# Mean, Median, Mode: What They Are, How to Find Them

2. How to find the mean, median and mode by hand.
3. Find the mean, median and mode with Technology:

## Overview

Stuck on how to find the mean, median, & mode in statistics?

1. The mean is the average of a data set.
2. The mode is the most common number in a data set.
3. The median is the middle of the set of numbers.

Of the three, the mean is the only one that requires a formula. I like to think of it in the other dictionary sense of the word (as in, it’s mean as opposed to nice!). That’s because, compared to the other two, it’s not as easy to work with.

### Hints to remember the difference

Having trouble remembering the difference between the mean, median and mode? Here’s a couple of hints that can help.

• “A la mode” is a French word that means fashionable and it also refers to a popular way of serving ice cream. So “Mode” is the most popular or fashionable member of a set of numbers. The word MOde is also like MOst.
• The “Mean” requires you do arithmetic (adding all the numbers and dividing) so that’s the “mean” one.
• “Median” has the same number of letters as “Middle”.

Still aren’t sure what the difference is between the three? Watch the video or read the mean, median and mode definitions below for a full explanation of each term.

## Definition

In statistics, the mean is the average of a set of data. In real life, you usually say the “average” of something (e.g. average pay, average height, or average weight), but in stats, we call it a mean. Essentially, they are the same thing. (There is a tiny difference, which you probably don’t care about, but if you do, read the mean vs. average section).

To find the mean, sum all the numbers and then divide by the number of items in the set. For example, to find the mean of the following set of numbers: 21, 23, 24, 26, 28, 29, 30, 31, 33

1. First add them all together:
21 + 23 + 24 + 26 + 28 + 29 + 30 + 31 + 33 = 245
2. Then divide your answer by the number of items in your set. There are 9 numbers, so:
245 / 9 = 27.222

Dividing the sum by the number of items to find the mean.

Note: The word “mean” can have other interpretations outside of statistics. For example, when the weather service reports that a “mean daily temperature” is 75 degrees, that number was obtained by taking the sum of the high daily temperature and the low daily temperature and dividing by 2. This is what is called a “Midrange“. While this can be a cause of confusion, remember that in statistics, the mean is the average.

The mean can be heavily influenced by outliers— numbers that are very small or very large. Take the example of 10, 10, 20, 40, 70. If we add 1500 to the set, the mean becomes 10 + 10 + 20 + 40 + 70 + 1500 / 6 = 275, which is a poor reflection of the center of the set. The median on the other hand is less affected by outliers; the middle of this set is 30 (in between the 20 and 40). In general, if you have outliers, use the median as a measure of central tendency. If you have a large data set with no outliers, use the mean.

## Mean vs Median

Both are measures of where the center of a data set lies, but they are usually different numbers. For example, take this list of numbers: 10,10,20,40,70.

• The mean (average) is found by adding all of the numbers together and dividing by the number of items in the set: 10 + 10 + 20 + 40 + 70 / 5 = 30.
• The median is found by ordering the set from lowest to highest and finding the exact middle. The median is just the middle number: 20.

Sometimes the two will be the same number. For example, the data set 1,2,4,6,7 has an average of 1 + 2 + 4 + 6 + 7 / 5 = 4 and a median (a middle) of 4.

## Mean vs Average: What’s the Difference?

When you first started out in mathematics, you were probably taught that an average was a “middling” amount for a set of numbers. You added up the numbers, divided by the number of items you can and voila! you get the average. For example, the average of 10, 5 and 20 is:
10 + 6 + 20 = 36 / 3 = 12.
The you started studying statistics and all of a sudden the “average” is now called the mean. What happened? The answer is that they are exactly the same word (they are synonyms).

That said, technically, the word mean is short for the arithmetic mean. We use different words in stats, because there are multiple different types of means, and they all do different things.

## Specific “Means” commonly used in Stats

You’ll probably come across these in your stats class. They have very narrow meanings:

## Other Types

There are other types of means, and you’ll use them in various branches of math. Most have very narrow applications to fields like finance or physics; if you’re in elementary statistics you probably won’t work with them.

