## Average : Mean, Median and Mode

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Average is used to represent a large set of numbers with a single number. In the data set, It is a representation of all the numbers available . The average is calculated by adding up all the data values and dividing by the number of data points. The age of the students in a class is taken and the average is calculated to give a single value of the average age of the students of a class. Average has many applications in our daily life. For quantities with the average is calculated , changing values and a unique value is used to represent the values.

Learning about averages helps us to quickly summarize available data. A large set of students’ scores, changing stock prices, weather data for a place, incomes of different people in a city are all examples for which we can calculate averages. Let’s take a look at the page to know more about Average.

### what is the mean?

Average is a numerical value that is a single representation of a large amount of data. The average of the marks of the students of a class in a particular subject is taken to give the class average marks. One needs to know the performance of the whole class rather than the performance of each student. Here, the average is helpful. The average of a set of values is equal to the sum of the values divided by the individual values. Also, averaging is used in situations of changing values. The temperature of a place is the average to indicate the temperature of a place throughout the season. To know the income of employees in a company, the average of the income of different employees in a company is taken.

It is sometimes difficult to make a decision based on a single data or a large set of data. Therefore, the average value is taken and it helps to represent all the values in a single value.

### Definition of Average:

Average is known as arithmetic mean which is the sum of all the numbers in the collection divided by the number of all the numbers present in the collection.

In other words, average is the ratio of the sum of all the given observations to the total number of observations.

**Average = Sum of observations/number of observations**

## Calculation of Average** **

Calculating the average follows three simple steps. In addition, it includes arithmetic operations: addition and division.

- Step 1: Sum of Numbers: The first step in the process of finding the average is to find the sum of the given numbers. As an example, let us take the weight of six children. Find the sum of all the individual weights of 6 children. 20lb, 25lb, 21lb, 30lb, 25lb, 26lb
- Step 2: Number of Observations. Here we need to know the count or data points. In our example of a load of children, we have 6 children. Count the total number of observations. Here, in this case, Zoe has a total of 6 observations which include the weight of Zoe and 5 of her friends. Total number of observations 6
- Step 3: Calculating Average: Substituting the values in Step 1 and Step 2 in the average formula, we get the following expression. I

Average = Sum of observations/number of observations = 147/6 = 24.5lbs

### Can the median be considered an average?

No, median is not considered as average. Average is the mean value of the data and is different from the mean value of the data. Median is the middle value of a set of data arranged in ascending order. The mean or median value is also known as central tendency. To find the measure of central tendency, we need to write the data points in increasing or decreasing order. Furthermore, the calculation of the median depends on the number of data points. Let us look at the following two cases for computing the median value.

- Case 1: n is odd. Here for an odd number of data points, there is only one middle data point. and the mean of the data is (n + 1)/2 observation.

**Case 2:**n is Even. Here for the even number of data points, there are two middle data points. And the median is the average of n/2 and (n/2 + 1) observation.

Calculating the average follows three simple steps. In addition, it includes arithmetic operations: addition and division.

Step 1: Sum of Numbers: The first step in the process of finding the average is to find the sum of the given numbers. let us take the weight of six children , As an example. Find the sum of all the individual weights of 6 children. 20lb, 25lb, 21lb, 30lb, 25lb, 26lb

Step 2: Number of Observations. Here we need to know the count or data points. In our example of a load of children, we have 6 children. Count the total number of observations. Here, in this case, Zoe has a total of 6 observations which include the weight of Zoe and 5 of her friends. Total number of observations 6

Step 3: Calculating Average: Substituting the values in Step 1 and Step 2 in the average formula, we get the following expression. I

Average = Sum of observations/number of observations = 147/6 = 24.5lbs

### Can the median be considered an average?

No, median is not considered as average. Average is the mean value of the data and is different from the mean value of the data. Median is the middle value of a set of data arranged in ascending order. The mean or median value is also known as central tendency. To find the measure of central tendency, we need to write the data points in increasing or decreasing order. Furthermore, the calculation of the median depends on the number of data points. Let us look at the following two cases for computing the median value.

