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Mean-Median-Mode Calculator


Solution:

Mean:

The mean of a data set is commonly known as the average. You find the mean by taking the sum of all the data values and dividing that sum by the total number of data values. The formula for the mean of a population is $$ \mu = \frac{{\sum}x}{N} $$ The formula for the mean of a sample is $$ \bar{x} = \frac{{\sum}x}{n} $$ Both of these formulas use the same mathematical process: find the sum of the data values and divide by the total. For the data values entered above, the solution is: $$ \frac{2851}{15} = 190.0667$$


Median:

The median of a data set is found by putting the data set in ascending numerical order and identifying the middle number. If there are an odd number of data values in the data set, the median is a single number. If there are an even number of data values in the data set, the median is the average of the two middle numbers. Sorting the data set for the values entered above we have:

$$ 137, 143, 145, 152, 160, 178, 188, 189, 191, 191, 218, 222, 227, 241, 269 $$

Since there is an odd number of data values in this data set, there is only one middle number. With 15 data values, the middle number is the data value at position 8. Therefore, the median is 189.


Mode:

The mode is the number that appears most frequently. A data set may have multiple modes. If it has two modes, the data set is called bimodal. If all the data values have the same frequency, all the data values are modes. Here, the mode(s) is/are

How to Use the Mean Median Mode Calculator

This mean median mode calculator finds the measures of central tendency.  First, the calculator will give you an answer, and then it will guide you through a step-by-step solution to easily learn how to do the problem yourself.  The mean is the average of a set of numbers.  The median is the middle number/s when arranged in numerical order.  Lastly, the mode is the number that appears most often.

This mean median mode calculator finds the measures of central tendency. 

Once you’ve finished calculating measures of central tendency, go and take a look at the Range, Variance, and Standard Deviation Calculator.  Statistics has never been easier… You GOT this!

What are Mean Median Mode?

The mean, median, and mode are the measures of central tendency in statistics.  In other words, they are numbers that we use to measure and describe the center of a distribution.  When students begin learning about distributions in statistics courses, instructors and textbooks teach the mean, median and mode as some of the first topics in the course.  Even elementary school teachers introduce their students to measures of central tendency, so elementary school students can use the Mean Median Mode Calculator too.  Here, we’ll explore how to find the mean median and mode by hand.  Then, you can use the mean median mode calculator to check your work.

How to Find the Mean

The mean is commonly what we know as the average of a set of numbers.  Most students are very familiar with averages and often use them to determine how they are performing in a class.  How do you find the mean in math?  We find the mean by adding up all the numbers of the data set, then dividing by the total number of values.

$$ \text{mean} = \frac{\sum{\text{data values}}}{\text{total}} $$

In math, we use the summation symbol, $\sum$, to note that we should add all the numbers together.

If you are studying AP statistics, college-level statistics, or anything past middle school mathematics, then, you’ll use specific symbols to denote the mean.  Symbolically, you’ll denote the population mean by “mu” $\mu$ and denote the population size by N.  You’ll denote the sample mean by “x-bar” $\bar{x}$ and you’ll denote the sample size by n.  The population mean formula is:

$$ \mu = \frac{\sum{x}}{N} $$

The sample mean formula is:

$$ \bar{x} = \frac{\sum{x}}{n} $$

In each of these formulas, x represents each of the numbers in the data set.  So, $\sum{x}$ is the sum of all the data values.

Here is how to find the mean in the data set 3, 5, 9, 15, 17.  The population mean is

$$ \mu = \frac{3 + 5 + 9 + 15+ 17}{5} $$

$$ \mu = \frac{49}{5} = 9.8 $$

You can use the Mean Median Mode Calculator above and enter values of 3, 5, 9,15, 17 to verify the mean is 9.8.

How Do You Find the Median?

What does the median mean in math?  When you arrange all the numbers of a data set in increasing numerical order, the median is the number that falls exactly in the middle.  If there are two numbers that fall in the middle, the median is the average of these two numbers.

3, 5, 9, 15, 17

The median of the data set above is 9.  9 is the number in the middle.

