What are mean, median and mode?

The mean, median and mode all represent some kind of average value. We use them in statistics, history, economics and many other fields. However, We should make a distinction between the three, as they all have different definitions and uses.

The mean

The mean is most often called the average, however this is not the case in all situations. There are multiple types of mean. The three most commonly used ones are called the Pythagorean means. These include the following three: the arithmetic (AM), the geometric (GM), and the harmonic mean(HM).

The arithmetic mean is most often used. In fact, you have probably used it many times yourself.

Mean Formula

We calculate the arithmetic mean by taking the sum of all values in a set and dividing it by the number of values.

\bar x = \frac {x_{1}+x_{2}+x_{3}...+x_{n}}{n}

Types of mean

I previously mentioned the three Pythagorean means, but there are actually a few more. I’ll give you a quick list and then we will go into more detail.

  • Arithmetic mean (AM)
  • Geometric mean (GM)
  • Harmonic mean (HM)
  • Weighted mean
  • Arithmetic-Geometric mean (AGM)
  • Root-Mean Square (RMS)
  • Heronian mean
  • Graphic mean

Arithmetic mean

The arithmetic mean is very widely used. It is what most people think when they say “average”. Other than in mathematics and statistics, we also use it in engineering, economics, geography, history and many other fields. For example, per capita income is the average income for a person in a certain region, and we calculate it using the arithmetic mean. However, it is not a perfect solution, as it is susceptible to great outliers, which are values that are way larger or smaller than the rest of the values in a given set. This will lead to something called skewed distributions. As mentioned before, the formula for calculating it is

\bar x_a = \frac {x_{1}+x_{2}+x_{3}...+x_{n}}{n}

Geometric mean

The geometric mean is used in business and finance for calculating growth rates. For example, it is used to calculate the compound annual growth rate (CAGR), portfolio returns and stock indexes. Because growth rates are expressed in percentages and not absolute values, the geometric mean will provide the most accurate results. The formula for calculating it is:

\bar x_g = \sqrt [n] {{x}_1 \cdot {x}_2 \cdot {x}_3… \cdot {x}_n}

Harmonic mean

The harmonic mean is used when dealing with fractions. You can calculate their multiplicative or divisor relationships without worrying about common denominators. It is also used in finances, but not for the same things the geometric mean is used for. For example, it can be used to average data, such as price multiples, and also to identify certain patterns, such as the Fibonacci sequence. The formula for calculating it is:

\bar x_h = (\frac {x_1^{-1}+ x_2^{-1} +...+ x_n^{-1}}{n})^{-1}

A simpler formula is:

/bar x_h = \frac {n}{x_1^{-1}+ x_2^{-1}+...+x_n^{-1} }

Weighted mean

This type of mean is used when, in a given set, there are numbers that are more valuable than others (i.e. carry more weight). The formula for calculating it is similar to the arithmetic mean formula, with one significant addition. Every number has its own multiplier. Those multipliers are called weighting factors.

{\bar {x}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}

Because it is a variation of the arithmetic mean, it has the same issues as the regular arithmetic mean, which are outliers.

Arithmetic-Geometric mean

To find the arithmetic-geometric mean (AGM) of to positive real numbers (a,b), you need to use the following formula:

a_{n + 1} = \frac {a_n+b_n}{2} \\ b_{n+1} = \sqrt {a_n \cdot b_n}

If it seems complicated, don’t worry, we’ll do a quick example to bring it closer to you.

