Definition
The Arithmetic Mean (often simply referred to as the “mean”) is a statistical measure that represents the average value of a set of numbers. It is calculated by summing all the values in the data set and then dividing this sum by the number of observations. Mathematically, it is expressed as:
\[ \text{Arithmetic Mean} (\overline{x}) = \frac{\sum_{i=1}^n x_i}{n} \]
where:
- \( x_i \) represents each individual value in the sample,
- \( n \) is the number of observations in the sample,
- \(\sum \) denotes the sum of all values.
Examples
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Example 1:
- Dataset: 5, 10, 15, 20, 25
- Calculation: \[ \overline{x} = \frac{5 + 10 + 15 + 20 + 25}{5} = \frac{75}{5} = 15 \]
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Example 2:
- Dataset: 12, 8, 10, 14, 18
- Calculation: \[ \overline{x} = \frac{12 + 8 + 10 + 14 + 18}{5} = \frac{62}{5} = 12.4 \]
Frequently Asked Questions
1. How is the Arithmetic Mean different from Median and Mode?
- The Arithmetic Mean calculates the average of all values. The median is the middle value when the data is ordered, and the mode is the most frequently occurring value.
2. When should I use the Arithmetic Mean?
- Use the Arithmetic Mean when you need to identify the central tendency of a data set with interval or ratio-level data that is symmetrically distributed and without significant outliers.
3. Can the Arithmetic Mean be used for nominal data?
- No, the Arithmetic Mean is not suitable for nominal data because it requires numerical values that can be summed.
4. How do outliers affect the Arithmetic Mean?
- Outliers can significantly skew the Arithmetic Mean, making it higher or lower than the central tendency of the majority of the data.
5. Is Arithmetic Mean the same as Average?
- Yes, in common usage, the terms “Arithmetic Mean” and “Average” are often used interchangeably.
Related Terms
Mean, Geometric
The Geometric Mean is another measure of central tendency, calculated as the nth root of the product of all the values in a data set, useful for data sets with multiplicative relationships.
Median
The Median is the middle value in a data set when it has been ordered from least to greatest, providing a measure of central tendency that is less affected by outliers.
Mode
The Mode is the value that appears most frequently in a data set, useful for categorical data analysis.
Online References
Suggested Books for Further Studies
- “Statistics for Business and Economics” by Paul Newbold, William L. Carlson, and Betty Thorne.
- “The Cartoon Guide to Statistics” by Larry Gonick and Woollcott Smith.
- “Practical Statistics for Data Scientists” by Peter Bruce and Andrew Bruce.
Fundamentals of Arithmetic Mean: Statistics Basics Quiz
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