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Mean Median and Mode

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Mean Median and Mode

The average income of the workforce in the UK in 2020 was estimated to be £38,600 according to the ONS. Notice how a single value is able to describe the entire income of the workforce in the UK.

In this article, we will learn about mean, median, and mode, and their applications.

Mean, median and mode definition

Mean, median, and mode are measures of central tendency that attempt to summarize a given data set into one single value by finding its central value.

We use thus that single value to represent what the whole data set says as it reflects what the data set is about.

Each of these three measures of central tendency, mean, mode, and median, provide different values for the same data set as they have different approaches to each measure.

Mean definition

The mean is the sum of all the data values divided by the number of data values.

Median definition

The median is the value separating the higher half from the lower half of the data set.

Mode definition

The mode denotes the most occurring data value in a data set. This measure of central tendency seeks to outline which data point occurs more.

Mean median and mode formula

In this section, we will go into the details of the calculation of the mean, median, and mode.

Mean formula

As stated earlier in this article, the mean of a list of numbers is the sum of these numbers divided by the number of these numbers. That is for a list of numbers , the mean denoted by is calculated through the formula

.

Median formula

As stated earlier in this article, the median is the value separating the higher half from the lower half of the data set.

The median of a finite list of numbers is the "middle" number when those numbers are listed in order from smallest to greatest.

The median of a finite set can be calculated while following the steps,

  • Arrange the numbers from smallest to greatest.
  • If the number of the numbers is odd, the middle value is the median.
  • If the number of numbers is even, the median is the average of the two middle values we have.

Mode formula

As stated earlier in this article, the mode denotes the most occurring data value in a data set.

A data set may have one mode, more than one mode, or no mode at all.

To find the mode, we follow these steps,

  • Rearrange the values of your data set from the lowest to the highest.
  • Note the most occurred data value.

Mean median and mode examples

Find the mean annual salary for a team put together by a company, where their respective annual salaries are as follows; £22,000, £45,000, £36,800, £70,000, £55,500, and £48,700.

Solution

We sum up the data values and divide them by the number of data values we have, as the formula says.

By this calculation, it means that the mean salary amongst the team is £46,333.

Find the mean of the data of salaries of a team of employees put together by a company including their supervisor as £22,000, £45,000, £36,800, £40,000, £70,000, £55,500, and £48,700, find the median.

Solution

We arrange our data values from lowest to the highest.

£22,000, £36,800, £40,000, £45,000, £48,700, £55,500, and £70,000.

We notice that the number of the data values is 7, which is an odd number, so the median is the middle between the lowest half (constituting of £22,000, £36,800, £40,000), and the highest half of the data set (constituting of £48,700, £55,500, and £70,000).

Thus, the middle value here is £45,000 , hence we deduce that

.

Now, supposing the supervisor is not included in the count and we have an even number as the total number of data points, how will we find the median? Let's take the next example.

The data set of the team put together by the company excluding their supervisor is as follows, £22,000, £45,000, £36,800, £40,000, £55,500, and £48,700, find the median.

Solution

We arrange these values from the lowest to the highest.

£22,000, £36,800, £40,000, £45,000, £48,700, £55,500.

We notice that the number of the data values is 6, which is an even number, so we have two numbers as our middle data point. Yet, to find the median, we find the average of those two numbers, £40,000 and £45,000.

Hence the median is £42,500.

Find the mode for the given data set, 45, 63, 1, 22, 63, 26, 13, 91, 19, 47.

Solution

We rearrange the data set from the lowest to the highest values.

1, 13, 19, 22, 26, 45, 47, 63, 63, 91

We count the occurrence of each data value and we see that all data values occur only once, while the data value 63 occurs twice. Thus the mode of the data set is

Suppose Mike wants to buy a property in London so he goes out to find out the prices of what exactly he might like. The data he gets on the pricing of all the properties he enquired about are as follows; £422,000, £250,000, £340,000, £510,000, and £180,000.

Find

  1. Mean
  2. Median
  3. Mode

Solution

1. To find the mean, we use the mean formula. We first find the sum of all the data values and divide it by the number of data values.

