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## Statistics and probability

### Course: Statistics and probability > Unit 3

Lesson 1: Measuring center in quantitative data# Mean, median, and mode review

## Mean, median, and mode

Mean, median, and mode are different measures of center in a numerical data set. They each try to summarize a dataset with a single number to represent a "typical" data point from the dataset.

**Mean:**The "average" number; found by adding all data points and dividing by the number of data points.

Example: The mean of 4, 1, and 7 is left parenthesis, 4, plus, 1, plus, 7, right parenthesis, slash, 3, equals, 12, slash, 3, equals, 4.

**Median:**The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).

Example: The median of 4, 1, and 7 is 4 because when the numbers are put in order left parenthesis, 1, 4, 7, right parenthesis, the number 4 is in the middle.

**Mode:**The most frequent number—that is, the number that occurs the highest number of times.

Example: The mode of left brace, 4, 2, 4, 3, 2, 2, right brace is 2 because it occurs three times, which is more than any other number.

*Want to learn more about mean, median, and mode? Check out the more in-depth examples below, or check out this video explanation.*

## Calculating the mean

There are many different types of mean, but usually when people say mean, they are talking about the arithmetic mean.

The arithmetic mean is the sum of all of the data points divided by the number of data points.

Here's the same formula written more formally:

### Example

**Find the mean of this data:**

1, 2, 4, 5

Start by adding the data:

1, plus, 2, plus, 4, plus, 5, equals, 12

1, plus, 2, plus, 4, plus, 5, equals, 12

There are 4 data points.

**The mean is 3.**

### Practice problems

*Want to practice more of these? Check out this exercise on calculating the mean.*

## Finding the median

The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.

To find the median:

- Arrange the data points from smallest to largest.
- If the number of data points is odd, the median is the middle data point in the list.
- If the number of data points is even, the median is the average of the two middle data points in the list.

### Example 1

**Find the median of this data:**

1, 4, 2, 5, 0

Put the data in order first:

0, 1, 2, 4, 5

0, 1, 2, 4, 5

There is an odd number of data points, so the median is the middle data point.

0, 1, 2, 4, 5

**The median is 2.**

### Example 2

**Find the median of this data:**

10, 40, 20, 50

Put the data in order first:

10, 20, 40, 50

10, 20, 40, 50

There is an even number of data points, so the median is the average of the middle two data points.

10, 20, 40, 50

**The median is 30.**

## Finding the mode

The mode is the most commonly occurring data point in a dataset. The mode is useful when there are a lot of repeated values in a dataset. There can be no mode, one mode, or multiple modes in a dataset.

### Example 1

Ms. Norris asked students in her class how many siblings they each had.

**Find the mode of the data:**

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5

Look for the value that occurs the most:

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5

0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 5

**The mode is 1 sibling.**

### Example 2

Ms. Rubin asked students in her class how many siblings they each had.

**Find the mode of the data:**

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4

Look for the value that occurs the most:

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4

There is a tie for the value that occurs the most often.

**The modes are 1 and 2 siblings.**

## Want to join the conversation?

- Is there any formula for figuring out the median? TIA(18 votes)
- It is not possible to create a formula for the median, because the median value depends on the position of the middle value of the set and the fact that it is an even or odd set of numbers.

It can, however, be explained like this:*median (odd set of numbers)*= ((n+1)/2)th term*median (even set of numbers)*= ((n/2)th term + ((n/2)+1)th term)/2

Source: http://formulas.tutorvista.com/math/mean-median-mode-formula.html(18 votes)

- I am having problems with the median questions. I have sorted and then chosen the answer but check failed. When I've opened the explanation and hint, I've seen there exactly the same answer I have entered. It happened previously on some of practice pages. Please fix these issues so I can proceed with the lessons. Thanks!(17 votes)
- Yes ecause once you know w what's in the middle that would be you median(0 votes)

- I have a problem with the "median" question. I've sorted and then chosen the answer but check failed. When I've opened the explanation, I've seen there exactly the same answer I've entered. It happened previously on some of practice pages.(13 votes)
- i am a teacher and it says sort the data from least to greatest in the quextion you need to arrange the values first than solve the answer will be 8(4 votes)

- Hey guys. Whenever I'm taking a test or quiz and I get asked for the mean, median, or mode I get confused and forget which is which. Does any one have a good way to memorize them? It would be greatly apreciated.(6 votes)
- "Hey, diddle diddle, the median's the middle,/You add then divide for the mean./The mode is the one that you see the most,/And the range is the difference between." This is a poem that i found in google when i started to learn this topic. Just memorize this poem...(20 votes)

- What if there would be same number of repeated number exist in the list? for example set: 5,10,10,8,8,6,3,2. What is the mode of this set?(5 votes)
- I believe you would list them both. So in your case, the modes would be 8 and 10 since they both occur at the same time. I hope this helps!(5 votes)

- Can there be negative infinity?(6 votes)
- Yep, you can just keep getting more and more negative.(3 votes)

- Why is it called Mean, Median, and Mode?(5 votes)
- No idea... good thing to research thougH!(1 vote)

- wow that makes more sense now!(5 votes)
- Can you guys respond and upvote this?(4 votes)
- under what circumstances that you should choose Mean over Median? or Median over Mode, or Mean over Mode?(2 votes)
- Choose MEAN if you need the average and you are working with numerical qualitative values.

Choose MEDIAN if you have outliers. Outliers impact the mean. This is why we have a 'median' house price. because expensive houses would skew the mean average so we don't use that.

Choose MODE if you are dealing with qualitative data where it doesn't make sense to use an average. Hair colour, gender.(4 votes)