[**PEREZGONZALEZ Jose D [ed] (2012).** *Arithmetic mean.* Journal of Knowledge Advancement & Integration (ISSN 1177-4576), 2012, pages 247-248.] [Printer friendly]

# The arithmetic mean

In statistics, the **arithmetic mean**, or simply the **mean**, is the value which is the most equidistant from all other values in a dataset. That is, when all values are considered in relation to each other, the mean is the value that represents the minimum possible distance among all of them. Thus, the mean is a measure of central tendency, and it is also the proper average when doing statistical analyses at interval or ratio levels.

## Interpreting a mean

The mean may be annotated as 'M', 'X', a bar 'X' or Greek 'M'. The mean is interpreted as the average value in a dataset, although, literally, it is the calculated value which is the most equidistant from all others in the dataset. Therefore, the mean does not need to be one of the "real" values in the sample. The mean is, simply, the result of a formula. This is why you can read things like the mean population growth of a country is 1.3 children per couple per year (even when nobody can actually have neither 1.3 of a child nor a full child and 1/3 of a second child).

For example, imaging that you read a report about how many times a person has flown to a particular destination and what was the average airfare she paid. Let's say, she flew 3 times and the mean price she paid was $150. What this 'mean' tells you is that, on average, each flight cost $150, not that the person necessarily paid $150 for all or any of those flights (eg, she may have paid $98, $212 and $140, respectively).

When used in a quasi-inferential manner, the mean informs of the most equidistant value in a future dataset.

For example, imaging that you need to fly to the same destination above in the near future and you want to save money for the ticket. Yet, you need to administer your money well: you don't want to save too little and find yourself with no money for the airfare, nor you want to save to much and go hungry. How much money should you save? If you have no other information than your past experience, then you can consider the mean as the most accurate value you can rely on and, thus, expect your future airfare to be $150. This is the amount of money that represents the best compromise for your financial goals, and, thus, the amount that you should save for your future ticket.

## Properties of the mean

- The mean is, probably, the most popular measure of central tendency, and it is used often, even when better measures are available. This is why you often find results such as that a population growth is 1.3 children per couple per year. We may understand the meaning but the interpretation is, nonetheless, unrealistic.
- Because the mean is the most equidistant value to all other values in the dataset, it is highly sensitive to extreme values when the dataset is skewed, to the point of results becoming ridiculous.
- One handy tip to remember is that when datasets are normally distributed, the mean equals both the mode and the median. Thus, if you are given both the median and the mean, you can assess whether the distribution of scores is skewed or not (thus, whether it is normal on that account). A great difference between both measures in either direction (ie, a mean which is either sensibly smaller or bigger than the median) indicates that the distribution is skewed and, thus, non-normal. In such cases, the median may be a better indicator of central tendency.

For example, imagine that you read a report regarding the average pay at two companies, each with 5 employees. In company A, the salaries are $4000, $2000, $3000, $3000 and 3000; mean = $3000, median = $3000. In company B, the salaries are $39000, $2000, $3000, $3000 and 3000; mean = $10000, median = $3000. As you can observe, the extreme salary of $39000 in company B has such an effect that the mean is increased by $7,000 for the remaining employees, even when none of them earns that much. Given such big difference between the mean and the median in company B, the median is probably the best measure of central tendency in such case.

- A good measure of dispersion around the mean is the standard deviation.
- A good graphic for the mean is the histogram.

## Editor

**Jose D PEREZGONZALEZ (2012).** Massey University, Turitea Campus, Private Bag 11-222, Palmerston North 4442, New Zealand. (JDPerezgonzalez).

# Want to know more?

**Stats4Students.com**- This page contains an article explaining more in detail measures of central tendency. The article is, LUND RESEARCH (2007).
*Measures of central tendency.*Stats4Students.com. **Wiki of Science - Descriptive statistics**- This Wiki of Science page provides you access to more descriptive statistics.
**Wiki of Science - Mean (calculation)**- This Wiki of Science page provides you with the tools to calculate the arithmetic mean.

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