Basic Concepts of Measures of Central Tendency:

We have learnt about representation of data in various forms like plotting bar graph, histogram, or by using frequency table but while using this method a question arises that is there a need to form certain important features, by considering representative data since studying data to make sense of it is a continuous process. Yes, this is possible by using the concept of averages or method measures of central tendency.

 A measure of central tendency (additionally alluded to as measures of focus or focal area) is an outline measure that endeavors to depict an entire arrangement of information with a solitary esteem that speaks to the centre or central point of its appropriation.

The mode, the median and the mean are the three main measures of central tendency which are used in describing a different mark of central point in the distribution.

Mean: The sum of the values of the observation divided by the total number of observations is the mean of a number of observations. It is generally denoted by the symbol of x bar.

Lets us consider an example. Suppose there are 5 students who scored 50,65,48,78 and 85 marks respectively in their mathematics exam. We need to find the mean of marks of these students.

Mean = Sum of the observations / Total Number of Observations

Mean = 50+65+48+78+85 / 5

Mean = 326 / 5 = 65.2, the mean marks of these 5 students is 65.2.

Direct Method for Mean = ∑f_ix_i /  ∑f_i , for values of i 1 to n. This formula is used in case of an ungrouped frequency distribution. 

Assumed Mean Method: It is a method where we do not calculate mean from mid points, instead we assume mean to find out mean. To start with we "assume" or accept a mean and afterward we apply an amendment to this expected mean so as to locate the correct mean.

Assumed Mean = a + ∑f_id_i /  ∑f_i , where a is the value of assumed mean, f_i  is frequency and d_i (deviation) is obtained by x_i - a. 

Step Deviation Method = a + (∑f_iu_i /  ∑f_i ) * h with the assumption that class frequency is at its central point also known as class mark. Here h is the class size and u_i  =  x_i - a / h .

Benefits of Mean:

1. Mean is unbendingly characterised so that there is no doubt of misconception about its importance and nature.

2. It is the most popular central tendency as it is straightforward.

3. It is anything but difficult to compute.

4. It incorporates every one of the scores of a distribution.

5. It is not influenced by sampling so that the outcome is solid. 

Median: The value of the given number of observations, dividing the observations into two parts. Median of data when arranged in an ascending or descending order is given in the following ways:

For the n odd number of observations, the median is the value of (n+1/ 2)^th  observation. For example we have 9 number of observations of marks scored in mathematics exam. Here the value of n is 9, then the median will be (9 + 1 /2) = 5. Thus the value of  median is 5th observation.

Similarly, for the n even number of observations, the value of the median is the mean of (n / 2)^th and (n/2 + 1)^th  observation. For example we have 18 number of observations of marks scored in mathematics exam. Here the value of n is 18 i.e. an even number, then the median will be mean of (18/2)th and (18/2 + 1)th observations i.e value of median is the mean of 9th and 10th observation.

Median of Grouped Data: In case of grouped data its get difficult to find median in its usual way, so we introduce the concept of Cumulative frequency to find the value of median.

Median = l + (n/2 - cf / f) * h

Where l is the lower limit of median class, n = number of observations, cf = cumulative frequency of preceding class to median class, f = frequency of median class, h = class size.

Note: Median class is a class whose frequency is greater than and nearest to n/2.

Frequency obtained by adding the frequencies of the preceding classes is known as the cumulative frequency of a class.

Benefits of Median:

1. It is anything but difficult to register and get it.

2. Every observation is not required for its calculation.

3. Extraordinary scores do not influence the median.

4. It can be resolved from the open-ended arrangement.

5. It can be resolved from unequal class interims. 

Mode: The value of observation which occurs most frequently or the observations which has highest number of frequency of occurrence is known as the mode. For example the value of the mode of the following marks obtained by 10 students : 55, 85, 75, 65, 85, 95, 85, 82, 80, 93 is 85, because 85 is the value with maximum frequency i,e, 3 times.

Mode of Grouped Data : In a grouped frequency, it gets difficult to determine the value of mode by looking at frequencies. Hence we observe a class with maximum frequency known as modal class. The mode is the value in a modal class, represented by a formula:

Mode =  l + (f₁ - f₀ / 2f₁ - f₀ - f₂) * h

where l = Modal class lower limit , h = class interval size , f₁ = Modal class frequency , f₀ = frequency of class preceding modal class , f₂ = frequency of class succeeding modal class.

Benefits of Mode:

1. Mode gives the most illustrative value of an arrangement.

2. The mode is not influenced by any extreme scores like mean.

3. It can be determined from an open-ended class interim.

4. It helps in analysing and evaluating qualitative data.

5. The mode can likewise be determined graphically through histogram or recurrence polygon.

6. The mode is straightforward.

Importance of Measures of Central Tendency: 

Mean is the regular "normal". In a few sections of the world, there are sets of activity cameras that take pictures of autos and measure the time between when the photos are taken. They then figure the rate (separate/time). The outcome is the mean travel speed. (Additionally, utilizes the Mean Value Theorem). In school, your last grade is a mean. Total them then gap by the number and you have your score. To improve it is a weighted mean where each score may "measure" more than the others, for example, an Exam. Any Per-Capita insights report is also a use case of mean. The mean, or the normal, is an imperative measurement utilised as a part of games. Mentors utilise midpoints to decide how well a player is performing. General Managers may utilise mean to decide how great a player is and how much cash that player is worth. Median is used in detailing earnings. The median pay in a territory discloses to you progressively what the "normal" individual procures. A couple cosmically high values, from CEOs, Bill Gates, and so on through the Mean off, so the BLS utilises Median. The median is utilised as a part of financial aspects. For instance, the U.S. Evaluation Bureau finds the median family unit wage. As indicated by the U.S. Enumeration Bureau, "median family unit salary" is characterised as "the sum which isolates the pay dissemination into two equivalent gatherings, half having wage over that sum, and half having wage underneath that sum." The mode might be helpful for a supervisor of a shoe store. For instance, you would not see measure 17 shoes supplied on the floor. Why? Since not very many individuals have a size 17 shoe measure. Thusly, store supervisors may take a gander at information and figure out which shoe size is sold the most. Chiefs would need to stock the floor with the top rated shoe measure.

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