Let’s say our two numbers are 28 and 7. Those numbers would be a0 and b0. To calculate a1 and b1, we would use the formula from above:

a_1 = \frac {28 + 7}{2} ; a_1 = \frac {35}{2} ; \boxed {a_1 = 17.5} \\ b_{1} = \sqrt {28 \cdot 7} ; b_{1} = \sqrt {196} ; \boxed {b_1 = 14} \\ a_2 = \frac {17.5 + 14}{2} ; a_2 = \frac {31.5}{2} ; \boxed {a_2 = 15.75} \\ b_{2} = \sqrt {17.5 \cdot 14} ; b_{2} = \sqrt {245} ; \boxed {b_2 = 15.6524758425} \\ a_3 = \frac {15.75 + 15.6524758425}{2} ; a_3 = \frac {31.4024758425}{2} ; \boxed {a_3 = 15.7012379212} \\ b_{3} = \sqrt {15.75 \cdot 15.6524758425} \\ b_{3} = \sqrt {246.526494519} \\ \boxed {b_3 = 15.7011622028} \\ a_4 = \frac {15.7012379212 + 15.7011622028}{2} \\ a_4 = \frac {31.402400124}{2} \\ \boxed {a_4 = 15.701200062} \\ b_{4} = \sqrt {15.7012379212 \cdot 15.7011622028} \\ b_{4} = \sqrt {246.527683386} \\ \boxed {b_4 = 15.701200062}

As we can see, they eventually converge into the same number. AGM is often used for calculating mathematical constants, notably π, as well as for calculating fast algorithms and exponential or trigonometric functions.

Root-Mean Square (RMS)

The root-mean square, often called the quadratic mean, is the square root of the mean square. The mean square is the arithmetic mean of the squares of a set of numbers. So, the formula for calculating RMS would be:

\bar x_{RMS} = \sqrt {\frac {x_1^2+x_2^2+...+x_n^2}{n}}

It is mainly used in electrical engineering, for calculating things such as the alternating electric current.

Heronian mean

It got its name from Hero of Alexandria, a Greek mathematician and engineer. For two positive real numbers, A and B, the heronian mean is equal to:

H= \frac {A+\sqrt {A \cdot B}+B}{3}

It is used in solid geometry for calculating the volume of a frustum of a pyramid or a cone.

The median

The median is quite literally the middle value in a set. To find the median of a set of values, you need to find the value that separates the set into two equal parts. In other words, re-write all the values in the set in numerical order (the smallest one at the beginning and the largest at the end). Then, depending on whether the set has an odd or an even number of values, use one of these two methods.

Median formula

For calculating the median, you may need to use one of two formulas. Which formula you use depends on how many numbers you have in the set. They’re pretty similar, with one slight adjustment.

Different ways of finding the median

If the set you want to find the median of has an odd number of values, the formula you need is:

median (x) = \frac {n+1} {2}

The result you get is the place of the number in your list that is the median. Let me give you an example:

37, 8, 59, 1, 16, 11, 5, 3, 25

Now if we rewrite this set, with the 9 values arranged in numerical order, we get:

1, 3, 5, 8, 11, 16, 25, 37, 59

Using the formula above, we get the following result:

\frac {9+1} {2} = 5

Therefore, the 5th number by size in this set is the median. In this case, it is the number 11.

1,3,5,8, \red {11}, 16, 25, 37, 59

If the set you want to find the median of has an even number of values, you actually need two results:

median (x_{1}) = \frac {n} {2}

as well as

median (x_{2}) = \frac {n} {2} + 1

Let’s see this method with actual numbers:

16, 15, 27, 23, 21, 20, 2, 11, 8, 5

Once we rearrange the numbers, we get this:

2, 5, 8, 11, 15, 16, 20, 21, 23, 27

Since this set has 10 values, we already know the result for the first step is 5. To get the second one, we just do a quick calculation:

\frac {10}{2}+1 = 6

So the results we got are 5 and 6. Now for the final step, we take the 5th and 6th values in the sequence, and calculate their arithmetic mean:

2, 5, 8, 11, \red {15}, \red {16}, 20, 21, 23, 27 \\ \frac {15+16} {2} = 15.5

So, for this set of numbers, the median is 15.5.

Mean vs median

As we mentioned before, even though the arithmetic mean is used most often, in some cases, it gives you the wrong picture of the situation. In those cases, it is better to use the median. In certain cases they can be equal, but it doesn’t matter since you will almost always calculate the arithmetic mean first.

The mode

The mode of a set is simply the value that appears most often in that given set.

