# Mean Median Mode Range Calculator

## How To Calculate Mean, Median, Mode & Range

Lets first understand, what is mean, median, mode and range?

Mean:

For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.

The arithmetic mean of a set of numbers x1, x2, x3, ......, xn is typically denoted by $$\bar{x}$$.

$$Mean(\bar{x}) = \frac{1}{n}\sum_{i=1}^n x_i$$ $$\bar{x}= \frac{1}{n}(x_1 + x_2 + x_3 + ... + x_n)$$ In other words, $$Mean = \frac{Sum \hspace{0.2cm}of \hspace{0.2cm}terms}{Number\hspace{0.2cm} of\hspace{0.2cm} terms}$$

Example 1: Calculate mean for the data set 20, 22, 28, 32, 56.

Solution: We know,

$$Mean = \frac{Sum \hspace{0.2cm}of \hspace{0.2cm}terms}{Number\hspace{0.2cm} of\hspace{0.2cm} terms}$$

Sum of terms = 20+22+28+32+56 = 158

Number of terms = 5

$$Mean = \frac{158}{5}$$ $$Mean = 31.6$$

If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted by $$\bar{x}$$) to distinguish it from the mean of the underlying distribution, the population mean (denoted by $$\mu$$).

The population mean, or expected value, is a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution.

In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving

$$Mean(\mu)=\sum xp(x)$$

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property, while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual—divided by the total number of individuals.

Types of Means

1. Arithmetic Mean (AM)

The arithmetic mean (or simply mean) of a list of numbers, is the sum of all of the numbers divided by the amount of numbers. Similarly, the mean of a sample $$x_{1},x_{2},\ldots ,x_{n}$$, usually denoted by $$\bar {x}$$ is the sum of the sampled values divided by the number of items in the sample

$$Mean(\bar{x})= \frac{1}{n}\sum_{i=1}^n x_i$$

2. Geometric Mean (GM)

The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product.

$$Mean(\bar{x}) = \biggl(\prod_{i=1}^n x_i\biggr)^{\frac{1}{n}}$$ $$Mean(\bar{x}) = (x_1x_2 \ldots x_n)^{\frac{1}{n}}$$

For example, the geometric mean of five values: 2, 4, 6, 8, 10 is

$$Mean=(2*4*6*8*10)^{\frac{1}{5}}$$ $$Mean=\sqrt[5]{3840} = 5.21$$

3. Harmonic Mean (HM)

$$Mean(\bar{x})= n\biggr(\sum_{i=1}^n \dfrac{1}{x_i}\biggr)^{-1}$$

Median:

The median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as the middle value.

For finding the Median, first sort the data in the Ascending or Descending Order.Then
if the number of terms is odd, then the median will be $$\bigl(\frac{n\hspace{0.1cm}+\hspace{0.1cm}1}{2}\bigr)$$th term or
if the number of terms is even, then the median will be $$\dfrac{1}{2}\biggl(\bigl(\frac{n}{2}\bigr)th\hspace{0.1cm}term + \bigl(\frac{n}{2}\hspace{0.1cm}+\hspace{0.1cm}1\bigr)th\hspace{0.1cm}term\biggr)$$.

Example 2: Find the median for data set 13, 16, 10, 5, 8.

Solution: First sort the data in the ascending order:

5, 8, 10, 13, 16

Number of terms is odd, so the median will be middle term (3rd term) will be 10.

Example 3: Find the median for the data set 13, 16, 10, 5, 8, 12.

Solution: First sort the data in the ascending order:

5, 8, 10, 12, 13, 16

Number of terms is even, so the median will be average of 3rd and 4th term.

$$Median=\frac{10+12}{2}$$ $$Median=11$$

Mode:

Mode is the value that has maximum number of occurences.

Mode is very easy to understand and calculate.

Example 4: Find the mode for the data set 2, 4, 2, 5, 6, 8, 10, 10, 2, 4.

Solution: Mode will be 2 because in this data set, 2 has maximum number of occurences.

Range:

Range will be the difference between the maximum value and the minimum value.

Example 5: Find the range for the data set 2, 4, 2, 5, 6, 8, 10, 10, 2, 4.

Solution: Range = 10 - 2 = 8

For mean, median, mode and range calculation using above calculator, you have to just write the comma separated values in the given input box and press the calculate button, you will get all the result.