##### Mean, Median and Mode

## What is a data set?

A data set is a set that is collected for a specific purpose, so it`s an organized collection of data.

The two most common types of data sets

1. Numerical data sets

2. Categorical data sets

1. Numerical data is a data type expressed in ** numbers**, is always collected in number form, it´s something that is measurable. For example the height, weight of a person, the number of car accident in a day, the level of the blood pressure of a healthy person, the normal range of iron for a normal person.

Numerical data can be grouped in two types of data:

– Discrete

– Continuous

- Discrete data are numerical data that can be counted, they can be listed. They can be finite or infinite but still can be countably.For example the number of the 5 multiples of a number is countably finite but if we ask for all the multiples of a number then it is countably infinite.
- Continuous are numerical data in which the data can´t be counted and can only be described using intervals with numerical values.

2. Categorical data is not numerical data,it represents the characteristic of something. Like the gender, types of books people like, hair color.

## What are the mean, median and mode?

Mean, median and mode are different measures of center in a numerical data set.

The mean – also called the average of the numbers is found by adding all the numbers on a data set and divided by the number of numbers of the data set.

The median – is the middle value we pick up after we have written all the elements of a data set on ascending order.

**The mode** – is the element in a data set that is repeated more than the other elements of the data set.

## How to calculate them?

The mean is calculated by adding all the numbers and then dividing the sum by the number of elements that are on the data set.

If we have the data set x1, x2, x3, x4,…….., xn

The mean is presented with the formula: sum of all the data / the number of data

$ \displaystyle m=\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+………{{x}_{n}}}{n}$

Example: Find the mean of the given data set: 1, 5, 7, 8, 6, 13, 21

We add all the numbers of the data set: 1 + 5 + 7 + 8 + 6 + 13 + 21 = 61

Count the number of elements we have on total on the data set: 7 elements

Then divide the sum of all the data with the number of the elements on the data set: $ \displaystyle \frac{61}{7}=8.7$

The median is calculated by writing all the elements of the data set on the ascending order and then picking up the element that is in the middle if the number of elements is odd. If the number of elements is even than we find the median by finding the mean of the two elements that are in middle.

**Median Formula. **If the number of the elements n is odd then the median is the value at the position: $ \displaystyle \frac{n+1}{2}$

If the number of the elements n is even then the median is found by finding the value at position $ \displaystyle \frac{n}{2}$ and $ \displaystyle \frac{n+1}{2}$

Find the mean of those two values and that is the median.

**Example 1**: Find the median in the given data set: 2, 3, 8, 15, 7, 13, 24

Write all the elements on the ascending order: 2, 3, 7, 8, 13, 15, 24

Then pick up the element in the middle: 8

So, the number 8 is the median of the given set

Example 2: Find the median in the given data set: 13, 58, 46, 23, 17, 9, 5, 4, 1, 3.

Write all the elements on the ascending order: 1, 3, 4, 5, 9, 13, 17, 23, 46, 58.

Since we have an even number of the given elements then we can’t pick up an element in the middle.

We are going to pick up the two elements in the middle: 9 and 13.

Then find the mean of those two numbers: $ \displaystyle \frac{9+13}{2}=11$

The number we find it’s the median of the data set: 11

The mode is the data that is repeated more in a data set. There can be one mode, no mode or more than one mode in a data set.

Example

**a)** Find the mode in the given data set: 2, 5, 6, 3, 5, 3, 8, 5, 9, 1

Write the data set in the ascending order: 1, 2, 3, 3, 5, 5, 5, 6, 8, 9

Then pick the data set that is repeated more: 5

**b)** Find the mode in the given data set: 1, 5, 7, 8, 6, 13, 21

Write the data set in the ascending order: 1, 5, 6, 7, 8, 13, 21

As we can see all the data is repeated in the same number of time

So, there is no data for this data set

**c)** Find the mode in the given data set: 2, 3, 2, 3, 6, 4, 6, 7, 9.

Write the data set in the ascending order: 2, 2, 3, 3, 4, 6, 6, 7, 9

As we can see there are three numbers that are repeated the same number of time: 2, 3 and 6

So, there are three modes: 2, 3 and 6.