These are some of the most common types you’ll come across.

1. ## Weighted Mean

These are fairly common in statistics, especially when studying populations. Instead of each data point contributing equally to the final average, some data points contribute more than others. If all the weights are equal, then this will equal the arithmetic mean. There are certain circumstances when this can give incorrect information, as shown by Simpson’s Paradox.

2. ## Harmonic Mean

The harmonic formula.

To find it:

1. Add the reciprocals of the numbers in the set. To find a reciprocal, flip the fraction so that the numerator becomes the denominator and the denominator becomes the numerator. For example, the reciprocal of 6/1 is 1/6.
2. Divide the answer by the number of items in the set.
3. Take the reciprocal of the result.

This is used quite a lot in physics. In some cases involving rates and ratios it gives a better average than the arithmetic mean. You’ll also find uses in geometry, finance and computer science.

3. ## Geometric Mean

This type has has very narrow and specific uses in finance, social sciences and technology. For example, let’s say you own stocks that earn 5% the first year, 20% the second year, and 10% the third year. If you want to know the average rate of return, you can’t use the arithmetic average. Why? Because when you are finding rates of return you are multiplying, not adding. For example, the first year you are multiplying by 1.05.

4. ## Arithmetic-Geometric Mean

This is used mostly in calculus and in machine computation (i.e. as the basic for many computer calculations). It’s related to the perimeter of an ellipse. When it was first developed by Gauss, it was used to calculate planetary orbits. The arithmetic-geometric is (not surprisingly!) a blend of the arithmetic and geometric averages. The math is quite complicated but you can find a relatively simple explanation of the math here.

5. ## Root-Mean Square

It is very useful in fields that study sine waves, like electrical engineering. This particular type is also called the quadratic average. See: Quadratic Mean / Root Mean Square.

6. ## Heronian Mean

Used in geometry to find the volume of a pyramidal frustrum. A pyramidal frustrum is basically a pyramid with the tip sliced off.

## How to Find the Mean

The mean is found by adding the numbers in a data set and then dividing by the number of items. For example, to find the mean of 1, 2, and 3:

1. 1 + 2 + 3 = 6
2. 6 / 3 = 2.

Example problem 1: Find the mean for the following set of numbers: 1, 5, 6, 8, 3, 5, 11, 19, 21, 34.

1 + 5 + 6 + 8 + 3 + 5 + 11 + 19 + 21 + 34 = 113.

Adding the numbers on a calculator is the first step to find the mean.

Step 2: Count the number of items in your set. There are 10 numbers in the set: 1, 5, 6, 8, 3, 5, 11, 19, 21, 34.

Step 3: Divide the number you calculated in Step 1 by the number you found in Step 2.
Mean = 113 / 10.

Dividing the sum by the number of items to find the mean.

The mean for this data set is 11.3.

Example problem 2: Find the mean for this set of numbers: 101, 135, 156, 156.1, 190.5:
101 + 135 + 156 + 156.1 + 190.5 = 738.6.

Step 2: Count the number of items in your set. There are 5 numbers in the set.

Step 3: Divide the number you calculated in Step 1 by the number you found in Step 2.
Mean = 738.6 / 5 = 147.72.
The mean for this data set is 147.72.

That’s it!

## Find the Mean on the TI 89 Graphing Calculator

Watch the video or read the steps below:

Example problem: Find the mean of the following set of data: 12, 23, 78, 98, 121, 342, 88, 7, 27.

Step 1: Get to the HOMEscreen by pressing the HOME button. You can also access the home screen from the APPS menu. Click APPS and then use the arrow keys to scroll to the HOME icon. Press ENTER.

Step 2: Press the CATALOG key (located below the APPS key).

Step 3: Press the ALPHA key (a white key to the left of the Apps key, then the number 5. This is you actually entering the letter ‘M’ which will bring you to the letter “M” in the catalog.

Step 4: Scroll down to “mean(” and press ENTER.

Step 5: Press the 2nd key (a light blue key at the top left) then the ( key. This enters a curly bracket.

