Empirical relationship between averages

Mean, Median and Mode

1. EMPIRICAL RELATIONSHIP AMONG AVERAGES Dr. RMKV Asst. Prof.

2. Illustration The following table gives the distribution of the number of workers according to the weekly wage in a company Obtain Mean, Median, Mode Weekly wage (in Rs.100’ s) 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Numbers of workers 5 10 15 18 7 8 5 3

3. Solution x m f fm cf 0-10 5 5 25 5 10-20 15 10 150 15 20-30 25 F0 15 375 CF 30 30-40 35 F F1 18 630 48 40-50 45 F2 7 315 55 50-60 55 8 440 63 60-70 65 5 325 68 70-80 75 3 225 71 71 2485 x m f fm cf 0-10 5 5 25 5 10-20 15 10 150 15 20-30 25 F0 15 375 CF 30 30-40 35 F F1 18 630 48 40-50 45 F2 7 315 55 50-60 55 8 440 63 60-70 65 5 325 68 70-80 75 3 225 71 Let us take weekly wages as ‘x’. No. of workers as ‘f’ X = = = 35 Median class = size of( ) th class = =35.5 th class = 30 -40 Median = L ; l =30; cf =30; f = 18;c=10 Med = 30+ x 10 =30+ x 10 = 30+ 3.06 = 33.06  

4. The highest frequency is 18 and corresponding class interval is 30 – 40, which is the modal class. Here L =30, f1=18,f0=15, f2=7, C= 10 = f1 - f0 = 18 -15 = 3 = f1 - f2 = 18 – 7 = 11 Mode = M0 = = 30 + x 10 = 3 x 10 = 3 0+0.214 x 10 = 3 0 + 2.14 = 32.14 Mode is 32.14   Mean = 35 Median =33.06 Mode= 32.14

5. Chart

6. EMPIRICAL RELATIONSHIP AMONG AVERAGES In a symmetrical distribution the three simple averages Mean = Median = Mode. For a moderately asymmetrical distribution, the relationship between them are brought by Prof. Karl Pearson as Mode = 3Median – 2Mean.

7. Illustration and Solution Illustration If the mean and median of a moderately asymmetrical series are 26.8 and 27.9 respectively, what would be its most probable mode? Solution: Using the empirical formula Mode = 3 median - 2 mean = 3 ( 27.9) - 2 ( 26.8) = 30.1 Mode = 30.1

8. Illustration and Solution Illustration I n a moderately skewed distribution, median = 10 and mean = 12. Using these values, find the approximate value of mode. Solution Using the empirical formula Mode = 3 median - 2 mean Mode = 3(10) -2(12) = 30 – 24 = 6 Mode = 6

9. Illustration and Solution Illustration Find out the mean when you are given that the median = 20.6 and the mode = 26. Solution Using the empirical formula Mode = 3 Median - 2 Mean 26 = 3(20.6) -2 Mean 26 = 61.8 – 2 mean 26 + 2mean = 61.8 2 mean = 61.8 – 26 2 mean = 35.8 Mean = 35.8/2 =17.9

10. Illustration and Solution Illustration In a moderately asymmetrical distribution the values of mode and mean are 32.1 and 35.4 respectively. Find the median value. Solution: Using the empirical formula Mode = 3 Median - 2 Mean 32.1 = 3 median -2(35.4) 32.1 = 3 median – 70.8 32.1 – 3 median = -70.8 -3 median = -70.8 -32.1 - 3median = -102.9 Median = -102.9/-3=34.3 Median = 34.3

11. DIFFERENCE AMONG AVERAGES Sl. No. Mean Median Mode 1. The average was taken for a set of numbers is called a mean. The middle value in the data set is called Median. The number that occurs the most in a given list of numbers is called a mode. 2. Add all of the numbers together and divide this sum of all numbers by a total number of numbers. Place all the given numbers in an ascending order It shows the frequency of occurrence. 3. The result is the mean or average score. The next step is to find the middle number on the list. It is called as the median. We can have more than one mode or no mode at all.

12. DIFFERENCE AMONG AVERAGES Sl. No. Mean Median Mode 4. Example: To find the average of the four numbers 2, 4, 6, 8, we need to add the number first. 2 + 4 + 6+ 8 = 20 Divide the sum by the total number of numbers, i . e 4. 20/4 = 5 is the average or mean Example: If the given list is 4, 2, 8, 10, 19. Arrange the numbers in ascending order i .e 2, 4, 8, 10, 19. As the total numbers are 5, so the middle number 8 is the median here. Example: In the given series 3,3,5,6,7,7,8,1,1,1,4,5,6 Find the frequency of each number. For number 3 it’s 2, for 5 it’s 2, for 6 it’s 2, for 7 it’s 2, for 8 it’s one, for 1 it’s 3, for 4 it’s 1. The number with the highest frequency is the mode.