Now consider the following data set:

3, 5, 9, 15, 17, 20

How do you find the median in this case?  Here, we’ll need to find the median of even numbers.  To do that, we take the average of the two middle numbers.  The median of this data set is the average of 9 and 15.  That is

$$ \frac{9 + 15}{2} = 12 $$

Now, use the Mean Median Mode Calculator above to check that the median is now 12.

What is the Mode?

What does the mode mean in math?  When describing a data set, the frequency of a data value is the number of times that data value occurs.  The data value with the highest frequency, or the one that occurs most often, is the mode.  A data set may have two modes if two numbers have the same highest frequency.  We say that the distribution is bimodal if it has two modes.  Sometimes, if there are more than two numbers with the greatest frequency, we’ll say that the data set has no mode.  However, with the Mean Median Mode Calculator above, the mode(s) found will include all the numbers with the greatest frequency.

In the data set 3, 5, 9, 15, 17, 9, what is the mode?  The data value 9 has a frequency of 2 while all the other numbers have a frequency of 1.  9 has the greatest frequency and therefore 9 is the mode.  You can use theMean Median Mode Calculator above to confirm that the mode is 9.

Mean Median and Mode Examples

Here are some examples of mean median and mode in statistics.

Example 1: Find the Mean

Find the population mean of the data set: 85, 78, 92, 65

Solution:

Since the problem tells us to find the population mean, the correct symbol to use for the mean is $ \mu $.  The population mean formula is

$$ \mu = \frac{85 + 78 + 92 + 65}{4} $$

$$ \mu = \frac{ 320}{4} = 80$$

Therefore, the population mean, $ \mu $, is 80.

You can cut and paste this data set into the Mean Median Mode Calculator above and verify that the mean is 80.

Example 2: Find the Median

Find the median for the following data set:

72, 75, 77, 80, 81, 81, 84, 86, 87, 88, 89, 91, 93, 94, 94, 94, 97, 99, 100

The median is the middle number.  Since there are 19 data values in this data set, the middle number is the number in the 10th position.  That is 88.  The median is 88.

Let’s add one more data value to the beginning of the data set to make a total of 20 data values.  We’ll include a 71.

71, 72, 75, 77, 80, 81, 81, 84, 86, 87, 88, 89, 91, 93, 94, 94, 94, 97, 99, 100

Since there is an even set of data values, the median is the average of the two middle numbers.  With 20 data values, the middle two values are in positions 10 and 11.  These two numbers are 87 and 88.  The average of 87 and 88 is:

$$ \text{median} = \frac{87 + 88}{2} $$

$$ \text{median} = 87.5 $$

Therefore, the median of this data set is 87.5.

You can cut and paste this data set into the Mean Median Mode Calculator above and verify that the median is 87.5.

Example 3: Find the Mode

71, 72, 75, 77, 80, 81, 81, 84, 86, 87, 88, 89, 91, 93, 94, 94, 94, 97, 99, 100

The mode is the data value that appears most often.  That is the data value with the greatest frequency.  In this data set, the data value 94 has a frequency of 3, which is greater than the frequency of any other data value.  Therefore, the mode is 94.

You can cut and paste this data set into the Mean Median Mode Calculator above and verify that the mode is 94.

Mean vs. Median – What’s the More Typical Measure of Central Tendency?

Find the Mean with an Additional Outlier Score

When we calculate the mean, we find that the outlier greatly affects the answer.  An outlier in a data set is a value that is much lower or much greater than all the other values.  Let’s look at an example of test grades.  Suppose the data values in Example 1 above represent 4 test grades.

85, 78, 92, 65

We found the mean of this data set to be 80.  Now, consider a 5th test grade of 20.  We would consider 20 an outlier because it is much smaller than the rest of the test grades.

85, 78, 92, 65, 20

So, what’s the mean now?

$$ \mu = \frac{85 + 78 + 92 + 65 + 20}{5} $$

$$ \mu = \frac{ 340}{5}  = 68 $$

Therefore, the new mean is 68.  An average that changes from 80 to 68 is a great difference.  The outlier value of 20 greatly affects the mean.  Do you think the score 68 represents a typical value in this data set?

