Median and Mode: Definition, Formulae, Relation, Videos with Examples

The number of students in your classroom, the money of money your parents earns, the temperature in your city are all important numbers. But how can you get the information of the number of students in your school or the amount earned by the citizen of your entire city? This is where median and mode comes is useful. So let us now study median and mode in detail.

Median

Median and Mode

To define the median in one sentence we can say that the median gives us the midpoint of the data. What do you mean by the midpoint? Suppose you have ‘ n ‘ number of data, then arrange these numbers in ascending or descending order. Just pick the midpoint from the particular series.

From the Raw Data

The very first thing to be done with raw data is to arrange them in ascending or descending order. In Layman’s terms:

Median = the middle number

The median number varies according to the total number being odd or even. Initially let us assume the number as the odd number. Now if we have numbers like 12, 15, 21, 27, 35. So here we can say that the midpoint here is 21,

Median = \( (\frac{n + 1}{2} )\)^thterm if n is odd

What if we have even number? Let us take the same set of series but this time just add one number to the series: 12, 15, 21, 27, 35, 40. So in this case when the numbers are even then taking the mean of the numbers,

Median = Mean( \( \frac{n }{2} \)^thand \( (\frac{n }{2} + 1)\)^thterms )

So in the above example, take the mean of 21 and 27 and divide it by 2 which will give you 24.

Case Study

Let us understand the median through another example. Suppose a restaurant collects the cans for two weeks and sends it to a recycling plant. The number of cans collected each day are: 84, 97, 77, 31, 84, 58, 63, 72, 47, 84, 64, 94, 43 and 68.

Now we need to find the median of these numbers. The first step to find the median is to arrange the numbers either in ascending order of descending order. So arranging the data in the ascending order, 31, 43, 47, 58, 63, 64, 68, 72, 77, 84, 84, 84, 94 and 97. Here the total numbers are even. So using the formula

Median = (\( (\frac{n }{2}) \)^th+ \( (\frac{n }{2} + 1)\)^thterms)/2

Since n = 14, 7^thterm = 68 and 8^thterm = 72. Now we have our two middle terms as 68 and 72, hence

Median = \( \frac{68 + 72}{2} \) = 70

Mode

The most frequent value in the data set is called as the mode. Suppose you have scored the following marks in your exams.

1^st sem	75
2^nd sem	59
3^rd sem	75
6^th sem	68
4^th sem	75
5^th sem	60

First, you need to arrange the numbers in the order

3^rd sem	55
6^th sem	60
4^th sem	68
1^st sem	75
2^nd sem	75
5^th sem	75

Now check for the number that occurs for the most number of time. Here the number 75 occurs the maximum times. So the number 75 is the mode of this data set. This is the unimodal data. Since it only has one mode. The data mode which has two modes is called as bimodal data. Similarly, if the data set has more than two modes, we call it as multimodal data.

Relation Between Mean, Median and Mode

If the value of the n=mode is equal to the value of the median and the mean then we call it as symmetrical data set. For such data sets, there is a simple relationship between the three M’s (mean, median and mode):

Mode = 3 Median – 2 Mean

But if the data set is asymmetrical then this relationship might not hold true.

Mode ≠ 3 Median – 2 Mean

Solved Examples for You

Question 1: The weights of students of a certain class are given below. 39, 42, 47, 38, 42, 40, 42, 38, 43, 42, 38, 44, 46, 39, 42, 40, 43, 42, 41. Find the mode.

Answer : The correct option is D. Weight of the students: (arranging them in ascending order) 38, 38, 38, 39, 39, 40, 40, 41, 4642, 42, 42, 42, 42, 42, 43, 43, 44, 46, 47. 42 occurs 6 times and a maximum number of times in the data. Hence, 42 is the mode weight.

Question 2: In a diagnostic test in mathematics given to students, the following marks (out of 100) are recorded: 46, 52, 48, 11, 41, 62, 53, 54, 96, 40, 98, 44. Which average will be a good representative of the above data?

Mean
Median
Can not be calculated
None of these

Answer : The correct option is B. Median will be the good representative of data. Because each value occurs once and the data is influenced by extreme values.

Question 3: How do you find the mode?

Answer: The mode of a data set is basically the number which takes place most commonly in the set. To find the mode in an easy manner, you must arrange the numbers in order from least to greatest. Then, we must count how many times is each number is occurring. Finally, the number that will occur the most will be your mode.

Question 4: How do you find the median in a frequency table?

Answer: The median refers to the middle number in an ordered set of data. You see that in a frequency table, they have already set the observations in ascending order. Thus, it will be easier to attain the median when we look for the value in the middle position. Moreover, if you have an odd number of observations, your middle number will be the median.

Question 5: What is the median of numbers?

Answer: The median refers to the middle value in the list of numbers. In order to find the median, you must list your numbers in numerical order from the smallest to the greatest. Thus, you will have to rewrite the list before finding the media. Moreover, the mode will be your value which takes place most often.

Question 6: How do you find the probability?

Answer: In order to find the probability, you first need to divide the number of events by the number of potential results. You will get the probability of single even taking place by this. If there is a case of rolling a 3 on a die, the number of events will be 1 as there is only a single 3 on each die. Thus, the number of results will be 6.