 1: Find the arithmetic mean of the following set of data: 3, 7, 11, 15, 19

Solution: The arithmetic mean is calculated by adding all the values in the data set and then dividing by the total number of values in the data set.

Mean = (3 + 7 + 11 + 15 + 19) / 5

Mean = 55 / 5 Mean = 11

Therefore, the arithmetic mean of the data set is 11.

##### Example 2: The scores of a student in five tests are 87, 91, 92, 95, and 98. Find the arithmetic mean of the scores.

Solution: The arithmetic mean is calculated by adding all the scores and then dividing by the total number of scores.

Mean = (87 + 91 + 92 + 95 + 98) / 5

Mean = 463 / 5 Mean = 92.6

Therefore, the arithmetic mean of the scores is 92.6.

##### Example 3: Find the arithmetic mean of the following continuous data set: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Solution: The arithmetic mean for continuous data is calculated by dividing the total area under the probability density function by the total range of the data.

In this case, the data is evenly distributed over the range, so the probability density function is constant.

Mean = (∫ x * f(x) dx) / (∫ f(x) dx)

= (1/10) * [(2 + 4 + 6 + … + 20) / 2]

Mean = (1/10) * [(2 + 20) * 5]

= 11

Therefore, the arithmetic mean of the continuous data set is 11.

## FAQs on Arithmetic Mean

What is Arithmetic Mean?

Arithmetic mean is a measure of central tendency, which is used to find the average of a set of data. It is calculated by adding up all the values in a data set and dividing by the total number of values.

How is the Arithmetic Mean calculated?

The arithmetic mean is calculated by dividing the sum of all values in a data set by the total number of values in the set.

Arithmetic Mean = (sum of all values) / (total number of values)

What are the properties of Arithmetic Mean?

The properties of arithmetic mean include:

• Distributive property
• Symmetry property
• Always exists
• Sensitive to outliers
• Can be calculated for continuous data
• Useful in comparing data sets

What is the difference between mean and average?

The terms mean and average are often used interchangeably. The mean is a type of average, which is calculated by adding up all the values in a data set and dividing by the total number of values.

##### What are the limitations of Arithmetic Mean?

The limitations of arithmetic mean include:

• It is sensitive to outliers.
• It may not accurately represent the entire data set, especially if the data is skewed.
• It may not provide enough information about the distribution of the data.

How can Arithmetic Mean be used in real life?

Arithmetic mean can be used in many real-life situations, such as calculating average test scores, average sales figures, and average temperatures. It can also be used to compare different sets of data and identify trends or patterns.

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