**Introduction**

There are different tools or methods for the determination of the presentative value or central value of a group of observations. In different situations, different methods or tools are used to find the central value of a group of observations. Also, for different requirements from a group of observations or data, we use different measures of central value. One such value is the median. Let us consider a group of 17 students of a school with the following height in class –

101, 106, 116, 110, 109, 123, 118, 127, 120, 117, 102, 122, 115, 112, 121, 125, 111

What will be the mean height of these students? The mean height of this groups of students will be given by

$\frac{101+106+116+110+109+123+118+127+120+117+102+122+115+112+121+125+111}{17}$

Hence, the mean height of this groups of students = $\frac{1955}{17}$ = 115

Now, we can clearly see that there are 7 students of heights less than the mean height and 9 students are of heights greater than the mean height. Let us now suppose that the sports teacher of the school wishes to divide this group of 17 students into two groups on the basis of the mean height so that one group has students of heights less than the mean height and the other group has students of height more than the mean height. Using the mean as the deciding factor would mean that one group has more students than the other group. Therefore, mean cannot be the only parameter in such situation. In other words, we cannot always use to divide a group into two equal groups. So, there must be some another representative value or central value that can help us divide a group into two equal sub – groups. Such a representative value or central value is known as the median of the data. Median is the value that lies in the middle of the data with half of the observations above it and the other half below it. Let us now learn more about the median.

**Definition**

**The median of a group of observations is the value of the variable which divides the group into two equal parts.** In other words + the median is the value that exceeds and is exceeded by the same number of observations + i.e. it is the value such that the number of observations above it is equal to the number of observations below it.

In simple terms + the median is the value that lies in the middle of the data with half of the observations above it and the other half below it.

**Median of Ungrouped ( Raw ) Data**

In the case of individual observations + i.e. ungrouped data + the following are used to find the median.

Step 1 Arrange the observations in ascending or descending order or magnitude.

Step 2 Determine the total number of observations + say + n.

Step 3 + If n is odd + then use the following formula –

Median = Value of ($\frac{n+1}{2}$)^{th} observation

If n is even + then use the following formula

Median = $\frac{Value\: of\: \frac{n}{2}^{th}\: observation+Value\: of\: (\frac{n}{2}+1)^{th}\: observation}{2}$

Let us understand both the scenarios with the help of examples

**Example 1**

Find the median of the data: 24, 36, 46, 17, 18, 25, 35

**Solution**

We will find the median using the above steps –

Step 1 Arrange the observations in ascending or descending order or magnitude.

Therefore + let us arrange the numbers in ascending order. We get –

17 , 18, 24 , 25 , 35 , 36 , 46

Step 2 Determine the total number of observations , say + n.

Here the total number of observations n is 7 , which is an odd number

Step 3 If n is odd , then use the following formula –

Median = Value of ($\frac{n+1}{2}$)^{th} observation

Hence , we find the value of ($\frac{n+1}{2}$)^{th} observation = ($\frac{7+1}{2}$)^{th} observation = 4^{th} observation

4^{th} observation in the list is 25

**Hence the median of the given set of observations is 25.**

**Example 2**

Find the median of the data – 25 , 34 , 31 , 23 , 22 , 26 , 35 , 28 , 20 , 32

**Solution**

We will find the median using the following steps –

Step 1 Arrange the observations in ascending or descending order or magnitude.

Therefore , let us arrange the numbers in ascending order. We get –

20 , 22 , 23 , 25 , 26 , 28 , 31 , 32 , 34 , 35

Step 2 Determine the total number of observations , say + n.

Here the total number of observations n is 10 , which is an even number

Step 3 + If n is even , then use the following formula

Median = $\frac{Value\: of\: \frac{n}{2}^{th}\: observation+Value\: of\: (\frac{n}{2}+1)^{th}\: observation}{2}$

Now , the value of $\frac{n}{2}$^{th} observation =value of $\frac{10}{2}$^{th} observation = value of 5^{th} observation = 26

and

Value of ($\frac{n}{2}$+1)^{th} observation = value of ($\frac{10}{2}$+1)^{th} observation = value of 6^{th} observation = 28

Therefore , Median = $\frac{26+28}{2} = \frac{54}{2}$ = 27

**Hence , the median of the given observations = 27**

**Uses of Median**

- Median is the only average that is used while dealing with qualitative data which cannot be measured quantitatively but still can be arranged in ascending or descending order of magnitude. For example + the median is used to find the average honesty + average intelligence + average beauty etc.
- Median is used for determining typical value in problems concerning wages distribution of wealth etc.