Case 1: n is odd. Here for an odd number of data points, there is only one middle data point. and the mean of the data is (n + 1)/2 observation.

## Solved Examples

### Quick guide:

#### To calculate mean

Add the numbers together and divide by the number of numbers.

(the sum of the values divided by the number of values).

#### To find the median

Arrange the numbers in sequence, find the middle number.

(middle value when values are ranked).

#### To set the mode

Count how many times each value occurs; The most frequently occurring value is the mode.

(most frequently occurring value)

## Mean, Median and Mode Calculator

Use this calculator to work out the mean, median and mode of a set of numbers.

## Mean

The mathematical symbol or notation for the mean is ‘x-bar’. This symbol appears in scientific calculators and in mathematical and statistical notation.

The ‘mean’ or ‘arithmetic mean’ is the most commonly used form of average. To calculate the mean, you need a set of related numbers (or data sets). At least two numbers are required to calculate the mean.

The numbers must be linked or linked to each other for any meaningful result to be made – for example, temperature readings, the price of coffee, number of days in the month, number of heartbeats per minute, students’ test grades, etc.

To find the average price (average) of a loaf of bread at the supermarket, for example, first record the price of each type of bread:

White: £1

Whole Meal: £1.20

Baguette: £1.10

Next, add the (+) prices together £1 + £1.20 + £1.10 = £3.30

Then divide your answer by the number of loaves (3).

£3.30 3 = £1.10.

The average price of a loaf of bread in our example is £1.10.

The same method applies with large sets of data:

To calculate the average number of days in a month we will first establish how many days are in each month (assuming it was not a leap year):

Month | Days |

January | 31 |

February | 28 |

March | 31 |

April | 30 |

May | 31 |

June | 30 |

July | 31 |

August | 31 |

September | 30 |

October | 31 |

November | 30 |

December | 31 |

Next we add all the numbers together: 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 30 + 31 = 365

Finally we divide the answer by the number of values in our data set, in this case 12 (one counts for each month).

So the mean average is 365 12 = 30.42.

Therefore, the average number of days in a month is 30.42.

The same calculation can be used to average any set of numbers, for example the average salary in an organization:

Let’s say the organization has 100 employees at one of the 5 grades:

Grade | Annual Salary | Number of Employees |

1 | £20,000 | 21 |

2 | £25,000 | 25 |

3 | £30,000 | 40 |

4 | £50,000 | 9 |

5 | £80,000 | 5 |

In this example we can avoid adding up each employee’s salary because we know how many are in each category. So instead of writing £20,000 twenty one times we can multiply to get our answer:

Grade | Annual Salary | Number of Employees |
Salary x Employees |

1 | £20,000 | 21 | £420,000 |

2 | £25,000 | 25 | £625,000 |

3 | £30,000 | 40 | £1,200,000 |

4 | £50,000 | 9 | £450,000 |

5 | £80,000 | 5 | £400,000 |

Next add the values in the Salary x Employees column to find a total: £3,095,000 and finally divide this number by the number of employees (100) to find the average salary:

£3,095,000 ÷ 100 = £30,950

Quick Tip:

The salaries, in the example above, are all multiples of £1,000 – they all end in 000.

You can ignore the ,000 when calculating, as long as you remember to add them back at the end.

In the first row of the table above we know that twenty-one people are paid £20,000 instead of working with £20,000:

21 x 20 = 420 Then substitute 000 to get 420,000.

Sometimes we can know the sum of our numbers, but not the individual numbers that make up their sum.

In this example, let’s say that selling lemonade makes £122.50 a week.

We don’t know how much money was earned each day, how much in total at the end of the week.

What we can work out is the daily average: £122.50 7 (total money divided by 7 days).

122.5 7 = 17.50.

So we can say that we earned £17.50 a day on average.

