ANA 809 - Measures of Central Tendency - Emmanuel Uchenna.pptx

In statistics, the central tendency is the descriptive summary of a data set.
The central tendency is the statistical measure representing the single value of the entire distribution or a dataset. It aims to accurately describe the entire data in the distribution(Mean, Mode, and Median -Measures of Central Tendency -When to Use With Different Types of Variable and Skewed Distributions | LaerdStatistics, n.d.).


3 Ways of Measuring Central Tendency:
● The Mean
● The Median
● The Mode

1. Measures of Central Tendency Presented by : Ebere, Uchenna Emmanuel Course code : ANA 809

2. Table of Content 01. 04. 02. 05. 03. 06. Introduction Mean Median Mode Group Data Conclusion

3. Introduction 01

4. Introduction In statistics, the central tendency is the descriptive summary of a data set. By definition, the central tendency is the statistical measure that represents the single value of the entire distribution or a dataset. It aims to provide an accurate description of the entire data in the distribution ( Mean, Mode and Median - Measures of Central Tendency - When to Use With Different Types of Variable and Skewed Distributions | Laerd Statistics , n.d.) . There are questions that are difficult to provide answers to . Questions such as: “ how many calories do I eat per day?” or “how much time do I spend talking per day? ” can be hard to answer because the answer will vary from day to day. It’s sometimes more sensible to ask “how many calories do I consume on a typical day?” or “on average, how much time do I spend talking per day?”. Measures of Central Tendency help us answer these type of questions.

5. Introduction (Cont‘d) 3 Ways* of Measuring Central Tendency: The Mean The Median The Mode *Each measure has its own strengths and weaknesses ( Mean, Mode and Median - Measures of Central Tendency - When to Use With Different Types of Variable and Skewed Distributions | Laerd Statistics , n.d.) . .

6. Mean 02

7. Mean The mean is the most commonly used measure of central tendency (Manikandan, 2011). It represents the average value of the dataset (Hurley & Tenny , 2023). It can be calculated as the sum of all the values in the dataset divided by the number of values. The mean is considered as the arithmetic mean. According to Reddy (2017), some other kinds of mean used to find the central tendency are as follows: Geometric Mean Harmonic Mean Weighted Mean For the purpose of this lesson, we would be taking a look at Arithmetic Mean .

8. Mean (Cont‘d) The Arithmetic Mean ( or mean) is nothing but the average. It is computed by adding all the values in the data set divided by the number of observations in it ( Arithmetic Mean - Definition, Examples, Calculating/Finding , n.d.). If we have the raw data, mean is given by the formula: Where, ∑ (the uppercase Greek letter sigma), X refers to summation, refers to the individual value and n is the number of observations in the sample (sample size). According to Javaid (n.d.), the population mean of m numbers x1, x2, . . . , xm is denoted by µ and is computed as follows: The sample mean of the numbers x1, x2, . . . , xn is denoted by x̄ and is computed similarly:

9. Calculating the Mean Efficiently We can calculate the mean more efficiently using frequencies. We can see from the Exercise 2 calculation above that: 3, 2, 4, 1, 1, 2, 3, 4, 2, 3, 2, 0, 0, 1, 3 # of books Frequency # books x Frequency 2 0 x 2 1 3 1 x 3 2 4 2 x 4 3 4 3 x 4 4 2 4 x 2 We can calculate the mean like so: x̄ =  

10. Calculating the Mean Efficiently We can calculate the mean more efficiently using frequencies. We can see from the Exercise 2 calculation above that: 3, 2, 4, 1, 1, 2, 3, 4, 2, 3, 2, 0, 0, 1, 3 Outcome O i Frequency f i Outcome x Frequency O 1 f 1 O 1 x f 1 O 2 f 2 O 2 x f 2 O 3 f 3 O 3 x f 3 ⋮ ⋮ ⋮ O R f R O R x f R We can calculate the mean like so: x̄ = Where: Σfx is the sum of the product of each frequency (f) and its corresponding outcome (value) (O i ) f is the total frequency  

11. Advantages and Disavantages: Mean

12. Exercise 2 Exercises: Mean The following data shows the results for the number of books that a random sample of 15 students were carrying in their book bags: 3, 2, 4, 1, 1, 2, 3, 4, 2, 3, 2, 0, 0, 1, 3 The mean of the sample is the average number of books carried per students: X = 3+2+4+1+1+2+3+4+2+3+2+0+0+1+3 / 15 = 31/15 = 2.06 Consider the following set of data, showing the number of times a sample of 8 students check their phones per day: 3, 4, 4, 8, 7, 2, 2, 2 Here, n =8; x 1 =3; x 2 =4; x 3 =4; x 4 =8; … x 8 =2 X = 3+4+4+8+7+2+2+2 / 8 = 32/8 = 4 Exercise 1 The mean does not necessarily have to be one of the values observed in our data; in this case, it is a value that could never be observed.

13. Median 03

14. Median The median of a set of quantitative data is the middle number when the measurements are arranged in ascending or descending order ( Library Guides: Statistics: Measures of Central Tendency , n.d.). To Calculate the Median: Arrange the n measurements in ascending (or descending) order. The median of the data is denoted by M ( Library Guides: Statistics: Measures of Central Tendency , n.d.). If n is odd, M is the middle number. If n is even, M is the average of the two middle numbers.