**Properties of Median**

- The median can be calculated graphically + while the mean cannot.
- The sum of the absolute deviations taken from the median is less than the sum of absolute deviations from any other observations in the data.
- The median is not much affected by extreme values and would therefore be a better representative as a centre or average.

**Relation between Mean, Median and Mode**

Recall that the mean, also known as the “Arithmetic Mean” of a group of observations is the value that is equally shared out among all the observations. In simple terms, it is the average of a set of numbers.

Ungrouped data refers to the data where the numbers are present in a simple manner, with each value defined separately. The arithmetic mean of ungrouped data is defined as

Arithmetic Mean = $\frac{Sum\: of\: all\: observations}{Number\: of\: observations}$

Thus, if x_{1}, x_{2}, x_{3}, ….…. x_{n} are the values of n observations then the arithmetic mean of these observations is given by

Arithmetic Mean = $\frac{x_1+ x_2+ x_3 + ……+ x_n}{n}$

Also, we know that **the mode is the most frequently occurring value in a data set.** In other words, the mode is of distribution is the value at the point around which the items tend to most heavily concentrated. Thus mode or the modal value is the value in a series of observations that occurs with the highest frequency.

For instance, let the scores 10 students be

45, 40, 44, 44, 43, 38, 44, 35, 18, 38

Let us see which scores appears how many times in the List

Score | Number of times it appears |

45 | 1 |

40 | 1 |

44 | 3 |

43 | 1 |

38 | 2 |

35 | 1 |

18 | 1 |

From the above table, we can see that the mode is 44 which occurs thrice and the other scores occur only once or twice.

Now, can we define a relation between the mean, median and the mode? Let us find out.

Mean , median and mode are correlated by the following formula –

**Mode = 3 Median – 2 Mean**

**Example**

If the ratio of mode and median of a certain data is 6: 5 , then find the ratio of its mean and median.

**Solution**

We know that the Empirical formula defining the relation between mean , median and mode is

Mode = 3 Median – 2 Mean

We have been given that the ratio of mode and median of a certain data is 6: 5.

This ratio can also be written as

$\frac{Mode}{Median} = \frac{6}{5}$

Or

Mode = $\frac{6 x Median}{5}$

Now + we replace this value of mode in the empirical formula.

We get

$\frac{6 x Median}{5}$ = 3 Median – 2 Mean

⇒ 6 x Median = 5 ( 3 Median – 2 Mean)

⇒ 6 x Median = 15 Median – 10 Mean

⇒ 10 Mean = 15 Median – 6 Median

⇒ 10 Mean = 9 Median

We are required to find the ratio of mean and median. Hence we will write the above equation as –

$\frac{Mean}{Median} = \frac{9}{10}$

Or Mean : Median = 9 : 10

**Hence , we can say that if the ratio of mode and median of a certain data is 6: 5 + then the ratio of its mean and median will be 9:10.**

**Solved Examples**

**Example 1** The following number of goals were scored by a team in a series of 10 matches: 2 , 3 , 4 , 5 , 0 , 1 , 3 , 3 , 4 , 3

Find the median of these scores.

**Solution**** **Here + the observations in this case are – 2 , 3 , 4 , 5 , 0 , 1 , 3 , 3 , 4 and 3

Total number of observations = 10

To calculate the median , we will first arrange the observations in ascending or descending order or magnitude.

We get ,

0 , 1 , 2 , 3 , 3 , 3 , 3 , 4 , 4 , 5

Now , we determine the total number of observations , say , n. Here the total number of observations n is 10 , which is an even number.

If n is even , then we use the following formula

Median = $\frac{Value\: of\: \frac{n}{2}^{th}\: observation+Value\: of\: (\frac{n}{2}+1)^{th}\: observation}{2}$

Now , the value of $\frac{n}{2}^{th}$ observation =value of $\frac{10}{2}$^{th} observation = value of 5^{th} observation = 3

and

Value of ($\frac{n}{2}$+1)^{th} observation = value of ($\frac{10}{2}$+1)^{th} observation = value of 6^{th} observation = 3

Therefore , Median = $\frac{3+3}{2} = \frac{6}{2}$ = 3

**Hence , median of the given observations = 3**

**Example 2**** **Find the median of the data 19 , 25 , 59 , 48 , 35 , 31 , 30 , 32 , 51. If 25 is replaced by 52. What will be the new median?

