## Understanding Central Tendency: Mean, Median, Mode, and Standard Deviation

Introduction Central tendency is a vital concept in statistics that helps us grasp the essence of a dataset by identifying a single value that represents the entire distribution. In this blog, we’ll delve into the fundamentals of central tendency, exploring measures such as mean, median, mode, and standard deviation, along with their merits, demerits, and real-world applications. Example 1: Suppose, one of your relative ask you about your exam result. If you explain the marks of every subjects separately, he/she will become boring and would not able understand your performance. What will you do? You will answer 90%. This is the central value of your marks score in different subjects Example 2: Suppose, you are a class teacher. Marksscore by five students in a test is given below. Will you able to compare theperformance of these students by comparing the marks score in differentsubjects separately. What will you do? You will compare by their percentage. This is the central value of marks score by the five students in the test. Measures of central tendency Mean: Definition: The mean is the average of a set of numbers. It is calculated by summing all the values in the dataset and dividing the sum by the number of values. Mean:For Ungroup Data Ifthe variable x assumes n values x1, x2 … xn, then the mean is given by x ̅=(x_1+x_2+x_3+ . . . +x_n)/n= 1/n ∑_(i=1)^n▒x_i Example:Suppose marks scored by five students in a test is 66, 72, 85, 52, and 75. x ̅=(66+72+85+52+75)/5=350/5=70 Mean:For Grouped data x ̅=(∑fx)/n Wherex = the mid-point of individual class, f = the frequency of individual class N = the sum of the frequencies or total frequencies in a sample. For example (∑fx)/n =1320/50=25.x ̅=38 Marks No. of Students (f) x Fx 0-10 6 5 30 10-20 8 15 120 20-30 17 25 425 30-40 11 35 385 40-50 8 45 360 N=50 1,320 Merits: 1. Easy to understand and calculate. 2. It is rigidly defined. 3. Utilizes all the data points in the dataset. 4. Sensitive to small changes in values. 5. Widely used in inferential statistics. 6. Applicable to both discrete and continuous data. 7. It provides a good basis for comparison. Demerits: 1. It cannot be obtained by inspection nor located through a frequency graph. 2. It cannot be in the study of qualitative phenomena not capable of numerical measurement i.e., Intelligence, beauty, honesty etc., 3. It can ignore any single item only at the risk of losing its accuracy. 4. It is affected very much by extreme values. 5. It cannot be calculated for open-end classes. 6. It may lead to fallacious conclusions, if the details of the data from which it is computed are not given. Some more examples: 1) Consider the following dataset representing the salaries of employees in a small company: {30000, 35000, 40000, 45000, 50000} Mean= 30,000+35,000+40,000+45,000+50,000/5 = 200,000/5 = 40,000 Now, let’s introduce an extreme value, such as the CEO’s salary, which is significantly higher: {30000, 35000, 40000, 45000, 500000} Recalculating the mean = (30,000+35,000+40,000+45,000+500,000)/5 =6,50,000/5 = 1,30,000 2) The Per capita Income of India is Rs. 1,72,000 and the BPL income limit in India is Rs. 27,000. India should not have any PBL population, But India has more than 145.71 million or 10.2% of the total population. Mean interpretation of SPSS data table : Interpretation of Mean: Variable name: 1) Number of Extracurricular Activities participated by the student The mean of 2.68 indicates that, on average, each student participates in approximately 2.68 extracurricular activities. This mean suggests a moderate level of participation in extracurricular activities among the students surveyed. While not exceptionally high, it indicates that students are somewhat involved in activities beyond their regular academic curriculum. Variable name: 2) Income Levels of Students Parent (In Thousand per month) The mean income of 33,800 rupees per month suggests that, on average, the parents of these 100 students earn approximately 33,800 rupees per month. The mean income level of 33,800 rupees per month provides valuable insight into the socioeconomic context of the student population, informing decision-making processes aimed at promoting equity, access, and inclusivity in education. Both variables value calculated is scale or pure numeric. Therefore mean or arithmetic mean or average is used in statistical analysis. Median Definition: The median is the middle value in a sorted list of numbers. If there is an even number of values, it is the average of the two middle values. The median is the middle most item that divides the group into two equal parts, one part comprising all values greater, and the other, all values less than that item. Suppose median salary of a company is Rs. 40,000. This measure of central tendency means that one one-half of all employee earn more than 40,000, and one-half earn less than 40,000. Ungrouped or Raw data Arrangethe given values in ascending order. If the number of values is odd,the median is the middle value. If the number of values are even, median is themean of middle two values. When n is odd, Median = Md = ((n+1)/2)^th value When n is even, Median = Average of (n/2)^th and(n/2+1)^th value Some examples: 1) The number of rooms in 7 hotels in Delhi is 713, 300, 618, 595, 311, 401, and 292. Find the median. Here n = 7, First arrange it in ascending order:292, 300, 311, 401, 595, 618, 713 Median =((n+1)/2)^th value=((7+1)/2)^th value=4^th value = 401 2) The number of floods that have occurred in India over an 8-year period follows. Find the median. 684, 764, 656,702, 856, 1133, 1132, 130 Here n = 8, First arrange it in ascending order: 656, 684, 702, 764, 856, 1132, 1133, 1303 Median = Average of (n/2)^th and(n/2+1)^th value= Average of (8/2)^th and(8/2+1)^th value = Average of 4^th and〖 5〗^th value = (764+856)/2 = 810 Groupeddata Ina grouped distribution, values are associated with frequencies.

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