We can also use the average to give clues to possible future events – if we know that we earned an average of £17.50 a day selling lemonade over a week, we can assume that in a month we would:

£17.50 × number of days in that month

17.50 × 31 = £542.50

We may record average sales figures each month to help predict sales and compare our performance for future months and years. We can use words like ‘above average’ – to refer to a time period when sales were above the average amount and similarly ‘below average’ when sales were below the average amount.

Using speed and time as data to find mean:

If you travel 85 miles in 1 hour 20 minutes, what was your average speed?

The first thing to do to deal with this problem is to convert the time to minutes – time on the decimal system doesn’t work because there are 60 minutes in an hour and not 100. So before we start we need to standardize our units:

1 hour 20 minutes = 60 minutes + 20 minutes = 80 minutes.

Then divide the distance covered by the time taken: 85 miles 80 minutes.

85 80 = 1.0625.

So our average speed was 1.0625 mph.

Convert this figure back to hours by multiplying by 60 (the number of minutes in an hour).

1.0625 × 60 = 63.75mph (mph).

For spreadsheet users:

Use the <average> function to calculate the mean average in a spreadsheet. The following example formula, assumes your data is in cells A1 to A10:

=average(A1:A10)

## Median

The median is the middle number in the list of ordered numbers.

To calculate the median: 6, 13, 67, 45, 2

First, arrange the numbers in order (also known as ranking)

2, 6, 13, 45, 67

then – find the middle number

Median = 13, the middle number in the sorted list.

When there are even numbers, there is no one middle number but a pair of middle numbers.

In such cases the median is the mean of the two middle numbers:

for example:

6, 13, 67, 45, 2, 7.

Arranged Ranking = 2, 6, 7, 13, 45, 67

The middle numbers are 7 and 13.

Median refers to a single number so we calculate the mean of two middle numbers:

7 + 13 = 20

20 2 = 10

So the median of 6, 13, 67, 45, 2, 7 is 10.

Method

Mode is the most frequently occurring value in a set of values. Mode is interesting because it can be used for any type of data, not just numbers.

In this example, let’s say you bought a pack of 100 balloons, the pack is made up of 5 different colors, you count each color and find that you have:

18 red

12 blue

24 orange

25 purple

21 green

Our balloon sampling mode is purple because there are more purple balloons (25) than any other color balloon.

To find the mode of the number of days in each month:

Month | Days |

January | 31 |

February | 28 |

March | 31 |

April | 30 |

May | 31 |

June | 30 |

July | 31 |

August | 31 |

September | 30 |

October | 31 |

November | 30 |

December | 31 |

7 months have 31 days, 4 months have 30 days and only 1 month has 28 days (29 in a leap year).

Hence the mode is 31.

Some data sets may have more than one mode:

1,3,3,4,4,5 – For example, the two most frequently occurring numbers are (3 and 4), this is known as a bimodal set. A data set with more than two modes is called a multi-modal data set.

Calculating the mode is more problematic if a data set contains only unique numbers.

It is generally perfectly acceptable to say that there is no mode, but if there is a mode to be found the usual way is to create number ranges and then count the one with the highest number of digits in it. For example from a set of data showing passing speed of cars we see that the recorded speeds of 10 cars are:

40, 34, 42, 38, 41, 50, 48, 49, 33, 47

All these numbers are unique (each occurs only once), there is no mode. To find a mode we create categories on a similar scale:

30–32 | 33–35 | 36–38 | 39–41 | 42–44 | 45–47 | 48–50

Then count how many values occur in each range, how many times a number between 30 and 32 occurs, etc.

30–32 = 0

33–35 = 2

36–38 = 1

39–41 = 2

42–44 = 1

45–47 = 1

48–50 = 3

The range with the most values is 48–50 3 with values.

We can take the mean value of the range as 49 to estimate the mode.

*-he met*-calculating the mode is not ideal as the mode may change based on the ranges you define

*-wS\X¥