# 12 Measures of Central Tendency

Jenna Lehmann

Central tendency is a statistical measure; a single score to define the center of a distribution. It is also used to find the single score that is most typical or best represents the entire group. No single measure is always best for both purposes. There are three main types:

• Mean: sum of all scores divided by the number of scores in the data, also referred to as the average.
• Median: the midpoint of the scores in a distribution when they are listen in order from smallest to largest. It divides the scores into two groups of equal size. With an even number of scores, you compute the average of the two middle scores.
• Mode: the most frequently occurring number(s) in a data set.

Here is a variety of videos to help you understand the concepts of these measures, finding the median using a histogram, and finding a missing value given the mean.

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There are properties that will change in the mean depending on how scores are modified. When every score has a number added to it, the mean also gets the same number added to it (ex. if the mean is 8 and every score within the distribution as a 3 added to is, the new mean will be 11). When all the numbers are multiplied by a something, the mean is also multiplied by that something (ex. if the mean is 2 and all the numbers in the distribution were multiplied by 3, the new mean would be 6). When only a few scores are greater or lower, the mean value follows with it but it needs to be recalculated.

The following videos detail what happens to the mean and median when increasing the highest value, the impact that removing the lowest value has on the mean and median, and estimating means and medians when given a graph.

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##### Computing Central Tendency Measures

Computing the mean: The mean is pretty straightforward. One should add up all the values and divide that sum by the number of values. For example, if I have a data set of 5 (2, 6, 3, 2, 2), I would add all the numbers up (15) and divide that by 5 to get a mean of 3.

Computing the median: Calculating the median involves lining up all the scores from smallest to biggest. The middle one is the median. If there are an even amount of numbers, the average of the 2 middle numbers is considered the median. Remember that the purpose of a median is to divide the data in half. When working with a discrete frequency distribution, please refer to the first video below. When working with a grouped or continuous frequency distribution, there are extra steps. Please refer to the second video included below.

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Computing the mode: Mode is the most frequent number which comes up. Whatever shows up the most in your frequency table, that’s the mode. There may be more than one mode, so keep this in mind.

Computing weighted means: Overall mean is the sum of all the scores of group one plus the sum of all the scores in group two. All of this is then divided by n1+n2. In some cases you’ll get something like “group 1 consists of 5 people with an average score of 10 and group 2 consists of 8 people with an average score of 7.” In this case you would multiply 5 and 10 and add that to 8 times 7. You would then divide that number by the total number of people to get the weighted mean. Here is a helpful video:

##### Central Tendency and How they Relate to Distribution Shape

The shape of a distribution can help you determine which measure of central tendency is greatest.

• Normal: The mean, median, and mode are all in the same spot
• Bimodal: The mean and median are together in the middle, while the two modes are on either side, represented by the two humps
• Skewed: The mean is going to be closest to the tail, median is between mean and mode (closer to the tail than in a normal distribution, but not as close as the mean), and the mode is found by the hump. This means that a positively skewed distribution will have a mean larger than its median and a median larger than its mode, while a negatively skewed distribution will have a mode larger than its median and a median lager than its mean.

##### When to Use Each Measure

In regards to the mean, no situation precludes it, but it shouldn’t be used when there are extreme scores, skewed distributions, undetermined values, open-ended distributions, ordinal scales, or nominal scales. With the median, it’s appropriate to use when there are extreme scores, skewed distributions, undetermined values, open-ended distributions, or ordinal scales. It is not to be used when there is a nominal scale. The mode is good to use with nominal scales, discrete variables, and in describing shape, but it shouldn’t be used with interval or ratio data, except to accompany the mean or median.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on June 4, 2019.