What measure of central tendency best describes the center of the distribution?

Measures of Central Tendency provide a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution.

There are three main measures of central tendency: the mean, the median and the mode.

When data is normally distributed, the mean, median and mode should be identical, and are all effective in showing the most typical value of a data set.

It's important to look at the dispersion of a data set when interpreting the measures of central tendency.

Mean

The mean of a data set is also known as the average value. It is calculated by dividing the sum of all values in a data set by the number of values.

So in a data set of 1, 2, 3, 4, 5, we would calculate the mean by adding the values (1+2+3+4+5) and dividing by the total number of values (5). Our mean then is 15/5, which equals 3. 

Disadvantages to the mean as a measure of central tendency are that it is highly susceptible to outliers (observations which are markedly distant from the bulk of observations in a data set), and that it is not appropriate to use when the data is skewed, rather than being of a normal distribution.

Median

The median of a data set is the value that is at the middle of a data set arranged from smallest to largest.

In the data set 1, 2, 3, 4, 5, the median is 3. 

In a data set with an even number of observations, the median is calculated by dividing the sum of the two middle values by two. So in: 1, 2, 3, 4, 5, 6, the median is (3+4)/2, which equals 3.5.

The median is appropriate to use with ordinal variables, and with interval variables with a skewed distribution.

Mode

The mode is the most common observation of a data set, or the value in the data set that occurs most frequently.

The mode has several disadvantages. It is possible for two modes to appear in the one data set (e.g. in: 1, 2, 2, 3, 4, 5, 5, both 2 and 5 are the modes).

The mode is an appropriate measure to use with categorical data.

Resources

• Module 22 (Page 28) of this WHO guide provides instruction on the use of measures of central tendency.

• Measures of Central Tendency - Laerd Statistics

  This webpage gives a concise and easy to follow explanation of the differences between the measures of central tendency, and when each is appropriate to use. It covers the mean, median and mode.

A measure of central tendency (measure of center) is a value that attempts to describe a set of data by identifying the central position of the data set (as representative of a "typical" value in the set). We are familiar with measures of central tendency called the mean, median and mode. There is another less popular measure of center called the midrange. The midrange is the average of the maximum and minimum values of the data set, which is actually the midpoint of the range. The midrange is not widely used since it is dependent upon only two values in the set. Consider: The mean of a distribution can also be thought of as a "balance point" for the distribution. The sum of the distances to the right of the mean equals the sum of the distances to the left of the mean. Think of the mean as the fulcrum for a seesaw. The balancing of the seesaw depends upon the number of items and their distances from the fulcrum. Example: Given the data set: {1, 4, 5, 5, 7, 8, 11, 11, 11} Graphically speaking, the center of a distribution is located at the median of the distribution. The median is the point where half of the data points are found on its left side and half on its right side. While the median indicates the "center", it may not always represent the most typical value in the data set. Let's see which measures of center represent the most typical values of the data given various situations. Which measures of center are representative of the most typical values in the data set? Distribution Central Tendency Typical Graph Bell-shaped Symmetric Distribution The mean, mode and median will be the same value. • Best measure of center: mean Distribution Skewed Right Typically has mean > median There may be exceptions to this statement. • Best measure of center: median Distribution Skewed Left Typically has mean < median There may be exceptions to this statement. • Best measure of center: median U-shaped Symmetric Distribution Neither the mean nor the median is a good indicator of typical values in the set. • Best measure of center: midrange - but the mean as a "balance point" is also a descriptor of the center of the distribution Unfortunately, measures of central tendency alone may be insufficient to truly describe the typical data in a set. It is possible that two data sets can have the same mean, but be very different kinds of sets. It is best to use measures of central tendency, along with other observations of the data set, to best describe the data set. NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".

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