Calculation of Mean, Median, and Mode

Mean, Median, and Mode

Mean, Median and Mode are the three statistical measures often used to encapsulate data sets. They are mainly known by the expected name “average”. There are many “averages” in statistics, but these three are the most common. 


The “mean” is mainly the “average” where all the numbers are added up and then divided by the number of digits. The "median" is the "middle" value in the number series, and the "mode” is the value that occurs on numerous occasions. Out of these three measures, the mode formula in statistics opens different ranges and aspects for the student's


Formulas of Mean and Median

The first theory to understand from the mean, median, and mode is mean. As described above, the mean is the ratio of summation of all the values to the number of items. There are two types of Mean - simple arithmetic mean, and weighted arithmetic mean. If you want to know what is the formula of mean, go through the section given below:


\[ Mean= \frac{\sum X}{ N}\]


Here, \[ \sum X\]= Sum of individual values and =Total number of Items


The simple arithmetic mean examine values in data as equal and permits equal importance to each value. On the other hand, in weighted arithmetic mean, weights or importance is assigned to the values.


To find the Median, the numbers should be arranged in numerical order from smallest to highest. Taking the number of observations as (n), and “th” signifies the (n)th number -


Median Formula = \[\frac{(n+1)}{2} th\]


Students who are looking for mean, median, mode formula pdf can wish to visit the official website of Vedantu. 


Mode Formula in Statistics for Discrete Series: 

Mode comes third in the concept of mean, median, and mode. As discussed earlier, it merely refers to the value that occurs most frequently in a number series. In discrete series, the values of items with their equivalent frequencies are found. Essentially, the value of the items with the highest frequency will be the mode for the distribution. Mode formulae in statistics help the students to solve different complicated statistical problems in their textbooks. Students can follow this method of mode in discrete series from an example shown below -


Example 1 - Find Mode from the following data

12

14

16

18

26

16

20

16

11

12

16

15

20

24


Ans: Arranging data in ascending order

11

12

12

14

15

16

16

16

16

18

20

20

24

26


The term that is occurring the maximum number of times is considered as Mode (Z). Here we get 16 four times, 12 and 20 two times each, and other terms once only. Thus Z=16


Students can find more examples of mode calculation in discrete series in Vedantu’s online classes.


Illustrating Measures of Central Tendency

Mean, median, and mode are together known as measures of central tendency. It signifies a synopsis of data that represents the center point or quintessential value of a dataset. These measures specify where most values in a distribution fall and are also referred to as the central location of a distribution. Best measures of central tendency can be chosen depending on the data you have. A “measure of central tendency” is either a location framework or a statistic used to evaluate a location parameter.


Several Measures of Central Tendency

  • Arithmetic mean

  • Median

  • Mode

  • Geometric mean 

  • Harmonic mean

  • Generalized mean

  • Weighted mean

  • Truncated mean

  • Interquartile mean

  • Midrange

Students can solve their statistical problems with measures of central tendency formulas that are given below -

Arithmetic Mean \[ (\overline{X} = \frac{\sum X}{N})\]

(Where \[\sum X\] is the summation of all the observations provided in a dataset and “n” is the number of observations)

Median formula= \[ \frac{{(n+1)}}{2}\]


Mode Calculation for Continuous Series

While demonstrating the mode formula for continuous series, it is only one step forward of the method for discrete series. We get the value of Mode by interposing as in the case with a median. Mode formula in statistics is easy to understand and pretty simple to calculate Formula provided below has been used to calculate Mode (Z):

Mode (Z) = \[L + \frac{f_{1} - f_{0}}{2f_{1} - f_{0} - f_{2}} * i\]

Where L= lower limit of modal interval

\[f_{1}\]= frequency correlating to modal interval

\[f_{0}\]= frequency heading modal interval

\[f_{2}\]= frequency succeeding modal interval

\[i\]= length of modal interval


Several Points While Calculating Mode -

  • Class intervals must be absolute

  • Length of the classes should be equal

  • Series should be arranged in ascending order

  • If the series is increasing, then convert it into continuous series

  • If the first class is considered as a modal class, then f0 will be zero

  • Similarly, if the last class is considered as a modal class, then fis zero

Students can have access to online resources where they can get innumerable solutions considering the formula of mode in continuous series. To know more about this topic, download Vedantu’s app now!