Find the Median with an Additional Outlier Score

If we were to order the original 4 test grades and find the median, we’d calculate it to be the average of 78 and 85.

65, 78, 85, 92

$$ \text{median} = \frac{78 + 85}{2} $$

$$ \text{median} = \frac{163}{2} = 81.5 $$

Therefore, the median for the original 4 test grades is 81.5.

Now, let’s include a 5th test grade of 20 to the data set and find the new median.  So now, our data set is

20, 65, 78, 85, 92

With an odd number of data values, the median is the middle number.  Here’s that value is 78.  So, the new median with the included outlier is 78.


So what does this mean?  The mean and median for the original data set were, respectively, 80 and 81.5.  But when we included an outlier in the data set, the mean and median became 68 and 78.  While the mean decreased 12 points with the included outlier, the median decreased 3.5 points.  We see that the outlier greatly affects the the mean, but the outlier only slightly affects the median.  Therefore, we say that the median more accurately describes the typical test grade because it is more resistant (not as affected) to any outliers in the data set.

Positions of the Mean Median and Mode on Various Distributions

For Normal, Bell-Shaped Distributions

The mean, median, and mode are all approximately in the center of a normal, bell-shaped distribution.  Therefore, we can estimate that they all have the same value.

The mean median and mode are all the same value on a normal bell-shaped distribution
The mean median and mode are all the same value on a normal bell-shaped distribution.

 

Here is an example of a data set that has the same mean, median and mode.  You can verify that the measures of central tendency are all the same by pasting this data set into the Mean Median Mode Calculator.

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9

For Skewed Left, or Negatively Skewed Distributions

In skewed left, or negatively skewed, distributions, there are low scores on the left side of the distribution, potentially outliers, and they pull the left tail out to the left.  As we saw in the example above, outliers greatly affect the mean, while outliers slightly affect the median. The mode remains the data value with the highest frequency.  Therefore, outliers do not affect the mode.

If we look at a picture of a skewed left distribution, we place the mean in a position furthest to the left.  Then, we put the median in a position somewhere in the middle of the mean and the mode.  Finally, we put the mode at a position where we see the highest peak of the graph, because it is the value of greatest frequency.  This is a visual example of how the mean is most influenced by outliers, the median is slightly influenced by outliers, and the mode is not influenced by outliers.  Therefore, the order of the three values along the horizontal axis are mean, then median, then mode.

mean median mode on a skewed left distribution
The position of the mean median and mode on a skewed left distribution. The mean is pulled out to the left in the direction of the skewed tail.

 

For Skewed Right, or Positively Skewed Distributions

When a distribution is skewed to the right, or positively skewed, there are high scores on the right side of the distribution, potentially outliers, dragging the right tail out to the right.  As with the skewed left distribution, the mean is greatly affected by outliers, while the median is slightly affected.  The mode is not affected by outliers.

If we look at a picture of a skewed right distribution, the mean will be positioned furthest to the right.  It’s value is being pulled in the direction of the skewed tail.  Because the outliers in the right tail only slightly affect the median, it will be positioned somewhere in the middle of the mean and the mode.  And again, the mode will be positioned at the highest peak of the graph, the position of greatest frequency.  Therefore, the order on the horizontal axis of the three values are mode, median, and mean.

mean median and mode on a skewed right distribution
The position of the mean median and mode on a skewed right distribution. The mean is pulled out to the right in the direction of the skewed tail.

What’s Next?

The Mean Median Mode Calculator above guides you through calculating the three measures of central tendency.  The next step is to develop an understanding of the measures of variability.  There are three measures of variability, the range, variance, and standard deviation.  Of these, the standard deviation will present itself extensively throughout a college level statistics course.

The Range, Variance, and Standard Deviation Calculator is the best resource online for learning how to calculate the three measures of variability by hand.  It gives clear steps on how to solve the problem yourself.  To develop an understanding of the measures of variability, see Variance and Standard Deviation Definition and How to Find the Standard Deviation and Variance.

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