Step 6: Enter your data set and separate each number by a comma; the comma key is above the 9 key. In our example, you’ll enter 12, 23, 78, 98, 121, 342, 88, 7, 27.

Step 7: Press the 2nd key, then the ).

Step 8: Press ENTER. Your answer may be given as a fraction: 796/9 (that’s the exact answer). To get a decimal answer, press the ♦ diamond key then ENTER to get 88.4444.

Find more help with statistics on our YouTube channel.

### 2. What is the Mode?

The mode is the most common number in a set. For example, the mode in this set of numbers is 21:
21, 21, 21, 23, 24, 26, 26, 28, 29, 30, 31, 33

### 3. What is the Median?

The median is the middle number in a data set. To find the median, list your data points in ascending order and then find the middle number. The middle number in this set is 28 as there are 4 numbers below it and 4 numbers above:
23, 24, 26, 26, 28, 29, 30, 31, 33

Note: If you have an even set of numbers, average the middle two to find the mean. For example, the mean of this set of numbers is 28.5 (28 + 29 / 2).
23, 24, 26, 26, 28, 29, 30, 31, 33, 34

## How to find the mean, median and mode by hand: Steps

### How to find the mean, median and mode: MODE

• Step 1: Put the numbers in order so that you can clearly see patterns.
For example, lets say we have 2, 19, 44, 44, 44, 51, 56, 78, 86, 99, 99. The mode is the number that appears the most often. In this case: 44, which appears three times.

### How to find the mean, median and mode: MEAN

• Step 2: Add the numbers up to get a total.
Example: 2 +19 + 44 + 44 +44 + 51 + 56 + 78 + 86 + 99 + 99 = 622.  Set this number aside for a moment.
• Step 3: Count the amount of numbers in the series.
In our example (2, 19, 44, 44, 44, 51, 56, 78, 86, 99, 99), we have 11 numbers.
• Step 4: Divide the number you found in step 2 by the number you found in step 3.
In our example: 622 / 11 = 56.5454545. This is the mean, sometimes called the average.

Dividing the sum by the number of items to find the mean.

### How to find the mean, median and mode: MEDIAN

If you had an odd number in step 3, go to step 5. If you had an even number, go to step 6.

• Step 5: Find the number in the middle of the series.
This is the median. 2, 19, 44, 44, 44, 51,56, 78, 86, 99, 99.
• Step 6: Find the middle two numbers.
For example, 1, 2, 5, 6, 7, 8, 12, 15, 16, 17. The median is the number that comes in the middle of those middle two numbers (7 and 8), so that number would be 7.5 in this case. (To do this mathematically, add the two numbers together and divide by 2).

Tip:  You can have more than one mode. For example, the mode of 1, 1, 5, 5, 6, 6 is 1, 5, and 6.

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step solutions, just like this one!

## SPSS Mean mode median

In order to find the SPSS mean mode median, you’ll need to use the Frequency tab. It seems a little counter-intuitive, but the Descriptive Statistics tab does not give you the option to find the mode or the median.

SPSS has a very similar interface to Microsoft Excel. Therefore, if you’ve used Microsoft Excel before, you will quickly adapt to SPSS.

## SPSS Mean Mode Median: Steps

Watch the video or read the steps below:

Sample question: Find the SPSS mean mode median for the following data set: 20,23,35,66,55,66

Step 1: Open SPSS. In the “What would you like to do?” dialog box, click the “type in data” radio button and then click “OK.” A new worksheet will open. Note: If you have opted out of the first help screen, you may not see this option. In that case, just start at Step 2.

Step 2: Type your data into the worksheet. You can type the data into one column or multiple columns if you have multiple data sets. For this example, type 20, 23, 35, 66, 55, 66 into column 1. Do not leave spaces between the data (i.e. don’t leave any empty rows).

Step 2: Click “Analyze,” hover over “Descriptive Statistics” and then click “Frequencies.”

Step 3: Click “Statistics” and then check the boxes “mean”, “mode” and “median.” Click “Continue” twice (select “none” as the chart type in the second window).
Note: In some versions of SPSS, you may only have to click “Continue” once and it may not give you an option for chart type.

