Relation Between Mean Median and Mode

Introduction

The realtion between mean , medina and mode is equal to the difference between 3 times the median and 2 times the mean. In statistics, data is a collection of information based on some natural or man-made mathematical phenomenon.

There are various methods of studying data and interpreting some properties of the mathematical phenomenon, but the most common is the central tendencies. Central tendencies, as the name suggests, is a method to find the centre of all the observations in the given data in many different ways, the first is to add all the observations and divide that sum by the number of observations, known as the mean, another is just to select the most common observation, known as mode, and there is another one in which we just select the middle most observation when arranged in an order, known as the median. In this tutorial, we are going to learn about Central Tendencies and their relationship with each other.

Central Tendencies

Central Tendencies are a method to find the most common observation or the region of the most common observations. There are 3 Central Tendencies in statistics,

• Mean

• Median

• Mode

Mean

Mean is the central tendency that selects the observation around which most of the other observations lie, it does so by breaking the data into two by the value, i.e. Mean is the middle most value of the given data such that the value of all the observations lesser or greater than mean is the same.

Mean is denoted by an overline above the symbol of the observations, i.e. most commonly $\mathrm{\overline{x}}$.

There are various means, such as Arithmetic Mean, Geometric Mean, and Harmonic Mean. The most commonly known and used Mean is the Arithmetic Mean, The Arithmetic mean is calculated by adding all the observations, and then dividing that sum by the total number of observations.

The formula for the mean of ungrouped data is,

$$\mathrm{\overline{x}\:=\:\frac{\Sigma\:x_{i}}{N}}$$

where N is the total number of Observations and 𝑥𝑖 is the ith observation.

The formula for the mean of a grouped data is,

$$\mathrm{\overline{x}\:=\:\frac{\Sigma\:f_{i}x_{i}}{\Sigma\:f_{i}}}$$

where $\mathrm{x_{i}}$ is the ith observation and $\mathrm{f_{i}}$ is its frequency, and $\mathrm{\Sigma\:f_{i}\:=\:N}$

Median

Median, as the name suggests, the middle-most observation, i.e. when arranged in an order the middle most observation is the median.

Median of Ungrouped data

Median of ungrouped data is simply calculated by arranging the observations in ascending order and if the number of observations is odd, then the median is the middle-most observation, and if the number of observations is even, then the median is the average of the two middle-most observations.

I.e. if the number of observations N is odd

$$\mathrm{Median\:=\;(\frac{N\:+\:1}{2})^{th}\:observation}$$

And if the number of observations N is even then,

$$\mathrm{Median\:=\frac{\;(\frac{N}{2})^{th}\:observation\:+\:(\frac{N}{2}\:+\:1)^{th}\:observation}{2}}$$

Median of a grouped data

To find the median of a grouped data, we construct a cumulative frequency table,

In the cumulative frequency table, the observation corresponding to the frequency just higher than the half of the total number of observations is the median,

And if the data is divided into classes the formula is as follows,

$$\mathrm{Median\:=\:l\:+\:\frac{\frac{n}{2}\:-\:cf}{f}\:\times\;h}$$

Where l is the lower limit of the median class, cf is the cumulative frequency which is just smaller than the half of the total number of observations, n, f is the frequency corresponding to the median class and h is the height of the class intervals.

Mode

Mode is the central tendency that is the most common observation.

Mode of ungrouped data

Mode of an ungrouped data is simply the term occurring most frequently,

Mode of a grouped data

Mode of a group data is the observation with the most frequency,

If the data is classified into class intervals, then the formula for the mode is,

$$\mathrm{Mode\:=\:l\:+\:\frac{f_{1}\:-\:f_{0}}{2^f_{1}\:-\:f_{0}\:-\:f_{2}}\:\times\:h}$$

Where l is the lower limit of the modal class, 𝑓1 is the modal frequency (highest frequency), 𝑓0 𝑎𝑛𝑑 𝑓2 are the frequency of classes above and below the modal class respectively and h is the height of the class.

Empirical Relationship

The relationship between the three central tendencies is known as the Empirical Relationship, it is given by the equation

$$\mathrm{3Median\:=\:2Mean\:+\:Mode}$$

The Empirical relationship becomes more and more accurate as the sample size of the data increases.

Solved Examples

1) Find the 3 Central Tendencies for the following data and verify the Empirical Relationship.

