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Lesson 3 – Measures of Central Tendency

Review Statistics that are used to organize and summarize the information so that the researcher can see what happened during the research study and can also communicate the results to others are called descriptive statistics.

Measures of central tendency are a type of descriptive statistic that uses one score to summarize an entire distribution of scores. In this way a researcher can communicate results of the study using just a few numbers.

Measures of Central Tendency A measure of central of tendency attempts to find one single score that defines the center of a distribution. This single score is the most typical or representative score of the sample or population of interest. Therefore, a measure of central tendency is a way to summarize a large set of numbers using one single score. We can use measure of central tendency to describe a single distribution or compare multiple sets of scores.

There are three measures of central tendency we will learn in this section:  Mean  Median  Mode

We then have to figure out which measure of central tendency best represents a given distribution. You might be thinking this process is rather simple. After all, finding the “center” of a distribution involves just looking at it. Let’s look at 3 frequency distributions below and decide subjectively what the most typical or representative “center” score would be.

Where is the “center” of this distribution?

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Where is the “center” of this distribution?

Where is the “center” of this distribution?

The Mean The mean, sometimes called the average, is the most commonly used measure of central tendency. The mean is essentially the balancing point of a distribution of scores. This means the distance to all scores below the mean equals the distance to all scores above the mean.

The mathematical definition of the mean: the point in a distribution at which the total distance to all the scores above that point equals the total distance to all scores below that point.

Calculating the Mean To calculate the mean, we use the following formulas. There are two formulas, one to calculate the mean of a sample, and one to calculate the mean of a population.

The two formulas are identical mathematically, but the symbols change depending on which type of data set we are describing.

ΣX M = ΣX µ = N n

The Median

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The median is the middle number in a distribution of scores that divides the distribution exactly in half. Median is the preferred measure of central tendency when a distribution has a few extreme scores and when a variable is measured using an ordinal level of measurement. In order to calculate median:

1. Arrange the numbers in the set from smallest to largest. 2. Determine N  If N or n is odd then the median is the middle number.  If N or n is even then the median is the average of the middle two numbers

The Mode The mode is the most frequently occurring value in a set of scores. If there are multiple values “tied” for most frequently occurring, the data set can have more than one mode. If all the values occur at the same rate, then there is no mode.

Choosing the Best Measure of Central Tendency

Measures of Central Tendency and Frequency Distributions A distribution is a graph that shows how scores are distributed along a measurement scale. The mean is the point on the x-axis that falls directly at the “balancing point” for the distribution. The median is the point on the x-axis at which half the area under the distribution curve lies below the median and half lies above the median. The mode is the point on the x-axis that falls directly below the tallest point on the distribution.

In a perfectly symmetrical (normal) distribution all three measures of central tendency are located at the same value. If a distribution is only roughly symmetrical, then the mean, median, and mode will be close to the same value. In this case, we might think that any measure of central tendency would be good enough. Not true!

The mean will inaccurately describe a skewed (non-symmetrical) distribution. You have seen this happen if you’ve ever received one very low grade in a class after receiving many high grades; your average drops like a rock. The one low grade produces a negatively skewed distribution, and the mean gets pulled away from where most of your grades are, toward that low grade. What hurts is then telling someone your average because it’s misleading. It gives the impression that all of your grades are relatively low, even though you have only that one F.

When a distribution is skewed (contains outliers):

 The mean is the most strongly affected and is “pulled” by outliers in the direction of the tail

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 The median is somewhat “pulled” in the direction of the tail

 The mode is not pulled at all

Photo Credit: Fabian van Den Berg; https://www.quora.com/What-does-SKEWED-DISTRIBUTION-mean

All measures of central tendency reflect something about the middle of a distribution; but each of the three most common measures of central tendency represents a different concept:  Mean: “average”  Median: “middle”  Mode: “most common”

Levels of measurement and measures of central tendency The level of measurement of a particular variable will determine which measure(s) of central tendency can be used. For example, the mean uses all the information available in a distribution and the preferred statistics to use with scale level data unless it clearly provides an inaccurate summary of the distribution. The median is preferred measure of central tendency when using ordinal data and the mode is the only appropriate measure of central tendency for nominal level data.

The goal of descriptive statistics is to summarize and organize large amounts of data and measures of central tendency tell us about the middle of a distribution but we need to select the measure that is most representative of the distribution.

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Remember there three things that determine which measure of central tendency to use 1. The concept it is trying to represent 2. Shape of the distribution 3. Level of measurement

Before deciding to report a mean, median or mode ask yourself what the data are trying to convey, what is the shape of the distribution (e.g., normal or skewed) and the level of measurement for the data.

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