3, 6, 7, 7, 8, 11, 15, 11, 8, 7, 3

We can see that many values are repeated in this set, however, the one that is repeated the most is 7, making it the mode of of this set.

3, 6, \red {7}, \red {7}, 8, 11, 15, 11, 8, \red {7}, 3

Find the mode

In terms of finding the mode, there is no shortcut or formula. There is, however, our calculator. If you’re not sure how exactly to use it, we’ll go over that too, don’t worry.

The range

The range is nothing more than the difference between the largest and smallest value in a given set. Our calculator will give you more than just the range though. It will also tell you the smallest and largest value in the set, as well as the midrange, which is essentially the arithmetic mean of the largest and smallest value in the set.

How to use the mean median mode calculator

So, if you’re still not sure how to use the calculator, we’ll quickly go over it now.

All you have to do is input all the values from your set into the fields, and the calculator will do the rest for you. So, let’s say we have the following set of numbers:

50, 23, 14, 9, 90, 18, 12, 23, 56, 17, 75, 14, 91, 90, 23

After we input those number into our calculator we would get the means:

The means

the median:

The median

the mode:

The mode

and the range:

The range

How to find the mean, median and mode by hand

If you want to practice your own skills, it could be useful to calculate all of these by yourself. As you saw before, that is also not difficult. You have all the formulas and guides for calculating them right here. Let’s take the set from the previous example and calculate everything by ourselves.

For the arithmetic mean, it is pretty simple:

AM = \frac {50+23+14+9+90+18+12+23+56+17+75+14+91+90+23}{15} = \frac {605}{15} = 40.333

For the geometric mean it would be:

GM = \sqrt [15]{50 \cdot 23 \cdot 14 \cdot 9 \cdot 90 \cdot 18 \cdot 12 \cdot 23 \cdot 56 \cdot 17 \cdot 75 \cdot 14 \cdot 91 \cdot 90 \cdot 23}{15} = \sqrt [15] {1.2199 \cdot 10^{22}} = 29.68

And finally, the harmonic mean:

HM = \frac {15}{50^{-1}+23 ^{-1} +14 ^{-1} +9 ^{-1} +90 ^{-1} +18 ^{-1} +12 ^{-1} +23 ^{-1} +56 ^{-1} +17 ^{-1} +75 ^{-1} +14 ^{-1} +91 ^{-1} +90 ^{-1} +23 ^{-1}} \\ HM= \frac {15}{\frac {1}{50}+ \frac {1}{23}+ \frac {1}{14}+ \frac {1}{9}+ \frac {1}{90} + \frac {1}{18} + \frac {1}{12}+ \frac {1}{23}+ \frac {1}{56}+ \frac {1}{17}+ \frac {1}{75}+ \frac {1}{14}+ \frac {1}{91}+ \frac {1}{90}+ \frac {1}{23}} \\ HM= \frac {15}{0.66637050199} \\ HM=22.51

Now, for the median, we need to arrange all the numbers in order from smallest to largest.

9, 12, 14, 14, 17, 18, 23, 23, 23, 50, 56, 75, 90, 90, 91

Since we have an odd number of values, we use the according formula:

median = \frac {15+1}{2} = \frac {16}{2} = 8

So, the 8th number in the sequence is the median.

9, 12, 14, 14, 17, 18, 23, \red {23}, 23, 50, 56, 75, 90, 90, 91

In this sequence we can also see the mode, which is the most common number in a given set. In this case, it’s 23:

9, 12, 14, 14, 17, 18, \red {23}, \red {23}, \red {23}, 50, 56, 75, 90, 90, 91

The range would be:

91-9=82

The largest number is 91, and the smallest one is 9, which would make the midrange:

\frac {91+9}{2} = \frac {100}{2} = 50

FAQ

How to calculate mean, median and mode of grouped data?

You can easily calculate all three with our calculator. If you want to do it by hand, we have a guide for that too.

What does mode mean in math?

In math, mode is the number that appears most in a given set of numbers.

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