$\mathrm{x_{i}}$	10	12	13	15	17	18	20	23	25
$\mathrm{f_{i}}$	3	5	6	7	9	8	6	4	2

Answer −

Mean

$\mathrm{x_{i}}$	10	12	13	15	17	18	20	23	25
$\mathrm{f_{i}}$	3	5	6	7	9	8	6	4	2	$\mathrm{\Sigma\:f_{i}\:=\:50}$
$\mathrm{f_{i}\:x_{i}}$	30	60	78	105	153	144	120	92	50	$\mathrm{\Sigma\:f_{i}\:x_{i}\:=\:832}$

$\mathrm{Mean\:=\:\overline{x}\:=\:\frac{\Sigma\:f_{i}x_{i}}{\Sigma\:f_{i}}\:=\:\frac{832}{50}\:=\:16.64}$

Median

$\mathrm{x_{i}}$	10	12	13	15	17	18	20	23	25
$\mathrm{f_{i}}$	3	5	6	7	9	8	6	4	2
C.F	3	8	14	21	30	38	44	47	50

Here, $\mathrm{N\:=\:50\:\Longrightarrow\:\frac{N}{2}\:=\:25}$

That means 𝑐𝑓 = 30

Median = 17

Mode

In the frequency table the highest frequency corresponds to 17

$\mathrm{\Longrightarrow\:Mode\:=\:17}$

The 3 central tendencies are,

Mean = 16.64

Median = 17

Mode = 17

Empirical Relationship

$$\mathrm{3Median\:=\:2Mean\:+\:Mode}$$

$$\mathrm{3\times\:17\:=\:2\times\:16.64\:+\:17}$$

$$\mathrm{51\:\approx\:50.28}$$

Conclusion

• Central Tendencies are a method to find a certain ‘common ground’ between the observations of data

• There are 3 different Central Tendencies

• Mean − It is the average value of the observations. It divides the data in half by the total value.

• Mode − It is the most common observation in the data. The observation that comes up most frequently is also a measure of central tendency

• Median − It is the centremost observation when arranged in order. The observation that is in the middle of the arranged data is also a measure of central tendency

• The relation between the three different central tendencies is known as the Empirical Relationship. It is as follows,

$$\mathrm{3Median\:=\:2Mean\:+\:Mode}$$

FAQs

1. What is a central tendency?

A central tendency is defined as a method to find a central ‘common ground’ between all the observations of a data.

There are 3 different types of central tendencies

• Mean

• Median

• Mode

2. What is the mean? What is the formula of Mean for Grouped and Ungrouped data?

Mean is defined as the number that divides the data into two parts of the equal total value.

There are 3 different types of Mean, Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). But, commonly when talking about the mean, we usually refer to the Arithmetic Mean.

The arithmetic mean is the average of all the data i.e the sum of all the observations divided by the total number of observations

For ungrouped data,

$\mathrm{\overline{x}\:=\:\frac{\Sigma\:x_{i}}{N}}$ where N is the total number of Observations and 𝑥𝑖 is the ith observation.

For grouped data,

$\mathrm{\overline{x}\:=\:\frac{\Sigma\:f_{i}x_{i}}{\Sigma\:f_{i}}}$ where $\mathrm{x_{i}}$ is the ith observation and $\mathrm{f_{i}}$ is its frequency, and $\mathrm{\Sigma\:f_{i}\:=\:N}$, the total number of observations.

3. What is Median? What is the formula to find the Median for Grouped and Ungrouped data?

Median is defined as the centre-most observation.

The Median of ungrouped data is simply the centremost observation or the average of the two centremost observations

Median of the grouped data is calculated by using a cumulative frequency table.

For simple grouped data, the median is the term that corresponds to slightly more than half of the total frequency.

For class intervals,

$$\mathrm{Median\:=\:l\:+\:\frac{\frac{n}{2}\:-\:cf}{f}\:\times\:h}$$

Where l is the lower limit of the median class, cf is the cumulative frequency which is just smaller than half of the total number of observations, n, f is the frequency corresponding to the median class and h is the height of the class intervals.

4. What is Mode? What is the formula for the Mode of Grouped and Ungrouped data?

Mode is defined as the most frequent observation.

The mode of ungrouped and simple grouped data is the observation with the highest frequency.

Mode for class intervals,

$$\mathrm{Mode\:=l\:+\:\frac{f_{1}\:-\:f_{0}}{2f_{1}\:-\:f_{0}\:-\:f_{2}}\times\:h}$$

Where, l is the lower limit of the modal class, 𝑓1 is the modal frequency (highest frequency), 𝑓0 𝑎𝑛𝑑 𝑓2 are the frequency of classes above and below the modal class respectively and h is the height of the class.

5. What is the Empirical Relationship?

The relationship between Mean, Median and Mode is known as the Empirical Relationship

$$\mathrm{3Median\:=\:2Mean\:+\:Mode}$$