Statistics for Modern Life: 05 Mean, Median, & Mode

Robert Ramstetter By Robert Ramstetter, 3rd Jun 2015 | Follow this author | RSS Feed | Short URL
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Statistics is full of terms and formulas, but three make up the backbone. They are Mean, Median, and Mode.

In Statistics, the Three Most Important Terms Are: Mean, Median, and Mode

There are three terms that formulate the backbone of Statistics. They are: mean, median, and mode. While they are not difficult concepts (I will explain them in the next several paragraphs), they can easily be confused by the novice statistician. They will, however, become part of larger and more complex equations, so you will need to completely familiarize yourself with these three terms.
In fig 1, I included some of the symbols that are needed for this lesson. Since mathematical symbols will not upload into the text of this article, I will refer to the terms by their actual names, but the formulas in the pictures will include the symbols.

Mean = Average

The first term is “Mean”. Mean is simply the average of a set of numbers. It may be easier to remember if you think of the two terms together. For instance, you often hear of the term “Average Mean Temperature” when describing the average temperature for a particular month in a particular location. So, when you hear “mean”, think “average”.
If you wanted to find the mean of the following set of numbers, you would find the average. Thus, the mean of: 2,4,6,8 would be (2+4+6+8)/4. You divide the sum (2+4+6+8) by 4 because there are 4 numbers (also called data points) that you are adding together. To find the mean, you first add the numbers together, so 2+4+6+8=20. Now, to find the mean, you divide 20 by 4. So, 20/4=5. The mean of 2,4,6,8 is, therefore, 5.

Median = Middle

The second term is “median”. This is a simple term, but it is often confused with “mean”. Remember, “mean” is average, like the “average mean temperature”. “Median”, on the other hand, designates the number that is in the middle of a group of numbers (data points). When you hear “Median”, think of the median in the road. When a report states that a car crossed the median and struck another, they are saying that it crossed the middle of the road. To put it in numerical terms, the following data points: 1, 2, 3, 4, 5, 8, 11, 18, 100 would have a median of 5 because it is the number that is in the middle of the group. Before the median can be recorded, the group of numbers must be placed in numerical order, otherwise it would fail to make sense. This can easily be accomplished by entering the numbers in a spreadsheet and using the “sort” feature.
To find the median, it is easy for small sets of data points like the example listed above. You just start counting at each end until you reach the middle. For larger groups of numbers, there is a convenient mathematical formula. It is a very easy formula once you try it a time or two.
Median = (+1)/2. In the example listed above, you would have (9+1)/2, which is 10/2 = 5. So, to find your median for the set of numbers listed above, you would start on either side and count over five positions. This gives you 5 as the median.
Notice what you have to do when there is an even number of data points in the next example:
2, 6, 9, 103. You first apply your formula (4+1)2 = 2.5. This means that you count two places in and take the average of the number before and after the median point. In this case, the numbers before and after two and a half places is 6 and 9. So, the median of this group of numbers is (6+9)/2, which is 7.5. You will notice that, for purposes of computing the median, the rest of the numbers beyond the median (or two surrounding if it is an even-numbered set) do not affect the median at all. If you changed the last number to anything larger, even considerably larger, the median would not change. In other words, the median would be the same for the following two sets of numbers:
2, 6, 9, 103
2, 6, 9, 19086
In both cases, the median is 7.5.

Finding the Median in a Large Data Set

Again, for large sets of numbers, you can add them to a spreadsheet and sort them. The cell number will easily help you to find the median, once you do the math. Fig 2 shows how you can use a spreadsheet to find the median for the following groups of data points:
6, 2, 18, 44, 28, 3, 19, 5, 17, 22, 53, 1, 66, 49, 22

The first thing that needs to be done in fig.2 is to sort the data points, which was done in the second column. Now, to find the median, we use the formula. We have 15 data points, so the formula is: (15+1)/2. which is 16/2 = 8. So, our median is the 8th data point, which is 19. (Remember, this is using the second column, which is the one that was sorted.) Take a minute to verify this and confirm that there are seven data points before and after the median.

Mode = Most Often

Mode: So far, we have learned that mean = average, and median = the middle point. The last term is the mode. While I do not have any memorization tips for remembering the mode, it is a relatively simple concept. In a group of numbers, the mode is the number value that occurs the most often. In the following group of data points: 2, 5, 5, 7, 5, 1, 5, 3, 5, the mode is obviously 5. it occurs five times in this example, whereas no other number is even repeated.
However, if there is no number that repeats, such as: 2, 4, 6, 8, 10, 54, the set of data points would have no mode. On the other hand, in the following set: 2, 2, 4, 4, 6, 7, 3, 10, the numbers 2 and 4 both repeat twice. In this case, It could be stated that the set of data points has either two modes 2 and 4), or that it does not have a mode. I would suggest you follow the guidelines set up by either your instructor or your employer and follow their wishes when such a circumstance occurs.

Sum and Standard Deviation

Finally, in figure 1, there are two more symbols that I would like to introduce. Statistics can be overwhelming with all of the terminology that bombards you, so I would like to gently acquaint you with two more topics. We will explore these in greater detail later on. The first is the sum. Simply put, it is the addition of a group of numbers or data points. There are quite a few variations of this, but for now, understand that the upper case Greek letter sigma indicates the sum of a group of numbers. The other is the lower case Greek letter for sigma. This represents “standard deviation”. Again, this will be explored in depth in other articles, but standard deviation represents the amount of variation in a set of numbers or data points.
The following set of numbers has very little standard deviation: 2, 2.8, 2,5 , 2, 2.3.
The next set of numbers has a greater amount of standard deviation: 2, 100, 851, -63.8. Get the idea? There is a mathematical formula, which we will explore in a later lesson, but it is good to understand the concept for now. Remember, when we were configuring the median? As long as the middle number or numbers does not change, the standard deviation will not affect the median, but it will affect the mean.

Outliers, Range, and Recap

Finally, there are two more terms that deal with mean, median, and mode. The first is the outliers. These are an extreme measurement at either the high end or the low end. If, for instance, you were measuring the mean time it takes for horses to complete a race and one takes three minutes just to leave the gate, that horse's time would be an outlier and you may or may not wish to exclude that time. The second is the range. The range is the difference between the smallest and largest numbers in a set of data points.

To recap:
Mean: average
Median: middle
Mode: most common
Outliers: extreme
Range: difference between smallest and largest
Sum: addition of data points
Standard deviation: amount of variation

I hope this helps you to understand these terms. If you study or use statistics, you will need to know these as a foundation for the rest of your education in Statistics. You will see how each of these comes into play when you calculate the more complex formulas.


Math, Mathematics, Mean, Median, Mode, Standard Deviation, Statistics, Sum

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author avatar Robert Ramstetter
Robert Ramstetter is a world traveler and writer of short stories, full length novels, and a vast array of technical articles.

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author avatar FRANCIS IDUMA
13th Jun 2015 (#)

such a good mathematician.

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author avatar Robert Ramstetter
13th Jun 2015 (#)

Thank you. I try to explain it in a way that everybody can understand. That is something that too many mathemeticians fail to do. I hope you benefit from my articles.

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