The mean price is £340,400

2. To find the median, we will need to arrange the data values in ascending order,

£180,000, £250,000, £340,000, £422,000, £510,000 .

The number of the data values is 5, which is odd, so we notice that the third data value is the middle between the lowest half and the highest half. So, we can now easily identify what the middle point value is

3. The mode is the most occurred data value. To find it, we will first rearrange the data values in ascending order.

£180,000, £250,000, £340,000, £422,000, £510,000

We notice that there is no most occurred data value. Thus, the data set has no mode.

The heights of students in grade 11 were collected and the data is given as

173cm, 151cm, 160cm, 151cm, 166cm, 149cm.

Find

  1. Mean
  2. Median
  3. Mode

Solution

1. To find the mean, we will use the mean formula, in which we add all the data values and divide the sum by the number of data values.

The mean height is 158.33cm.

2. The median is the middle point value of the data set. To find it, we will rearrange the data values in ascending order first, to get

149cm, 151cm, 151cm, 160cm, 166cm, 173cm

We notice that the number of the data values is 6, which is an even number, and hence we have two values in the middle. They are 151cm and 160cm. We will find the average of these values by adding them and dividing them by 2.

Thus, the median is

.

3. The mode is the most occurring value in the data set. We can rearrange the data values in ascending order to get,

149cm, 151cm, 151cm, 160cm, 166cm, 173cm.

We can identify that 151cm is the most commonly occurring value, thus

.

Mean Median and Mode - Key takeaways

  • Mean, median, and mode are measures of central tendency that attempt to summarize a given data set into one single value by finding its central value by some metric.
  • The mean is the sum of all the data values divided by the number of data values.
  • The median is the middle point value of the data set when arranged in ascending order.
  • The mode denotes the most occurring data value in a data set.

Frequently Asked Questions about Mean Median and Mode

Mean, median, and mode are measures of central tendency that attempt to summarize a given data set into one single value by finding its central value. 

The mean is the sum of all the data values divided by the number of data values. 


The median is the value separating the higher half from the lower half of the data set. 


The mode denotes the most occurring data value in a data set. 

To find the mean, sum the data values and divide by the number of data values.


To find the median, first order your data. Then calculate the middle position based on n, the number of values in your data set.


To find the mode, order the numbers lowest to highest and see which number appears the most often.

The mean formula is given by: the sum of a list of numbers/ the number of these numbers.


The median formula can be calculated while following the steps: 

  • Arrange the numbers from smallest to greatest. 
  • If the number of the numbers is odd, the middle value is the median.
  • If the number of numbers is even, the median is the average of the two middle values we have.


The mode formula can be calculated while following the steps: 

  • Rearrange the values of your data set from the lowest to the highest.
  • Note the most occurred data value.


Final Mean Median and Mode Quiz

Question

What are measures of central tendency that attempt to summarise a broader picture? 

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Answer

Mean, median, mode

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Question

What does the symbol μ denote in statistics?

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Answer

Population mean

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Question

To find the median, the data values involved need to be arranged from lowest to highest so what can be figured out much quicker? 


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Answer

The middle point value

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Question

What do you do what you have two middle point values?

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Answer

Find the average of the numbers

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Question

Find the mean of the given data set

56, 78, 22, 43, 19, 35, 54

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Answer

43.9

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Question

What is the median of a data set?

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Answer

The median is the middle point value of the data set when arranged in ascending order. 

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Question

The most occurring data value in a data set is known as ______


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Answer

Mode

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Question

What is the mode for the data set given 56, 78, 22, 43, 78, 19, 35, 54 

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Answer

78

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Question

Rearrange the values in the data set from lowest to highest when finding the median. 

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Answer

True

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Question

Find the mean for the data set 3, 9 ,7, 5 ,3, 4

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Answer

5.16

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Question

Find the median for the data set 3, 9 ,7, 5 ,3, 4 

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Answer

4.5

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Question

Find the mode for the data set 3, 9 ,7, 5 ,3, 4 

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Answer

3

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Question

What does N represent in the formula for finding the mean?

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Answer

Population/ Total number of data

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