Mode Calculation

# The mode

In statistics, the mode is the most frequent value in a distribution of scores. The mode can be considered as a frequency statistic which identifies the most typical value in a distribution, and as a suitable average when doing statistical analysis at nominal levels.

## Interpreting the mode

There seems to be no standard notation for the mode, thus it may be annotated as 'mode', 'Mo', etc. In any case, the mode is interpreted as the "most frequent value" or "typical value" in a distribution of scores, or as the "average value" of a nominal or categorical variable2.

For example, imaging that you are going to visit a primary school. You want to bring pens as presents for the students. However, you can only bring either blue or pink pens. You want to play safe and bring the color that would make more children happy following the convention that pink is for girls and blue is for boys. If you only knew the most typical (ie frequent) gender in the school, then you could be reassured in your choice of pen color. For example, if you knew that there were more girls than boys in that school, then you should bring pink pens. In this example, 'girls' is the mode (or most frequent value) for gender in that school. Notice that the mode tells you which is the most frequent value, not necessarily how much more frequent it is. Maybe only 51% of the children are girls, maybe 99% of them are girls. Yet, for the purpose of your pen choice strategy, the mode provides you with enough information as for making a good decision.

When used in a quasi-inferential manner, the mode informs of the most likely value in a future dataset (eg a sample or population).

For example, imaging that you are going to above school to chair an academic competition among students, with the winner receiving either a pink or a blue "magic" pen as price. You can only bring one pen. If you know that the modal gender for that school is "girls"3, then you also know that a girl will be the likely winner (simply because there are more girls than boys). Thus, you are better off bringing a pink pen as price item instead of a blue pen.

The mode is not unique. This means that a particular dataset may have more than one mode (ie, equal number of scores in two or more values). Distributions with two modes are called bimodal. Distributions with more than two modes are called multimodal.

For example, imaging that you go to a car showroom and you see that two car colors, such as red and black, are the most typical, and equally so, in the exhibition. Then, you can say that the (variable) color of the (population of) cars in that showroom the day of your visit was bimodal, with red and black being equally representative colors.

## Properties

• The mode is a statistical technology for identifying the most frequent value in a distribution. It is based on a procedure (counting score frequencies per value), not on a mathematical formula.
• The mode is a descriptive statistic which identifies the most frequent value (or values) although not its actual frequency.
• The mode can be used with all type of variables (nominal, ordinal, interval and ratio) as a summary statistic for frequency.
• The mode may also be used as a measure of central tendency. The mode is particularly useful for identifying the average for nominal variables such as gender, color, geographical location, etc. In fact, the mode should be the preferred central tendency statistic for describing nominal variables.
• The mode needs to be interpreted with care when using it with ordinal and interval variables. Using the mode as a measure of central tendency for these variables may be misleading (see, for example, Lund Research, 20071). In those cases, it is better to use the mode as a summary statistic for frequency, instead.
• When samples are normally distributed, the mode equals both the median and the mean. Otherwise, it may differ from those two statistics, sometimes greatly.
• The scarce information provided by the mode could be improved if it were accompanied by a measure of dispersion, such as the modal dispersion.
References
1. LUND RESEARCH (2007). Measures of central tendency. Retrieved from Stats4Students.com on 15 May 2010.
+++ Footnotes +++
2. When scores are independent, thus coming from different individuals, we could also say that the mode actually represents the value with the most cases (eg, with the most individuals in a sample or the population). This is not the case if scores are not independent, of course.
3. And, or course, the competition is not going to give an advantage to either gender.

# Want to know more?

Wikipedia - Mode
Khan Academy - Mean, median and mode
Khan teaches about central tendency measures (mean, median and mode) in this video.
RobertNiles.com - Median