The mean, median, and mode are the three classifications of central tendency.           


Mean

The arithmetic average of the given data set which is found by the division of the data set by the total number of observations in the data set is referred to as the mean of the data set.

\[ \overline{x} = \frac{\sum x}{N}\]

where the summation is represented by \[ \sum\]

observations are represented by x

number of observations is represented by N

If the data is presented in a tabular form, in this case, the formula used to compute the mean is

Mean=\[ \frac{\sum f x}{\sum f} \]

Where \[ \sum f = N\]


Median

The middle number in the data set is referred to as the median when the numbers are listed in either ascending or descending order.

In the case when the total number of observations (n) is an odd number, then the formula of the median is given by-

\[Median = \left ( \frac{n+1}{2} \right )^{th}observation\]

In the case when the total number of observations (n) is an even number, then the formula for the median is given by-

\[ Median = \frac{(\frac{n}{2})^{th}observation + (\frac{n}{2} + 1)^{th} observation}{2} \]

When the data is continuous and is presented in form of the frequency distribution, then the median given by-

\[Median=1 + \left [ \frac{(\frac{n}{2}) - c}{f} \right ] \times h\]

where the lesser limit which belongs to the median class= l

cumulative frequency value before the median class= c

frequency that is possessed by the median class= f

size of the class= h


Mode

The value in a data set that has occurred the most number times is referred to as the mode and the difference between the highest and lowest values of the data set is referred to as the range.

To calculate mode in a data set, the following steps are followed-

  1. Determination of the modal class that possesses the maximum frequency.

  2. Calculation of the mode by the formula- \[Mode= 1+ \left [ \frac{f_{m}-f_{1}}{2fm-f_{1}-f_{2}} \right ] \times h\]

where frequency possessed by modal class= \[f_{m}\]

frequency possessed by the class that is before the modal class= \[f_{1}\]

frequency possessed by the class that is after the modal class= \[f_{2}\] 

Lesser limit of modal class= l

Width of the class= h

FAQs on Calculation of Mean, Median, and Mode

1. What is the relation between the mean, mode, and median?

The central tendency has three estimates that include the mean, mode, and the median, and all three are related by an empirical relationship which is 2 Mean + Mode = 3 Median.

2. What is meant by the range?

For discrete random variables the range of distribution is referred to as the difference between the maximum and the minimum values while for distribution with continuous random variables, the range is referred to as the difference between the two extreme points that are present on the distribution curve where the value of the function falls to zero. To learn more about Range students can visit vedantu.

3. What is the mean value? Illustrate with a numerical example.

The most common form of “average” used is mean. To find the mean value, we have to add up all the numbers and then divide them by the number of data that you are equipped with. 

For example, what is the meaning of 2, 7, and 9? The solution is –

Adding the numbers: 2+7+9 = 18

No. of data = 3

Mean = 6

4. How can we calculate the median in a discrete series?

To find out the median in a discrete series, we need to follow a few steps - 

  1. Firstly we have to arrange a given number series in an ascending or descending order

  2. Secondly, we need to calculate the increasing frequencies

  3. Thirdly, we need to use this formula to calculate median

Median= Size of \[ \frac{N + 1}{2^{th}}\] item, where N (total of frequencies)

In the cumulative frequency, we need to search for the cumulative frequency that is equal to (N+1)/2 or the increasing frequency that is higher than this.

  1. The corresponding value of the variable is the value of the median.

5. What is the mode value?

Mode refers to the value that is frequently appearing in a set of values or number series. Note: If no number is often occurring in the series of items, then the answer will be "No mode." Students can find solutions based on mode formulas in statistics online.

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