Measures of Central Tendency

Measures of Central Tendency - Mean, Median , Mode

1. Measures of Central Tendency Suresh Babu G Assistant Professor CTE CPAS Paippad , Kottayam

2. Measures of Central Tendency A measure of central tendency is a summary statistic that represents the centre point or typical value of a dataset . In statistics , the three most common measures of central tendency are Arithmetic Mean Median Mode There are two more types of average ie , Geometric Mean and Harmonic Mean Each of these measures calculates the location of the central point using a different method.

3. Arithmetic Mean It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X̅. If there are N observations as X 1 , X 2 , X 3 ,….. X n Where Σ X = Sum of all observation N = Total number of observation

4. Arithmetic Mean for Ungrouped Data Individual Series Direct Method Example: Calculate Arithmetic Mean from the data showing marks of students in a class in an psychology test : 40, 50, 55, 78, 58.

5. Arithmetic Mean for Ungrouped Data Discrete Series Direct Method

6. Example Calculate the AM from the following: Mark No: of students fX 22 5 110 25 10 250 30 15 450 37 7 259 45 3 135 50 10 500 N = 50 Σ fx = 1704 Marks 22 25 30 37 45 50 No: of students 5 10 15 7 3 10

7. Continuous Series Direct Method Where M = Mid value

8. Example: From the following data calculate AM Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No: of Students 5 3 7 25 20 Class Frequency (f) Mid value (x) fx 0 – 10 5 5 25 10 – 20 3 15 45 20 – 30 7 25 175 30 – 40 25 35 875 40 - 50 20 45 900 N = 60 Σ fx = 2020 33.67

9. Short Cut Method Where A = Assumed Mean d = deveation of mid values from the assumed mean d = m-A N = Number of observations

10. Example: From the following data calculate AM Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No: of Students 5 3 7 25 20 Class Frequency (f) Mid value (x) d= x-A fd 0 – 10 5 5 -20 -100 10 – 20 3 15 -10 -30 20 – 30 7 25 30 – 40 25 35 10 250 40 - 50 20 45 20 400 N = 60 Σ fd =520 25 A = 25

11. Median Median is the middle element when the data set is arranged in order of the magnitude. Median of Ungrouped Data Individual Series Median = Size of item Where N is number of observations th

12. Example : The following data provides marks of seven students. Calculate median 110, 115, 140, 117, 109, 113, 120 Arrange the data in ascending order 109, 110, 113, 115, 117, 120, 140 Median = Size of item Median = Size of item = Size of item Median = Size of 4 item Median = 115 th th th th

13. Example 2 : Calculate the median 38, 24, 45, 50, 85, 60, 95, 40, 56, 63 Ascending order 24, 38, 40, 45, 50, 56, 60, 63, 85, 95 Median = Size of item Median = Size of item = Size of item Median = Size of 5.5 item Median = 50 + 0.5(56-50) = 50 + 0.5 X 6 = 53 Median = 53 th th th th

14. Median of Grouped Data Discrete Series Median = Size of item Example: Calculate Median th Marks No: of students 10 2 20 4 30 10 40 4 N = Total frequency

15. Marks No: of students Cumulative Frequency 10 2 2 20 4 6 30 10 16 40 4 20 Median = Size of item Median = Size of item = Size of item Median = Size of 10.5 item 10.5 is easily located at 16 of cf corresponding mark is 30 so Median =30 th th th th

16. Continues Series Median = Where L = Lower limit of the median class cf = Cumulative frequency of the classes preceding the median class f = frequency of the median class h = magnitude of the median class interval

17. Example: Find median Marks No of Students 0 – 10 4 10 – 20 12 20 – 30 24 30 – 40 36 40 – 50 20 50 – 60 16 60 – 70 8 70 - 80 5

18. Median = Median = = = 30 + 6.25 Medan = 36.25 Marks No of Students Cumulative Frequency 0 – 10 4 4 10 – 20 12 16 20 – 30 24 40 30 – 40 36 76 40 – 50 20 96 50 – 60 16 112 60 – 70 8 120 70 - 80 5 125 Size of N/2 th item = Size of 125/2 th item = Size of 62.5 th item Median class = 30 – 40 L = 30 cf = 40 f = 36 h = 10

19. Mode Mode is the most frequently observed data value. Ungrouped data – Individual Series Example Find mode 1, 2, 3, 4, 4, 5 Mode is 4 ( as 4 repeats 2 times )

20. Mode of Grouped Data Discrete Series Example : Find Mode By inspecting the data value, the maximum frequency is 20 ie , 30 mark repeats 20 times so the mode value is 30 mark Marks No. of Students 10 2 20 8 30 20 40 10 50 5 30

21. Continuous Series: Inspection Method Mode = Where L = lower limit of the modal class D1 = Difference between the frequency of the model class and the frequency of the class preceding the modal class (ignoring signs) D2 = Difference between the frequency of the model class and the frequency of the class succeeding the modal class (ignoring signs) h = class interval if the distribution

22. Example Find mode value Marks No of Students 0 – 10 5 10 – 20 7 20 – 30 8 30 – 40 20 40 – 50 10 50 – 60 6 60 – 70 2 70 - 80 2

23. The most frequently occurring data value is between 30-40 which occurs 20 times. The model class is 30 – 40 . Mode = D1 = 20 – 8 = 12 Mode = D2 = 20 – 10 = 10 Mode = 30 + 5.45 L = 30 Mode = 30.45 h = 10 Marks No of Students 0 – 10 5 10 – 20 7 20 – 30 8 30 – 40 20 40 – 50 10 50 – 60 6 60 – 70 2 70 - 80 2

24. Another Method to find mode value is Mode = 3Median – 2 Mean