15. Advantages and Disavantages: Median

16. Exercise 2 Exercises: Median The following data shows a sample of 5 students who were asked how much money they were carrying. Their answer are shown below: ₦0, ₦75, ₦25, ₦5, ₦0 Find the median money. To calculate the median, arrange in asc. order: ₦0, ₦0, ₦5 , ₦25, ₦75 The median therefore is: ₦5 The number of goals scored by the 32 teams in the 2014 world cup are shown below: 18, 15, 12, 11, 10, 8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4 , 4 , 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1 a) Find the median of the above set of data Notice that the data is arranged in descending order. There are 32 events and half of 32 is 16. The Median therefore, is   Exercise 1

17. Mode 04

18. Mode The mode of a set of measurements is the most frequently occurring value; it is the value having the highest frequency among the measurements ( WRKDEV100-20011 - Measures of Central Tendency: Mean, Median, and Mode , n.d.). It represents the frequently occurring value in the dataset. The downside to using the mode as a measure of central tendency is that a set of data may have no mode, or it may have more than one mode.

19. Advantages and Disavantages: Mode

20. Exercise 2 Exercises: Mode The following data shows a sample of 5 students who were asked how much money they were carrying. Their answers are shown below: ₦0, ₦75, ₦25, ₦5, ₦0 Find the mode money. Since ₦0 occurs twice and all the other events are unique, the mode is ₦0. The mode money therefore is: ₦0 The number of goals scored by C. Ronaldo at age 14 for different matches are shown below: 5, 4, 2, 3, 2, 1, 5, 4, 5 a) Find the mode of the above set of data Since the mode represents the most common value. Hence, the most frequently repeated value in the given dataset is 5. Exercise 1

21. Group Data 05

22. Group Data Grouping of data plays a significant role when we have to deal with large data. This information can also be displayed using a pictograph or a bar chart. Consider the marks of 50 students of Anatomy Department obtained in ANA 701. The maximum marks of the exam are 50. 23, 8, 13, 18, 32, 44, 19, 8, 25, 27, 10, 30, 22, 40, 39, 17, 25, 9, 15, 20, 30, 24, 29, 19, 16, 33, 38, 46, 43, 22, 37, 27, 17, 11, 34, 41, 35, 45, 31, 26, 42, 18, 28, 30, 22, 20, 33, 39, 40, 32 A frequency distribution table will help us to properly summarize, and analyze these marks for better understanding. The table below shows a group of observations of 0 to 10, 10 to 20 etc.

23. Group Data - Cont‘d This distribution is known as grouped frequency distribution. From this, we can make the following assertions: Many students got between 20 – 40 marks 8 students got higher than 40 marks The groups 0-10, 10-20, 20-30 ,… are known as class intervals (or classes).

24. Mean – Group Data How to find the mean of a group data Example : The following table ( below ) gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean. Solution : X ( midpoint ) is the midpoint of the class. It is gotten by adding the upper class limits and the lower class limit and divide the sum by 2.

25. Median – Group Data How to find the median of a group data Step 1 : Construct the cumulative frequency distribution. Step 2 : Determine the class width ( i ) , lower boundary median class ( Lm ) Step 3 : Decide the class that contain the median. Class Median is the first class with the value of cumulative frequency equal at least n/2. Step 4 : Find the median by using the following formula: Formula for median (grouped data)

26. Median – Group Data - Cont‘d How to find the median of a group data

27. Median – Group Data - Cont‘d How to find the median of a group data

28. Mode – Group Data How to find the mode of a group data above it below it

29. Mode – Group Data - Cont‘d Find the mode of a group data

30. How to Select a Suitable Measure Always select the mean whenever there is no special reason for choosing the other two measures. ( i ) Always select the mean whenever there is no special reason for choosing the other two measures. (ii) Select the median is the distribution consists of substantial amount of extreme large or small values. (iii) Select the mode if integral result is preferred as in cases the data are in ordinal scales.

31. Conclusion 06

32. Conclusion Measures of central tendency, including the mean, median, and mode, are crucial in statistics as they help summarize, simplify, and describe the characteristics of a dataset. They provide a sense of the typical value or average of a set of observations, enabling researchers, and analysts to make informed decisions and identify patterns. Measures of central tendency are often used in combination with dispersion measures, such as variance and standard deviation, to gain a comprehensive understanding of a dataset's distribution. In addition, they are essential in finance, economics, and the social sciences to analyze and interpret data, making them a fundamental concept in statistics.

33. References 07

34. References Arithmetic Mean - Definition, Examples, Calculating/Finding . (n.d.). Cuemath . Retrieved June 30, 2024, from https://www.cuemath.com/data/arithmetic-mean Javaid, M. A. (n.d.). Research Methods and Statistics . Library Guides: Statistics: Measures of Central Tendency . (n.d.). https://libraryguides.centennialcollege.ca/c.php?g=717168&p=5116867 WRKDEV100-20011 - Measures of Central Tendency: Mean, Median, and Mode . (n.d.). https://www.riosalado.edu/web/oer/WRKDEV100-20011_INTER_0000_v1/lessons/Mod05_MeanMedianMode.shtml

35. References – Cont‘d Mean, Mode and Median - Measures of Central Tendency - When to use with Different Types of Variable and Skewed Distributions | Laerd Statistics . (n.d.). https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php Manikandan, S. (2011). Measures of central tendency: The mean. Journal of Pharmacology and Pharmacotherapeutics , 2 (2), 140–142. https://doi.org/10.4103/0976-500x.81920 Hurley, M., & Tenny , S. (2023, July 17). Mean . StatPearls - NCBI Bookshelf. https://www.ncbi.nlm.nih.gov/books/NBK546702 Reddy, R. (2017, August 31). Measures of central tendency . SlideShare. https://www.slideshare.net/slideshow/measures-of-central-tendency-79313963/79313963

36. T hank You For Listening!