The frequency results will appear as output. The top part of the output will display the mean, mode and median.

If you scroll down, the frequency table will also show you the mode. The mode is defined in statistics as the number with the highest frequency (for this sample data set, the number appearing the most is 66, with two results in the frequency column).

## Mean Median Mode TI 89

TI-89

This article and short video explains how to use your calculator to enter a list of data and quickly find the mean, median, and mode on the TI 89. The mean is the average of a data set, the median is the “middle” of the data set (the number that would fall in the middle if you were to write the numbers in order) and the mode is the number that appears most often.

## Mean Median Mode TI 89 Steps

Watch the video or read the steps below:

Sample problem: Find the mean, mode, and median for the following list of numbers: 1, 9, 2, 3, 7, 8, 9, 2.

Step 1: Press APPS then scroll to Stats/List Editor (scroll with the arrow keys at the top right of the keypad). Press ENTER.

Step 2: Clear any data in the list editor by pressing F1 then 8.

Step 3: Press ALPHA 5 ENTER. This names your list “m.” Make sure m appears in the field: if 5 appears, it means the Alpha key didn’t work: try it again. Note: you can name the list anything you like.

Step 4: Enter your numbers, one at a time. Follow each entry by pressing the ENTER key. For our group of numbers, enter
1 ENTER
9 ENTER
2 ENTER
3 ENTER
7 ENTER
8 ENTER
9 ENTER
2 ENTER

Step 5: Press F4, then ENTER (for the 1-var stats screen).

Step 6: Tell the calculator you want stats for the list called “m” by entering ALPHA 5 into the “List:” box. The calculator should automatically put the cursor there for you. Press ENTER ENTER.

Step 7: Read the results for the mean. The mean is the first in the list (an x with a bar on top),= 5.125.

Step 8: Read the results for the median:The median is about half way down the list: scroll with the down arrow and look for MedX = 5.

Step 9: Find the mode: Return to the list editor. Press F3 2 ENTER ENTER to access “sort list”. Make sure “m” is in the “List:” box and the order is “Ascending.” Press ENTER. Your data is now sorted. Just count which number appears the most: that’s your mode.

Tip: You can name your list anything you want, but keep it simple and don’t use common variables like t,x,y, or z.

That’s it!

## TI 83 Mean, Median, and Mode

Finding the TI 83 mean or TI 83 median from a list of data can be accomplished in two ways: by entering a list of data, or by using the home screen to type the commands. Using the list feature is just as easy as entering the data onto the home screen, and it has the added advantage that you can use the data for other purposes after you have calculated your mean, mode and median (for example, you might want to create a TI 83 histogram).

## Steps for the Mean, Median and Mode on the TI 83

Watch the video for the mean and median or read the steps underneath (for the mode, see this note):

Example problem: Find the mean and the median for the height of the top 20 buildings in NYC. the heights, (in feet) are: 1250, 1200, 1046, 1046, 952, 927, 915, 861, 850, 814, 813, 809, 808, 806, 792, 778, 757, 755, 752, and 750.

Step 1: Enter the above data into a list. Press the STAT button and then press ENTER. Enter the first number (1250), and then press ENTER. Continue entering numbers, pressing the ENTER button after each entry.

Step 2: Press the STAT button.

Step 3: Press the right arrow button to highlight “Calc.”

Step 4: Press ENTER to choose “1-Var Stats” and then type in the list name. For example, to enter L1 press [2nd] and [1].

Step 5: Press ENTER again. The calculator will return the mean, x̄. For this list of data, the TI 83 mean is 884.05 feet (rounded to 3 decimal places).

Step 6: Arrow down until you see “Med.” This is the TI 83 median; for the above data, the median is 813.05 feet.

Note: The TI-83 plus doesn’t have a built in mode function, but once you’ve entered your list, it’s pretty easy to spot the mode: it’s just the number that occurs most often in the set. Not sure? Read more about the mode here.

That’s it!