**Solution** We have been given the set of observations as 19 , 25 , 59 , 48 , 35 , 31 , 30 , 32 , 51. Also, we have been given that 25 is replaced by 52 to obtain a new set of observations. We are required to find the median of this new set of observations.

Let us first arrange the given data in ascending order. Arranging the data in ascending order we will have,

19 , 25 , 30 , 31 , 32 , 35 , 48 , 51 , 59

Now, we can see that the number of observations in the given set of observations is 9. Hence, we can say that n = 9 which is an odd number.

Now, we know that in case of odd number of observations,

Median = Value of ( $\frac{n+1}{2}$ ) ^{th} observation

So, we have,

Median = Value of ( $\frac{9+1}{2}$ ) ^{th} observation = Value of ( $\frac{10}{2}$ ) ^{th} observation = Value of 5 ^{th} observation =

Now, from the ascending order of the data we can see that the 5^{th} value of the given set of observations is 32. Hence, median = 32

If 25 is replaced by 52 then the new set of observations will be 19 , 52 , 59 , 48 , 35 , 31 , 30 , 32 , 51. Let us arrange this new set of observations in ascending order. We will have,

19 , 30 , 31 , 32 , 35 , 48 , 51 , 52 , 59

Since the total number of observations remains the same, we shall use the same formula, i.e. Median = Value of ( $\frac{n+1}{2}$ ) ^{th} observation = 5^{th} observation.

Now, the 5^{th} observation in the new set of observations is 35. Hence the new median will be 35.

**Example 3** The number of rooms in the seven five stars hotel in a city is 71 , 30 , 61 , 59 , 31 , 40 and 29. Find the median number of rooms

**Solution**** **We have been given the set of observations as 71 , 30 , 61 , 59 , 31 , 40 and 29. We need to find the median of this data.

Let us first arrange the given set of observations in ascending order. Arranging the data in ascending order we will have,

29 , 30 , 31 , 40 , 59 , 61 , 71

Now, we can see that the number of observations in the given set of observations is 7. Hence, we can say that n = 7 which is an odd number.

Now, we know that in case of odd number of observations,

Median = Value of ( $\frac{n+1}{2}$ ) ^{th} observation

So, we have,

Median = Value of ( $\frac{7+1}{2}$ ) ^{th} observation = Value of ( $\frac{8}{2}$ ) ^{th} observation = Value of 4 ^{th} observation =

Now, from the ascending order of the data we can see that the 4^{th} value of the given set of observations is 40. Hence, median = 40

**Key Facts and Summary**

- The median of a group of observations is the value of the variable which divides the group into two equal parts.
- In the case of individual observations + i.e. ungrouped data + if n is odd + then use the following formula – Median = Value of ($\frac{n+1}{2}$)
^{th}observation. If n is even + then use the following formula – Median = $\frac{Value\: of\: \frac{n}{2}^{th}\: observation+Value\: of\: (\frac{n}{2}+1)^{th}\: observation}{2}$ - Median is the only average that is used while dealing with qualitative data which cannot be measured quantitatively but still can be arranged in ascending or descending order of magnitude. For example + the median is used to find the average honesty + average intelligence + average beauty etc.
- Median is used for determining typical value in problems concerning wages distribution of wealth etc.
- The median can be calculated graphically + while the mean cannot.
- The sum of the absolute deviations taken from the median is less than the sum of absolute deviations from any other observations in the data.
- The median is not much affected by extreme values and would therefore be a better representative as a centre or average.
- Mean, also known as the “Arithmetic Mean” of a group of observations is the value that is equally shared out among all the observations. In simple terms, it is the average of a set of numbers.
- The mode is the most frequently occurring value in a data set. In other words, the mode is of distribution is the value at the point around which the items tend to most heavily concentrated. Thus mode or the modal value is the value in a series of observations that occurs with the highest frequency.
- Mean + median and mode are correlated by the formula Mode = 3 Median – 2 Mean

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