What is the median for the following set of numbers? 10, 8, 5, 10, 7, 8, 9, 10, 11

Knowing how to find the mean, median, and mode can help you interpret data collected through psychological research. These values provide more insight into what may be considered "normal" or "abnormal" for a specific group of people in terms of cognitive processes or behaviors, for instance.

Because they are all measures of central tendency, psychology students often find it easy to confuse the three. Yet, there are differences in what each one is and how it is found. Here are some useful tips to help you distinguish between these measures, as well as how to calculate mean, median, and mode.

Definition of Mean, Median, and Mode

To understand the differences between the mean, median, and mode, let's start by defining these three terms.

• The mean is the arithmetic average of a set of given numbers. Therefore, the mean in math is often referred to as simply the "average."
• The median is the middle score in a set of given numbers. As the median, half of the scores are above this number and half are below.
• The mode is the most frequently occurring score in a set of given numbers. Put another way, it is the score that appears the greatest number of times.

How to Find the Mean

Take these two steps to calculate the mean:

• Step 1: Add all the scores together
• Step 2: Divide the sum by the number of scores used

As an example, imagine that your psychology experiment returned the following number set: 3, 11, 4, 6, 8, 9, 6. To calculate the mean, you first add all the numbers together (3 + 11 + 4 + 6 + 8 + 9 + 6 = 47). Then you divide the total sum by the number of scores used (47 / 7 = 6.7). In this example, the mean or average of the number set is 6.7.

Recap of How to Find the Mean

The mean is calculated by adding all the scores together, then dividing by the number of scores you added.

How to Find the Median

The median is the middle score in the set. To find the median, start by arranging all of the data points from smallest to largest. In an odd-numbered set, the median will be the number in the very middle of the list. In an even-numbered set, you will need to calculate the average of the two middle numbers. To do this:

• Step 1: Take the two middle numbers of the even-numbered set
• Step 2: Add the two numbers together
• Step 3: Divide the total by 2

As an example, consider this set of numbers: 5, 9, 11, 9, 7. First, you arrange them in numerical order (5, 7, 9, 9, 11). Since you have an odd number of scores, the number in the third position of the data set is the median which, in this case, is 9 (5, 7, 9, 9, 11).

To calculate the median for an even number of scores, imagine that your research revealed this set of data: 2, 5, 1, 4, 2, 7. Your first step is to put them in numerical order (1, 2, 2, 4, 5, 7). The two middle scores are 2 and 4, so you should add them together (2+4=6) and then divide 6 by 2, which equals 3. In this data set, the median score is 3.

Recap of How to Find the Median

The median is calculated by arranging the scores in numerical order, dividing the total number of scores by two, then rounding that number up if using an odd number of scores to get the position of the median or, if using an even number of scores, by averaging the number in that position and the next position.

How to Find the Mode

Of all the measures, finding the mode requires the least amount of mathematical calculation. Instead, since the mode is simply the most frequently occurring score in a distribution, all you do is look at all your scores and select the most common one.

• Step 1: Look at all the data scores
• Step 2: Identify the data score that appears most often

As an example, consider the following number distribution: 2, 3, 6, 3, 7, 5, 1, 2, 3, 9. The mode of these numbers would be 3 since this is the most frequently occurring number (2, 3, 6, 3, 7, 5, 1, 2, 3, 9).

If no number in a set occurs more than once, there is no mode for that set of data. It's also possible for a data set to have two modes. This is known as bi-modal distribution.

Bi-modal distribution occurs when there are two numbers that are tied in frequency. For example, consider the following set of numbers: 13, 17, 20, 20, 21, 23, 23, 26, 29, 30. In this set, both 20 and 23 occur twice (13, 17, 20, 20, 21, 23, 23, 26, 29, 30). Therefore, they are both modes.

Recap of How to Find the Mode

To find the mode, you identify the score that occurs most often within the data set. In cases where you have a large number of scores, creating a frequency distribution can be helpful in determining the mode.

Pros and Cons of Mean, Median, and Mode

Each measure of central tendency has its own strengths and weaknesses. Here are a few to consider.

• The mean utilizes all numbers in a set to express the measure of central tendency. However, outliers—or data that lies well outside of the data set—can distort the overall measure. For example, a couple of extremely high scores can skew the mean, so that the average score appears much higher than most of the scores actually are.
• The median gets rid of outliers or disproportionately high or low scores. At the same time, this could be an issue because it may not adequately represent the full set of numbers.
• The mode may be less influenced by outliers as well and is good at representing what is "typical" for a given group of numbers. But it also may be less useful in cases where no number occurs more than once.

While the mean in math is theoretically neutral, some contend that the use of the mean in psychology can lead to inappropriate conclusions if care is not taken with its application. This is due, in part, to behavior and cognition being both complex and variable in nature.

When to Use Mean, Median, and Mode

How do you determine whether to use the mean, median, or mode when analyzing psychology research? The one you select can depend on the data scores themselves.

If there are no outliers in your data set, the mean may be the best choice in terms of accuracy since it takes into account each individual score and finds the average. Conversely, if outliers exist, the median or mode may be more accurate since the results won't be skewed.

Also consider what you are trying to measure. Are you looking for the average (the mean), do you want to identify the middle score (the median), or are you looking for the score that appears most often (the mode)? While they are all measures of central tendency, each one looks at this tendency from a slightly different point of view.

An Example of Mean, Median, and Mode in Psychology

Imagine a research study in which psychologists are interested in learning the typical age at which someone might be diagnosed with schizophrenia. To collect this data, they send a questionnaire to mental health providers, asking that they share their patients' ages upon formal diagnosis.

The responses received indicate that the practitioners' patients were the following ages:

• 20
• 25
• 35
• 27
• 29
• 27
• 23
• 31

Using the calculations above, you would find that the mean, median, and mode for this data set are all around 27 years (27.1 years, 27 years, and 27 years respectively). In this case, any of these measures could be used to help you arrive at the typical age of onset.

But what if you had an additional score of 13? In this case, the calculation of the mean would be 25.6, while the median and mode would both be 27. Since the mean includes an outlier, median and mode would be more accurate as they aren't skewed by this number.

In case you are curious, the National Alliance on Mental Health reports that the average age of schizophrenia onset for men is late teens to early 20s, while women tend to be diagnosed with this condition in their late 20s to early 30s.

A Word From Verywell

Mean, median, and mode all serve a valuable purpose in analyzing psychological data. They all also have their pros and cons. Knowing how to find mean, median, and mode—as well as their strengths and weaknesses—can help you better interpret data collected via psychology research.

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