Students often find that it is easy to confuse the mean, median, and mode. While all are measures of central tendency, there are important differences in what each one means and how they are calculated. Explore some useful tips to help you distinguish between the mean, median, and mode and learn how to calculate each measure correctly.

### What Do We Mean By Mean, Median, and Mode?

In order to understand the differences between the mean, median, and mode, start by defining the terms.

**The mean**is the arithmetic average of a set of given numbers.**The median**is the middle score in a set of given numbers.**The mode**is the most frequently occurring score in a set of given numbers.

### How to Calculate the Mean

The mean, or average, is calculated by adding up the scores and dividing the total by the number of scores. Consider the following number set: 3, 4, 6, 6, 8, 9, 11. The mean is calculated in the following manner:

- 3 + 4 + 6 + 6 + 8 + 9 + 11 = 47
- 47 / 7 = 6.7
- The mean (average) of the number set is 6.7.

### How to Calculate the Median

The median is the middle score of a distribution. To calculate the median

- Arrange your numbers in numerical order.
- Count how many numbers you have.
- If you have an odd number, divide by 2 and round up to get the position of the median number.
- If you have an even number, divide by 2. Go to the number in that position and average it with the number in the next higher position to get the median.

Consider this set of numbers: 5, 7, 9, 9, 11. Since you have an odd number of scores, the median would be 9. You have five numbers, so you divide 5 by 2 to get 2.5, and round up to 3. The number in the third position is the median.

What happens when you have an even number of scores so there is no single middle score?

Consider this set of numbers: 1, 2, 2, 4, 5, 7. Since there is an even number of scores, you need to take the average of the middle two scores, calculating their mean.

Remember, the mean is calculated by adding the scores together and then dividing by the number of scores you added. In this case, the mean would be 2 + 4 (add the two middle numbers), which equals 6. Then, you take 6 and divide it by 2 (the total number of scores you added together), which equals 3. So, for this example, the median is 3.

### Calculating the Mode

Since the mode is the most frequently occurring score in a distribution, simply select the most common score as your mode. Consider the following number distribution of 2, 3, 6, 3, 7, 5, 1, 2, 3, 9. The mode of these numbers would be 3, since three is the most frequently occurring number. In cases where you have a very large number of scores, creating a frequency distribution can be helpful in determining the mode.

In some number sets, there may actually be two modes. This is known as bi-modal distribution and it 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.

If no number in a set occurs more than once, then there is no mode for that set of data.

### Applications of the Mean, Median or Mode

How do you determine whether to use the mean, median or mode? Each measure of central tendency has its own strengths and weaknesses, so the one you choose to use may depend largely on the unique situation and how you are trying to express your data.

- The mean utilizes all numbers in a set to express the measure of central tendency; however, outliers 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 disproportionately high or low scores, but it may not adequately represent the full set of numbers.
- The mode may be less influenced by outliers and is good at representing what is "typical" for a given group of numbers, but may be less useful in cases where no number occurs more than once.

Imagine a situation where a real estate agent wants a measure of the central tendency of homes she has sold in the last year. She makes a list of all of the totals:

- $75,000
- $75,000
- $150,000
- $155,000
- $165,000
- $203,000
- $750,000
- $755,000

The mean for this group is $291,000, the median is $160,000 and the mode is $75,000. Which would you say is the best measure of central tendency of the set of sales numbers? If she wants the highest number, the mean is clearly the best option even though the total is skewed by the two very high numbers. The mode, however, would not be a good choice because it is disproportionately low and not a good representation of her sales for the year. The median, on the other hand, seems to be a fairly good indicator of the "typical" sales prices of her real estate listings.

Sources:

Hogg RV, McKean JW, Craig AT. *Introduction to Mathematical Statistics*. Boston: Pearson; 2013.

Measures of central tendency. Aerd Statistics. https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php .